## 1. Introduction



Logics of conditionals deal with inferences involving sentences of the
form "if *A*, (then) *B*" of natural
language. Despite the overwhelming presence of such sentences in
everyday discourse and reasoning, there is surprisingly little
agreement about what the right logic of conditionals might be, or even
about whether a unified theory can be given for all kinds of
conditionals. The problem is not new, but can be traced back to
debates between Megarian and Stoic logicians (see Sextus Empiricus,
*Outlines of Scepticism* II, 110-112; Kneale & Kneale
1962; Sanford 1989; Weiss 2019). Famously, Philo of Megara proposed
that a conditional of the form "if *A* then
*B*" is true exactly when it is not the case that
*A* is true and *B* is false. The same definition is at
the core of the modern treatment of the conditional as so-called
"material implication" in the two-valued propositional
framework of Frege (1879) and Whitehead and Russell (1910).





  
|  | \(B\) |
| --- | --- |
| \(A \supset B\) | 1 | 0 |
| \(A\) | 1 | 1 | 0 |
| 0 | 1 | 1 |



Table 1: The material conditional




This understanding of the conditional has considerable virtues of
simplicity, and in that regard the material conditional analysis
provides a benchmark for other theories. Probably its main virtue is
that it lends itself to a truth-functional treatment (the truth value
of a conditional is a function of the truth values of antecedent and
consequent). A related one is that it makes the conditional
interdefinable with Boolean negation, disjunction, and conjunction. A
third, among the driving motivations for Frege, Russell and Whitehead,
is that it appears adequate to regiment mathematical proofs involving
conditional sentences. Despite that, as granted by Frege himself in
the *Begriffsschrift*, it fails to do justice to some of the
ways in which conditionals are ordinarily understood and used in
natural language.



An oddity pointed out early by MacColl (1908) is the observation that
of any two sentences of the form "not *A* or
*B*" and "not *B* or *A*", at
least one must be true. Assuming the equivalence with the material
conditional, this implies that either "if John is a physician,
then he is red-haired" or "if John is red-haired, then he
is a physician" must be true. Intuitively, however, one may be
inclined to reject both conditionals. Similar complications, known as
the *paradoxes of material implication*, concern the fact that
for any sentences *A* and *B*, "if *A* then
*B*" follows from "not *A*", but also
from "*B*", thereby allowing true and false
sentences to create true conditionals irrespective of their content
(C. I. Lewis 1912). Another peculiarity looms large: the negation of
"if *A* then *B*" is predicted to be
"*A* and not *B*", but intuitively one may
deny that "if God exists, all criminals will go to heaven"
without committing oneself to the existence of God (cited in Lycan
2001).



A fourth complication is that conditional sentences in natural
language are not limited to indicative conditionals ("if I
strike this match, it will light"), but also include subjunctive
conditionals used to express counterfactual hypotheses ("if I
had struck this match, it would have lit"). All counterfactual
conditionals would be vacuously true if analyzed as material
conditionals with a false antecedent, as pointed out by Quine (1950),
an obviously inadequate result, suggesting that the interplay of
grammatical tense and grammatical mood should also be of concern to
understand the logic of conditionals.



To a large extent, the development of conditional logics over the past
century has thus been driven by the quest for a more sophisticated
account of the connection between antecedent and consequent in
conditionals. A number of conditional logics have thus been developed
within the framework of *modal logic*, to capture the idea that
conditional sentences express a necessary connection of sort between
antecedent and consequent, whether metaphysical, epistemic, or deontic
(see C. I. Lewis 1912; Stalnaker 1968; D. Lewis 1973; Kratzer 2012). A
related wave of work encompasses the so-called cotenability theories
of conditionals, in which the idea is that a conditional is assertable
if its antecedent, together with suitable (co-tenable) premises,
entails its consequent (see Chisholm 1946; Goodman 1955; Rescher
1964).



Another strand of influential logics has been motivated by the idea
that conditionals have probabilities or express a probabilistic
dependence between their antecedents and their consequents (Ramsey
1931; de Finetti 1936; Adams 1965; Edgington 1995; Douven 2016). A
very influential source of this is a passage by Ramsey (1931), in
which he writes:



> 
> 
> If two people are arguing "If \(p\), then \(q\)?" and are
> both in doubt as to \(p\), they are adding \(p\) hypothetically to
> their stock of knowledge and arguing on that basis about \(q\); so
> that in a sense "If \(p\), \(q\)" and "If \(p\),
> \(\neg q\)" are contradictories. We can say that they are fixing
> their degree of belief in \(q\) given \(p\). If \(p\) turns out false,
> these degrees of belief are rendered *void*. If either party
> believes *not \(p\)* for certain, the question ceases to mean
> anything to him except as a question about what follows from certain
> laws or hypothesis. [Notation adapted]
> 
> 
> 



Importantly, the so-called Ramsey Test (adding the antecedent
hypothetically to one's beliefs) has inspired a number of
approaches that stand as some of the cornerstones of conditional
logics. This includes Stalnaker's modal logic of conditionals
based on the idea that the antecedent of a conditional selects a
hypothetical possible world (Stalnaker 1968). It also includes
Adams's approach based on studying formally the idea that the
probability of (non-nested) conditionals is given by the corresponding
conditional probability (Adams 1965, 1975), as well as more
qualitative theories of the acceptability of conditionals in terms of
(non-probabilistic) belief revision policies (Gardenfors 1986).
Finally, Ramsey's idea that conditionals with a false antecedent
are void underlies the main motivation behind three-valued accounts of
conditionals, developed independently by various authors (in
particular by de Finetti 1936).



In this entry, our presentation of conditional logics will be
structured by the centrality of those various frameworks. Our goal in
particular will be to allow readers to orient themselves easily
between those frameworks, and also to show how they communicate. We
start with a presentation of three-valued logics of conditionals in
 section 2.
 Section 3 surveys the main analyses of the conditional in modal
logics (possible worlds semantics). In
 section 4,
 we discuss so-called premise semantics, which formalize the
co-tenability idea, and which shed a different light on standard modal
analyses.
 Section 5
 moves on to probabilistic logics of conditionals, and
 section 6
 to qualitative approaches in terms of belief revision.
 Section 7
 and
 section 8,
 finally, will present more recent developments or refinements: first,
logics based on the idea that the antecedent ought to make a relevant
difference to the consequent, then some extensions of propositional
logics of conditionals addressing the interaction of conditionals with
modalities as well as with questions and commands.



One caveat is that in this entry we will try to keep a unified
perspective on both indicative and subjunctive conditionals. Although
some authors, notably David Lewis, have insisted on treating them
differently, differences can be handled parametrically relative to a
common semantic core (viz., Stalnaker 1975 or Pearl 2009). We will
point to differences when relevant, but refer the reader to
Starr's (2019) SEP entry on counterfactuals that deals with the
topic rather extensively. On some aspects our entry also overlaps with
Edgington's (2020) SEP entry on indicative conditionals, and
readers are invited to consult both of those entries for
complements.



In what follows, we will represent the conditional operator by the
symbol \(>\). We distinguish three main layers of syntactic
complexity. Let \(L\_0\) be a propositional language capable of
building *factual* sentences from atoms with the help of the
usual Boolean connectives (\(\neg\), \(\wedge\), \(\vee\), \(\supset\)
and \(\equiv\) for negation, conjunction, disjunction, material
conditional and material equivalence, respectively). Then \(L\_1\) is
the extension of \(L\_0\) with *flat conditionals*, such that if
\(A\) and \(B\) are in \(L\_0\) then \(A>B\) is in \(L\_1\). And
\(L\_2\) is the extension of \(L\_0\) and \(L\_1\) allowing free
combinations of all connectives, including arbitrary nestings of
conditionals.


## 2. Three-Valued Conditionals


### 2.1 Motivations



Logics of indicative conditionals, in so far as they are based on
truth evaluations, all agree that a conditional is false if its
antecedent is true and its consequent is false. This is captured by
the two-valued analysis. Less obvious is the assumption that a
conditional is true when both antecedent and consequent are true, for
this predicts that \(A\wedge B\) entails \(A >B\). And even more
controversial is the proviso that a conditional is true whenever its
antecedent is false, which indeed is one of the paradoxes of the
material conditional.



C. I. Lewis's (1912) response to the paradoxes of the material
conditional is his proposal of a strict conditional (aka "strict
implication", section
 3).
 The strict conditional is no longer truth-functional, however. A
closer kin to standard two-valued logic in this regard is the family
of three-valued logics originating from the work of Lukasiewicz
(1920), which retain truth-functionality. Lukasiewicz's
main motivation was not to deal specifically with conditionals,
however, but with the notion of possibility. Lukasiewicz's
conditional is shown in
 Table 2.
 Lukasiewicz defines validity to be the preservation of the value
1 from premises to conclusion, as in two-valued logic. His operator
retains the law of Identity \(A>A\) and various classic schemata
(such as Modus Ponens), but it also fails some classical properties
(it is non-contractive, for example, failing the inference from
\(A>(A>B)\) to \(A>B\)). It agrees with the two-valued
conditional on classical input, and it does not block the paradoxes of
the material conditional.





  
|  | \(B\) |
| --- | --- |
| \(A > B\) | 1 | 1/2 | 0 |
|  | 1 | 1 | 1/2 | 0 |
| \(A\) | 1/2 | 1 | 1 | 1/2 |
|  | 0 | 1 | 1 | 1 |



Table 2: Lukasiewicz's
conditional




Lukasiewicz's work was a source of inspiration to
Reichenbach, and subsequently to de Finetti, both of whom put forward
the idea expressed earlier by Ramsey that a conditional must be
indeterminate when its antecedent is false (see Reichenbach 1935,
1944; de Finetti 1936). Reichenbach deems *meaningless* the
evaluation of a conditional \(A>B\) when \(A\) concerns some
measurement rendered impossible by some other measurement (as
predicted in quantum physics). For de Finetti, the idea is that a
conditional sentence is like a bet. When someone bets that the card
will be a king if it is a club, the bet should simply be called off if
the card turns out to be of a different suit. On that basis, de
Finetti proposed a table (sometimes called the "defective"
conditional, see Baratgin et al. 2013, Over & Baratgin 2017) for
what he calls the logic of tri-events, where a tri-event refers to
"the event \(B\) conditional upon event \(A\)". The
tri-event is true when both events are true, false when \(A\) is true
and \(B\) false, and *null* if \(B\) is false.





  
|  | \(B\) |
| --- | --- |
| \(A > B\) | 1 | 1/2 | 0 |
|  | 1 | 1 | 1/2 | 0 |
| \(A\) | 1/2 | 1/2 | 1/2 | 1/2 |
|  | 0 | 1/2 | 1/2 | 1/2 |



Table 3: De Finetti's
conditional




As
 Table 3
 shows, there is now a third truth value 1/2 (for
"null" or "void" or
"indeterminate") beyond the classical values 1 (for
"true") and 0 (for "false"). This is why these
logics are called *trivalent*. The conditional is no longer
classically valued for classical input. Although Reichenbach and de
Finetti considered the interplay of this conditional with other
trivalent connectives, they did not investigate the logics resulting
from that choice. Investigations of logics resulting from de
Finetti's table spread in a scattered way, often oblivious of
the work of Reichenbach and de Finetti. One path was paved by remarks
due to Quine (1950), crediting the work of Ph. Rhinelander on the idea
that the assertion of a sentence of the form "if \(A\) then
\(C\)" is commonly felt "less as the assertion of a
conditional than as a conditional assertion of the consequent"
(see Jeffrey 1963; Cooper 1968; Belnap 1970, 1973; Manor 1975; Farrell
1979, 1986; Olkhovikov 2002; Huitink 2008). This view is known more
broadly as the suppositional view (Adams 1975; Edgington 1995).
Another line of work was pursued in connection with algebraic
treatments of probability, much in the spirit of de Finetti's
original work, who viewed trivalence as a way of representing the
notion of conditional probability more qualitatively (see Schay 1968;
Calabrese 1991; Dubois & Prade 1994; McDermott 1996; Milne 1997;
Cantwell 2008; Rothschild 2014; Lassiter 2020).


### 2.2 Some trivalent logics



Amongst trivalent logics based on variations of de Finetti's
table, some systems stand out. First, there is Cooper's logic
**OL** and a close variant of it, Cantwell's logic
**CC/TT** (as named in Egre, Rossi, &
Sprenger 2021, after Cooper and Cantwell), itself a fragment of the
logic **LImp** of Olkhovikov (2002), which also
incorporates a modal necessity operator.
**OL** and **CC/TT**are
defined for \(L\_2\), the full language allowing nested conditionals.
Like de Finetti, Cooper (1968) handles the conditional as
indeterminate when the antecedent is false (0). However, the
conditional takes the value of the consequent when the antecedent is
*not false* (either 1 or 1/2). Hence, instead of grouping
1/2 with 0, as de Finetti does, it groups 1/2 with 1.
 Table 4
 was rediscovered independently by Olkhovikov (2002) and by Cantwell
(2008), with very similar motivations in mind. It was also proposed by
Belnap (1973), who presents it as a suitable modification from Belnap
(1970), who first used the same table as De Finetti's.





  
|  | \(B\) |
| --- | --- |
| \(A > B\) | 1 | 1/2 | 0 |
|  | 1 | 1 | 1/2 | 0 |
| \(A\) | 1/2 | 1 | 1/2 | 0 |
|  | 0 | 1/2 | 1/2 | 1/2 |



Table 4: Cooper's conditional




Cooper, Olkhovikov, and Cantwell define an inference to be valid
if the conclusion is not false whenever all the premises are not
false, for all trivalent valuations of the language. A central feature
of their systems is that the full deduction theorem holds (a feature
also stressed by Jeffrey 1963 and Belnap 1973):

 

\[\tag{DT} \Gamma, A\vDash B \tiff \Gamma\vDash A>B.\]


One difference between their systems is that Cooper defines validity
relative to trivalent valuations that are atom-classical; another is
that Cooper uses so-called quasi-conjunction and disjunction (see
below), whereas Olkhovikov and Cantwell use the common min and max
rules for conjunction and disjunction, respectively. For Cooper and
Cantwell, in particular, and similarly for Belnap (1973), a central
motivation for this conditional is the validation of the principle of
Negation Commutation for the conditional, namely the conjunction
of:

 

\[\tag{EX} A>\neg B \;\vDash\; \neg (A>B). \] \[\tag{IN} \neg (A>B) \;\vDash\; A>\neg B.\]


Principle EX in particular is characteristic of connexive logics (see
Section
 7
 below). Another important feature of the resulting logics is that
whereas Modus Ponens is a valid argument schema, Modus Tollens and
Contraposition fail in general (though they remain valid over atomic
variants of the schemata in Cooper's **OL**):



\[\tag{MP} A,\ A>B \;\vDash\; B.\] \[\tag{MT} \neg B,\ A>B \;\not\vDash\; \neg A.\] \[\tag{CP} A>B \;\not\vDash\; \neg B >\neg A.\]


Since \(A\not \vDash \neg A >B\), the Cooper conditional blocks one
of the paradoxes of the material conditional, but it retains another
one, namely \(A\vdash B>A\). Relatedly, although Cooper and
Cantwell or Olkhovikov use different semantics for conjunction,
on their approaches, conjunction entails the conditional. This
principle is called *Conjunctive Sufficiency*:

 

\[\tag{CS} A \wedge B \;\vDash\; A>B.\]


Cooper provides a complete axiomatization for his logic, using a
system of inference rules close to a natural deduction system (system
**OL**). Using tableaux and sequent calculi,
Egre, Rossi, and Sprenger (2021) compare the
Cooper-Cantwell logic **CC/TT** with the logic
**DF/TT** that results from de Finetti's table and
the same notion of validity (preservation of non-zero values). The
logic **DF/TT** is connexive, too, and supports Negation
Commutation. A central difference, however, is that de Finetti's
table fails to support MP.



Another system worthy of consideration is the trivalent logic of
"conditional objects" investigated by Dubois and Prade
(1994). This system is itself closely related to a three-valued
treatment of conditionals presented in Adams (1986). The system rests
on de Finetti's original table, but only formulae from the flat
fragment \(L\_1\) are admitted. Validity in Dubois and Prade's
approach is defined as follows. Assume \(\Gamma\) is a set of flat
conditionals of the form \(A\_i>B\_i\). Then define \(\Gamma \models
A>B\) iff *either* \(A\) classically entails \(B\)
*or* for every trivalent valuation the value of the
quasi-conjunction of some subset of conditionals in \(\Gamma\) is less
than or equal to the value of \(A>B\) (for the usual ordering of
truth-values: \(0<1/2<1\)). Quasi-conjunction--a
connective proposed by Cooper (1968: 305) and in a different guise by
Adams (1966: 306), also proposed by Belnap (1973:60), and earlier by
Sobocinski 1952 in relation to a different trivalent conditional
--differs from min conjunction in that the conjunction of
classical values is classical, and the conjunction of 1/2 with a
classical value is that classical value. Quite remarkably, this
relation of logical consequence can be axiomatized by the system
**P** of preferential entailment of Kraus, Lehmann and
Magidor, which is discussed in detail below. Thus, the associated
notion of consequence does not support Monotonicity, Transitivity, and
Contraposition of the conditional (see
 section 3.2).


### 2.3 Open issues



Despite its simplicity, the use of trivalence raises several issues.
One open problem is how to deal with counterfactual
conditionals--the problem is not without options, but trivalence
so far has been used mostly to deal with indicative conditionals. A
related issue is how to make three-valued conditionals interact with
modalities and other operators more generally.



Another problem concerns the expressiveness of a finitely-valued
truth-functional account. McGee (1981) shows that the common
"flat" \(L\_1\)-fragment of the logics of conditionals of
Adams, Stalnaker and Lewis, i.e., the system **P**,
cannot be characterized in terms of the preservation of a designated
set of truth-values in a finite-valued logic. Prima facie, this
results appears in tension with Adams's as well as Dubois and
Prade's trivalent characterization of **P**. As
pointed out by M. Schulz (2009), however, the trivalent
characterization proposed by Adams--or indeed Dubois and
Prade--differs from the one entertained by McGee (it asks for
more than the preservation of a fixed set of designated values, and
allows quantification over subsets of the
 premises).[1]
 As discussed by Schulz, whether a similar characterization can be
given for the non-flat logics of Stalnaker and Lewis is an open
question.


## 3. Possible Worlds Models



Trivalent accounts suppose that the truth value of a conditional
depends only on the truth values of its parts. However, this
assumption does not account for a connection of the contents of the
antecedent and the consequent, a point granted by Frege (1879)
himself. In this section we review accounts that preserve the idea
that conditionals are truth-evaluable, but assume that their truth
value cannot be settled just by the truth values of the antecedent and
the consequent at the actual world. Instead, the antecedent has a
certain modal force, it makes its consequent in some sense necessary.
Possible-worlds models give an explication of the necessity
involved.


### 3.1 Strict conditionals



The first step toward an intensional treatment of conditionals was
made with C. I. Lewis's introduction of the strict conditional.
Although Lewis's treatment was axiomatic, its semantic content
can be made transparent using possible worlds semantics. On
Lewis's approach, "If \(A\), then \(C\)" means that
the material conditional is not just true, but necessarily true:



\[\tag{SI} (A >B ) \equiv \Box (A\supset B).\]


From a semantic view point, reference is made here not just to the
actual world, but also to other possible worlds, in fact to all
"accessible" worlds. Thus, the meaning of this conditional
cannot be explained by a simple truth table.



In modal logic, a *Kripke frame* is a pair \(\langle W,
R\rangle\) where \(W\) is a non-empty set of *possible worlds*
(possible states, possible situations or possible ways the world might
be) and \(R\) is a binary relation on \(W\). \(R\) is called the
*accessibility relation* of the model, and intuitively \(Rwv\)
means that world \(v\) is (in some sense) possible from the point of
view of \(w\). A *model* is a triple \(\langle W, R, V\rangle\)
such that \(\langle W, R\rangle\) is a frame and \(V\) is a valuation
function assigning truth values to all propositional variables of the
language at all possible worlds in \(W\). We write \(M, w \Vdash A\)
for "\(A\) is true at world \(w\) in model \(M\)" and let
\(\lvert A\rvert^M\) denote the set \(\{w\in W: M, w \Vdash A\}\) of
possible worlds at which \(A\) is true. Let \(R(w) := \{v\in W:
Rwv\}\). The truth conditions for the Boolean connectives are the
usual ones for propositional logic, extending \(V\) to complex
sentences. Those for the strict conditional are as follows:



\[\tag{SC } \begin{align} M, w \Vdash A >B & \quad\text{iff}\quad \text{for all } v \text{ such that } Rwv,\quad M, v \Vdash A \supset B\\ & \quad\text{iff}\quad R(w) \cap \lvert A\rvert^M \subseteq \lvert B\rvert^M.\\ \end{align} \]


Validity (relative to a class of frames) is defined as
truth-preservation at all worlds of all models (based on any frame in
the class). It is easy to see that the strict conditional invalidates
the paradoxes of the material conditional, relative to models built on
arbitrary Kripke frames. The strict conditional retains Modus Ponens
and a number of classic properties, such as Monotonicity,
Transitivity, and Contraposition (see below). Starting with Goodman
(1955), however, such inferences have been regarded with suspicion,
both for indicative conditionals (see Adams 1965) and for
counterfactuals conditionals (see Stalnaker 1968, Lewis 1973). For
instance, in the case of Transitivity, we have the following
counterexamples:



(1)
If Brown wins the election, Smith will retire to private life. If
Smith dies before the election, Brown will win it. #Therefore, if
Smith dies before the election, then he will retire to private life.
(Adams 1965, p. 166)
(2)
If J. Edgar Hoover were today a communist, then he would be a
traitor. If J. Edgar Hoover had been born a Russian, then he would
today be a communist. #Therefore, if J. Edgar Hoover had been born a
Russian, he would be a traitor. (Stalnaker 1968, p. 106)


Such examples inspired Adams to propose a probabilistic treatment of
conditionals. They inspired Stalnaker, Lewis, Nute, and a number of
others to weaken the strict conditional analysis. But recently there
has been a revival of the strict conditional analysis, based on the
idea that many phenomena can be accounted for pragmatically by
construing the accessibility relation that determines which possible
worlds are relevant as massively context-dependent (Daniels &
Freeman 1980; von Fintel 2001; Gillies 2007; Starr 2014a, and
Starr's entry on counterfactuals).


### 3.2 Variably strict conditionals



The basic idea behind "variably strict conditionals" (an
expression due to D. Lewis 1973) is that the standard of necessity
relevant to evaluate a conditional sentence depends on the antecedent.
Roughly speaking, in order to evaluate a conditional \(A>B\), one
has to extend the reach of the accessibility relation to the
"closest" or "most similar" worlds in which
the antecedent is true. In what follows we present three related ways
of stating the truth conditions of the variably strict
conditional.


#### 3.2.1 Main semantics


##### Relative modalities



Though not historically the first, arguably the simplest way of
expressing the idea of a strict conditional whose necessity varies
depending on the antecedent can be found in Chellas (1975) and
Segerberg (1989), who describe a conditional \(A>B\) as a relative
modality, namely a sententially indexed modality \(\Box\_{A}B\), with
the corresponding accessibility relation \(R\_{\lvert A\rvert^{M}}\)
for every sentence \(A\). A relative-modality frame or
*rm-frame* can be defined as a pair \(\langle W, R\rangle\)
where \(R\) is a ternary relation on \(W\times W\times \cP(W)\). In
practice we can write \(R\_Uvw\) instead of \(RvwU\), and see \(R\) as
assigning a binary relation over worlds relative to a proposition
\(U\). An *rm-model* is a triple \(M = \langle W, R, V\rangle\)
such that \(\langle W, R\rangle\) is an rm-frame and \(V\) a valuation
function. Given an rm-model \(M\), the truth condition for the
conditional is this:

 

\[\tag\*{(VSC)-rm } \begin{align} \ M, w \Vdash A >B & \quad\text{iff}\ \text{for all } v \text{ such that } R\_{\lvert A\rvert^{M}} wv, \ M, v \Vdash B\\ & \quad\text{iff}\ R\_{\lvert A\rvert^{M}}(w) \subseteq \lvert B\rvert^M.\\ \end{align} \]


Note that \(R\) could be defined on \(W\times W\times L\_2\), letting
\(R\_A(w)\) depend on the syntactic form of \(A\) (instead of the
proposition \(\lvert A\rvert^M\) expressed by \(A\)). The main effect
of this modification is to block the substitution of equivalent
antecedents, a feature called for by several theorists on separate
grounds (see
 section 3.5).


##### Selection functions



An equivalent definition is in terms of selection functions (Stalnaker
1968; Nute 1980). An *sf-frame* is a pair \(\langle W,
f\rangle\) where \(W\) is a non-empty set of *possible worlds*
and \(f: W \times \cP(W) \to \cP(W)\) is a function taking a world and
a proposition (i.e., a set of worlds) and yielding a proposition.
\(f\) is called the *selection function* of the model, and
intuitively \(v \in f(w,U)\), where \(U\subseteq W\), means that world
\(v\) is a world in \(U\) that is (in some sense) *closest* or
*most similar* to \(w\), as compared to all the other worlds in
\(U\). The correspondence with the previous approach is
straightforward, if we let \(f(w,\lvert A\rvert^M)\) coincide with
\(R\_{\lvert A\rvert^M}(w)\). An *sf-model* is a triple \(M =
\langle W, f, V\rangle\) such that \(\langle W, f\rangle\) is an
sf-frame and \(V\) is a valuation function. The truth condition for
the variably strict conditional becomes:

 

\[\tag\*{(VSC)-sf} \begin{align} \ M, w \Vdash A >B & \quad\text{iff}\ \text{for all } v \in f(w,\lvert A\rvert^M), \ M, v \Vdash B\\ & \quad\text{iff}\ f(w,\lvert A\rvert^M) \subseteq \lvert B\rvert^M.\\ \end{align} \]


As with rm-frames, the definition of an sf-frame can be modified to
let \(f\) depend on \(w\) and \(A\), instead of \(\lvert
A\rvert^M\).


##### Similarity orderings



There is a closely related alternative to this set-up which uses
comparative similarity orderings in place of selection functions (D.
Lewis 1973; Burgess 1981). Here, an *o-frame* is a pair
\(\langle W, \prec \rangle\) where \(W\) is a non-empty set of
*possible worlds* and \(\prec\) is a ternary relation on the
set of worlds, assumed to be transitive and irreflexive relative to
its first argument. The relation \(u \prec\_w v\) is supposed to mean
that world \(u\) is (in some sense) *more similar* or
*closer* to \(w\) than world \(v\). Let \(R(w) := \{y\in W:
\exists z \in W(y\prec\_wz)\}\). This corresponds to the set of
accessible worlds from \(w\). An *o-model* is a triple
\(\langle W, \prec, V\rangle\) such that \(\langle W, \prec\rangle\)
is an o-frame, and \(V\) is a valuation function. We use the
abbreviation \(\lvert A\rvert^M\_w := \lvert A\rvert^M \cap R(w)\), and
let \(min\_{\prec\_{w}}(U)\) denote the \(U\)-worlds most similar to
\(w\) (such that no \(U\)-world is strictly more similar to \(w\)). A
simple form of the truth-condition for the variably strict conditional
reads as follows:

 

\[\tag\*{(VSC)-o-L } \begin{align} \ M, w \Vdash A >B & \quad\text{iff}\ \text{for all } v \in \min\_{\prec\_w} (\lvert A\rvert^M\_w), \ M, v \Vdash B\\ & \quad\text{iff}\ \min\_{\prec\_w} (\lvert A\rvert^M\_w) \subseteq \lvert B\rvert^M. \end{align} \]


These truth-conditions, however, make sense only if \(\lvert
A\rvert^M\_w\) possesses minimal elements under \(\prec\_w\), that is,
only if we can assume (L), in which we write \(v \preceq\_w u\) for \(v
\prec\_w u\) or \(v = u\):

 

\[\tag{L } \forall w \in W \;\forall u \in \lvert A\rvert^M\_w \;\exists v \in \lvert A\rvert^M\_w ((v\preceq\_w u) \amp \neg\exists v' \in \lvert A\rvert^M\_w (v'\prec\_w v)).\]


(L) is a slight generalization of the *Limit Assumption* of
Lewis (1973, 1981). While most researchers are ready to endorse the
Limit Assumption, Lewis rejects it. According to him, for a
counterfactual such as "If Fred was taller than 2 meters, he
would be in the university basketball team" there is no closest
world that would make the antecedent true. Ordering semantics can deal
with failures of (L) by stating more complex truth conditions.



(VSC)-o
\(M, w \Vdash A >B\quad \) iff for all \(v \in \lvert
A\rvert^M\_w\) there is a \(u\in \lvert A\rvert^M\_w\) such that \(u
\preceq\_w v\) and for all \(u'\in \lvert A\rvert^M\_w\) such that \(u'
\preceq\_w u,\ M, u' \Vdash B\).


This intuitively means that the \(A\wedge \neg B\) worlds are more
remote from \(w\) than the \(A\wedge B\)-worlds.


##### Comparisons



First of all, it is easy to see how all three semantics can invalidate
inferences like
 (1)
 and
 (2),
 which assume Transitivity. In an o-model, for example, from



\[\min\_{\prec\_w} (\lvert A\rvert^M\_w)\subseteq \lvert B\rvert^M\]


and

 

\[\min\_{\prec\_w} (\lvert B\rvert^M\_w)\subseteq \lvert C\rvert^M,\]


it need not follow that

 

\[\min\_{\prec\_w} (\lvert A\rvert^M\_w)\subseteq \lvert C\rvert^M.\]


See the three-world model in
 Fig. 1,
 where the arrows represent \(\prec\_w\) so that \(x\!\leftarrow\!y\)
means the same thing as \(x\prec\_w y\).



![a diagram: link to extended description below](figure1.svg)

Figure 1: Counterexample against Transitivity in an
o-model [An
 extended description of Figure 1
 is in the supplement.]




Similar counterexamples to Monotonicity and Contraposition are easy to
come by.



The correspondence between selection function models and relative
modality models is very direct. The correspondence between selection
function models and ordering models is less obvious. How can we
explain it? If we interpret similarity relations as preference
relations, we can import the insights gained in rational choice theory
to the analysis of conditionals (Sen 1970; Suzumura 1983; Aleskerov,
Bouyssou, & Monjardet 2007; applications to nonmonotonic reasoning
are in Lindstrom 1994 and Rott 2001). If the Limit Assumption is
satisfied, then we can simply identify \(f(w,U)\) with
\(\min\_{\prec\_w} (U)\), for all \(U\subseteq W\). For the converse
direction, we can use the revealed preferences, defined by



\[u\prec\_w v \ \textrm{ iff for all } U \subseteq W \text{ such that } u,v \in U, v\notin f(w,U)\]


or, if \(f\) can take propositions with two elements,

 

\[u\prec\_w v \tiff v\notin f(w,\{u,v\}).\]


These preferences "rationalize" the selections made by
\(f\). This, however, works only under certain constraints. In this
respect sf-models are more general than o-models. They are also easier
to work with, and giving up the Limit Assumption is generally viewed
as inadequate from a linguistic point of view (see Stalnaker 1980;
Schlenker 2004; S. Kaufmann 2017 for more on (L)).


#### 3.2.2 Frame correspondence properties



For each semantics, specific conditions can be imposed on frames in
order to secure the validity of particular axioms or argument
schemata. Their effects on the logic of conditionals have been studied
in detail. For illustration we give the examples of two specific
axioms and their correspondence properties in the three frameworks.
Identity requires that every \(A\)-closest world be indeed an
\(A\)-world. Conditional excluded middle corresponds to a uniqueness
condition: there is at most one closest \(A\)-world.





|  |  |  |
| --- | --- | --- |
|  | \(A>A\) | \((A>B) \vee (A >\neg B)\) |
| rm | \(R\_Avw \supset w\in A\) | \(wR\_Aw\_1 \wedge wR\_Aw\_2 \supset w\_1=w\_2\) |
| sf | \(f(w,A)\subseteq A\) | \(card(f(w,A))\leq 1\) |
| o | \(\forall x\,\forall y\,(\exists z(x\preceq\_y z) \supset
x\preceq\_y x)\) | \(\forall y\,\forall y'\,\forall w\,((y\in \lvert A\rvert^M\_w
\wedge \forall z(z\prec\_{w}y \supset z\notin \lvert
A\rvert^M\_w))\supset {}\) \(((y'\in \lvert A\rvert^M\_w \wedge \forall
z(z\prec\_{w}y' \supset z\notin \lvert A\rvert^M\_w))\supset
y=y'))\) |



Table 5: Some frame correspondence
properties




We refer to Unterhuber and Schurz (2014) for a systematic presentation
of frame correspondence properties for rm-frames, to Nute (1980),
Girard (2007), and Raidl (2021) on the correspondence properties of
sf-frames, and to Friedman and Halpern (1994) and Herzig (1996) on the
correspondence properties of o-frames.


### 3.3 Logics for conditionals



In order to present some central logics of conditionals based on the
possible-worlds framework, we adopt a Frege-Hilbert axiomatic
perspective and introduce a list of axioms and rules of inference. Our
presentation and choice of terminology relies on Nute (1980) as well
as Herzig (1996), Unterhuber and Schurz (2014), and Crupi and Iacona
(forthcoming-b).


* As a first layer for all systems, we let **PC**
consist of all tautologies of the classical propositional
calculus.
* We also assume the systems below to be closed under material Modus Ponens, but distinguish moreover the following inference rules (Left Logical Equivalence, Right Weakening, and Rule of Conditional K). Those mean that if the premises are theorems, then so is the conclusion. \[\tag\*{LLE} \frac{B \equiv C}{(B >A) \equiv(C >A)} \] \[\tag\*{RW} \frac{B \supset C}{(A >B) \supset(A >C)} \] \[\tag\*{RCK} \frac{(B\_1 \wedge \ldots \wedge B\_n) \supset C} {(A >B\_1) \wedge \ldots (A >B\_n) \supset(A >C)\quad (n\geq 0)} \]
* Axioms



|  |  |
| --- | --- |
| LT | (Logical Truth)

\(A>\top\) |
| ID | (Identity)

\(A>A\) |
| AND | (And)

\(((A>B)\wedge (A>C))\supset(A>(B\wedge C))\) |
| OR | (Or)

\(((A>C)\wedge (B>C))\supset((A\vee B)>C)\) |
| CCut | (Cautious Transitivity)

\(((A>B)\wedge ((A\wedge B)>C))\supset(A>C)\) |
| CMon | (Cautious Monotonicity)

\(((A>B)\wedge (A>C))\supset((A\wedge B)>C)\) |
| Rec | (Reciprocity)

\(((A>B)\wedge (B>A)) \supset((A>C) \equiv(B>C))\) |
| SM | (Stronger-than-Material)

\((B>C)\supset(B\supset C)\) |
| CS | (Conjunctive Sufficiency)

\((B\wedge C)\supset(B>C)\) |
| RMon | (Rational Monotonicity)

\(((A>B)\wedge \neg(A>\neg C))\supset((A\wedge C)>B)\) |
| CEM | (Conditional Excluded Middle)

\((A>B)\vee (A>\neg B)\) |





  
| Rules and

axioms | Systems |
| --- | --- |
| Ck | CK | B | SS | NP | V | VW | VC | C2 |
| LLE |  |  |  |  |  |  |  |  |  |
| RW |  |  |  |  |  |  |  |  |  |
| RCK |  |  |  |  |  |  |  |  |  |
| LT |  |  |  |  |  |  |  |  |  |
| ID |  |  |  |  |  |  |  |  |  |
| AND |  |  |  |  |  |  |  |  |  |
| OR |  |  |  |  |  |  |  |  |  |
| CCut |  |  |  |  |  |  |  |  |  |
| CMon |  |  |  |  |  |  |  |  |  |
| Rec |  |  |  |  |  |  |  |  |  |
| RMon |  |  |  |  |  |  |  |  |  |
| SM |  |  |  |  |  |  |  |  |  |
| CS |  |  |  |  |  |  |  |  |  |
| CEM |  |  |  |  |  |  |  |  |  |



Table 6: Salient logics of conditionals.
**Ck** and **CK** correspond to basic
conditional logics; System **B** is due to Burgess;
system **SS** to Pollock; system **NP** to
Delgrande; systems **V**, **VW** and
**VC** to D. Lewis; system **C2** to
Stalnaker.




We highlight nine systems of conditional logic that have drawn
particular attention.



System **CK** axiomatizes the truth-conditions laid out
in (VSC)-rm. **CK** has been called the basic conditional
logic by Chellas (1980), as it is meant to be the counterpart to the
basic system **K** of modal logic. In particular it
yields the same theorems as system **K** relative to a
fixed modality \(\Box\_{A}\). Thus, it satisfies conditional analogues
of the axiom K of standard modal logic (viz., the axiom CK: \((A>B)
\supset((A > (B\supset C)) \supset(A > C))\)) and of the rule of
Necessitation (viz., the rule CNec: from \(C\) infer \(A>C\)). It
can be described as the smallest logic closed under LLE and RCK (the
latter entailing LT, RW and AND), and is also called a normal
conditional logic because of that. The logic **Ck**
corresponds to its hyperintensional variant, in which LLE is
dropped.



System **B**, proposed by Burgess, is the smallest
extension of **CK** containing ID, OR and CMon. System
**B** axiomatizes the truth conditions stated under
(VSC)-o assuming \(\preceq\) is a preorder (reflexive and transitive).
Another specific interest of system **B** is that its
flat fragment corresponds to the system **P** of
nonmonotonic logic due to Kraus, Lehmann and Magidor (see
 section 3.4).
 System **P** is also sound and complete for Adams'
probabilistic semantics of conditionals (see
 section 5),
 and a three-valued characterization can be given for it too, as
mentioned above, see Dubois and Prade (1994). In that sense, system
**P** is quite central amongst logic of conditionals.



Particularly noteworthy is the fact that none of these logics
validates Monotonicity (also known as Strengthening the Antecedent),
Transitivity or Contraposition for conditionals:




|  |  |
| --- | --- |
| Monotonicity | \((A>C)\supset ((A\wedge B) >C)\) |
| Transitivity | \((A>B) \supset ((B>C) \supset (A>C))\) |
| Contraposition | \((A>C)\supset (\neg C>\neg A)\) |



These principles have in many contexts been considered as paradigmatic
invalidities of conditional logic.



If we add RMon to **B** we get the system
**V** of David Lewis. John Pollock (1976) proposed the
system **SS** obtained by adding SM and CS to
**B**. Lewis' "official" axiomatization
of the logic of counterfactuals is the system **VC** that
is obtained by adding SM and C to **V**. The extension of
**V** with just SM is the system **VW**.
Finally, system **C2** with the incorporation of CEM into
**VC** is Stalnaker's logic of conditionals. As
 Table 6
 makes clear, principles like CS, CMon, RMon, and CEM in particular
are controversial, they have all been subject to philosophical
discussions.



James Delgrande (1987) proposed the system **NP** as a
different extension of **CK**. The main difference is
that **B** satisfies Cautious Monotonicity, whereas
**NP** satisfies Rational Monotonicity. In
Delgrande's interpretation, an instance of Rational Monotonicity
is the inference from the premises "ravens normally fly"
and "it is not the case that normally ravens are not
black" to "black ravens normally fly". Note that
Cautious Monotonicity predicts the inference from "ravens
normally fly" and "ravens normally are black" to
"black ravens normally fly". The existence of systems
**B** and **NP** makes clear that strictly
speaking, RMon and CMon are logically independent. But it is fair to
say that RMon is *essentially* stronger than CMon, because in
most environments one can infer \(\neg(A>\neg C)\) from \(A>C\).
In the ordering semantics, RMon needs weak orders which satisfy
Modularity (for all worlds \(t, u, v, w \in W\), if \(t\prec\_w u\),
then \(t\prec\_w v\) or \(v\prec\_w u\)), while CMon needs only
transitive relations. But RMon may be satisfied without CMon being
satisfied if one has a weak order over an infinite set without
anything like the limit assumption and uses a selection function or a
minimal models approach. This is the case in Delgrande
 (1987).[2]



Beside the systems mentioned, other intermediate systems between
**CK** and **C2** have been studied, in
particular by proof theorists. Olivetti, Pozzato, and Schwind (2007)
and Poggiolesi (2016) present sequent calculi for the systems
**CK**, **CK**+ID, **CK**+SM
and **CK**+SM+ID. Poggiolesi also investigates systems of
natural deduction for the same systems.


### 3.4 Nonmonotonic logic and preferential models



Earlier we mentioned that system **P** corresponds to the
flat fragment of system **B**. Given the centrality of
system **P**, here we say more about the way Kraus,
Lehmann and Magidor defined it as a system of nonmonotonic logic. KLM
do not aim at characterizing which formulae are logically valid, but
at specifying closure conditions for rational sets of conditionals.
They read the conditional \(A>C\) (actually they use the notation
\(A\pipsim C\)), with \(A\) and \(B\) only from \(L\_0\), as expressing
a default of the form "if \(A\) then normally \(C\)" or
"\(C\) is a plausible consequence of \(A\)". They start
with models right away, but their models are slightly different from
the ones we have seen so far. Importantly, KLM have no use for a truth
value of a conditional at a single world, but only look at the
"consequence relation" determined by a full model. What
KLM call "states" are more like information states than
like possible worlds, what they call "worlds" are
valuations, i.e., assignments of truth values to the propositional
variables of the language. KLM's preferential models are similar
to the o-models discussed above, except there is only one binary
relation for the full model rather than one for each world in the
model.



A *cumulative model* \(M\) for a universe \(W\) is a triple
\(\langle S, l, < \rangle\) where \(S\) is a set of states, \(l: S
\rightarrow \cP(W)\) is a labeling function assigning to each state a
non-empty set of worlds from the universe \(W\), and \(<\) is a
binary relation on \(S\) satisfying the *smoothness condition*:
for all \(A\in L\_0\) and all \(s\) such that
\(s\mathrel|\mathrel{\mkern-3mu}\equiv A\) either \(s\) is minimal in
\(\hat{A} = \{s \in S: s\mathrel|\mathrel{\mkern-3mu}\equiv A\}\) or
there is an \(s' < s\) that is minimal in \(\hat{A}\). Here
\(s\mathrel|\mathrel{\mkern-3mu}\equiv A\) (read: \(s\) satisfies
\(A\)) means that for every world \(w\) in \(l(s)\), \(w \Vdash
A\).



The smoothness condition is almost the same as condition (L) of
 section 3.2.1
 above. This notion of a model allows for a modification of the
classical definition of entailment in such a way that it becomes
compatible with the idea of defeasibility or nonmonotonicity:



Given a cumulative model \(M = \langle S, l, < \rangle\), the
entailment relation \(\pipsim \_M\) defined by \(M\) is given by: \(A
\pipsim \_{M} B\) iff for all states \(s\) minimal in \(\hat{A}\),
\(s\mathrel|\mathrel{\mkern-3mu}\equiv B\).



Thus every single cumulative model defines a preferential entaliment
relation--one with respect to which a sentence *A*
typically entails much more than just its classical logical
consequences.



Kraus, Lehmann, and Magidor (1990) focused on a particular class of
cumulative models, the so-called preferential models. They require the
binary relation to be a strict partial order and the labeling function
to assign just one world to each state. A *preferential model*
for a universe \(W\) is a triple \(\langle S, l, < \rangle\) where
\(S\) is a set of states, \(l: S \rightarrow W\) is a labeling
function assigning to each state a single world from the universe
\(W\), and \(<\) is a strict partial order on \(S\) (irreflexive,
transitive) satisfying the *smoothness condition*.



Lehmann and Magidor (1992) focused on a subfamily of preferential
models, the so-called ranked models. A *ranked model* \(R\) is
a preferential model \(\langle S, l, < \rangle\) where the strict
partial order \(<\) satisfies Modularity: for all \(s, t, u \in
S\), if \(s<t\), then \(u<t\) or \(s<u\).



These modes are called "ranked" because the property of
Modularity of the strict partial order < is equivalent to there
being a totally ordered set \(\Omega\) (the strict order on \(\Omega\)
will be denoted by \(\bangle\)) and a ranking function \(r : S
\rightarrow \Omega\) such that \(s < t\) iff \(r(s) \bangle r(t)\).
Intuitively, a state of smaller rank is "more normal" than
a state of higher rank.



The main representation theorem of Kraus, Lehmann, and Magidor
concerns the system **P** in terms of preferential
models. Below we repeat the rules of the system, except that here
axioms come in argument form (the entailment symbol
"\(\pipsim\)" is not part of the object language), and
inference rules in meta-argument form. The rules in the systems they
consider are not thought of as validity preserving. They are rather
taken to be closure conditions: any reasonable nonnomotonic inference
relation \(\pipsim \) is such that when it includes the premises
(sometimes, though, the premises are *exclusion* conditions,
see RMon, DRat and NRat below), then it includes the conclusion, too.
Kraus, Lehmann and Magidor's representation results then state
that an inference relations \(\pipsim \) is reasonable in this
syntactically specified sense just in case it can be defined by a
model with certain properties.

 

\[\tag\*{ID\(\_{\pipsim}\)} A \pipsim A \] \[\tag\*{LLE\(\_{\pipsim}\)} \frac{\models A \equiv B \qquad A \pipsim C}{B \pipsim C}\] \[\tag\*{RW\(\_{\pipsim}\)} \frac{\models A \supset B \qquad C \pipsim A}{C \pipsim B}\] \[\tag\*{CCut\(\_{\pipsim}\)} \frac{A \pipsim B \qquad A \wedge B \pipsim C}{A \pipsim C}\] \[\tag\*{CMon\(\_{\pipsim}\)} \frac{A \pipsim B \qquad A \pipsim C}{A \wedge B \pipsim C}\] \[\tag\*{AND\(\_{\pipsim}\)} \frac{A \pipsim B \qquad A \pipsim C}{A \pipsim B \wedge C}\] \[\tag\*{OR\(\_{\pipsim}\)} \frac{A \pipsim C \qquad B \pipsim C}{A \vee B \pipsim C}\]


In Kraus, Lehmann, and Magidor (1990), rules ID\(\_{\pipsim}\),
LLE\(\_{\pipsim}\), RW\(\_{\pipsim}\), CCut\(\_{\pipsim}\) and
CMon\(\_{\pipsim}\) together define the system **C** of
*cumulative reasoning*; AND\(\_{\pipsim}\) is a derived rule of
**C**. If OR\(\_{\pipsim}\) is added, one gets system
**P** of *preferential reasoning*; if one keeps
AND\(\_{\pipsim}\), CCut\(\_{\pipsim}\) can be derived from the other
rules of **P**.



Lehmann and Magidor (1992) prove that the system **R** of
*rational reasoning* that is obtained by adding the following
rule of Rational Monotonicity to the above set of rules is complete
with respect to ranked models.

 

\[\tag\*{RMon\(\_{\pipsim}\)} \frac{A \pipsim C \qquad A \npipsim \neg B} {A \wedge B \pipsim C} \]


This rule has an untypical form in that one of its premises is the
denial of a nonmonotonic implication; such rules are called
"non-Horn". If RMon\(\_{\pipsim}\) is added to
**P**, then CM\(\_{\pipsim}\) can be replaced by the rule
that infers \(A\wedge B\pipsim \bot\) from \(A\pipsim \bot\).
Arlo-Costa and Shapiro (1992) proved that the system obtained
by dropping CMon\(\_{\pipsim}\) from **R** is sound and
complete with respect to "rough" ranked models in which
the ordering of the states is no longer subject to the smoothness
condition.



Other important non-Horn rules in the literature are
"Disjunctive Rationality" and "Negation
Rationality":

 

\[\tag\*{DRat\(\_{\pipsim}\)} \frac{A\vee B \pipsim C \qquad A \npipsim C} {B \pipsim C} \] \[\tag\*{NRat\(\_{\pipsim}\)} \frac{A \pipsim C \qquad A\wedge B \npipsim C}{A \wedge \neg B \pipsim C} \]


Negation Rationality is weaker than Disjunctive Rationality, which in
turn is weaker than Rational Monotonicity (see Lehmann & Magidor
1992).


### 3.5 Or-to-if, import-export, simplification of disjunctive antecedents



To conclude this section we point out three axioms that are valid for
the material conditional but which fail in the extensions of
**CK** mentioned earlier, even in the strong logics
**C2** and **VC** of Stalnaker and
 Lewis.[3]
 These principles concern the interaction of conditionals with
disjunction and conjunction. They are:




|  |  |  |
| --- | --- | --- |
| OI | \((A\vee B) \supset(\neg A >B)\) | (Or to If) |
| IE | \((A >(B >C)) \equiv (A\wedge B >C)\) | (Import-Export) |
| SDA | \(((A \vee B) >C) \supset((A >C) \wedge (B >C))\) | (Simplification of Disjunctive

Antecedents) |



The principle OI is intuitively plausible, but as noted by C. I.
Lewis, assuming it leads to one of the paradoxes of the material
conditional. For assuming that \(A\) entails \(A\vee B\) (disjunction
introduction), and that the latter entails \(\neg A>B\), by
transitivity it should follow that \(A\) entails \(\neg A>B\).
Various options exist in the face of the paradox: to deny disjunction
introduction, to deny transitivity, or indeed to deny the validity of
OI. Stalnaker (1975) for one defends the invalidity of OI, arguing
that OI is a *pragmatically reasonable* principle, but that the
inference is context-sensitive.



A similar collapse result concerns the principle IE. Gibbard (1980)
shows that provided we have Modus Ponens, IE, LLE, and the principle
of Supraclassicality--whereby \(A>C\) must hold if \(A\)
classically entails \(C\)--, then if the conditional entails the
material conditional, the reciprocal also holds. McGee (1989) modified
Stalnaker's semantics in such a way that IE is valid, but MP is
no longer valid. McGee's rejection of MP is independently
motivated by natural language examples, in particular by the famous
example of the 1980 US elections, in which Reagan was ahead of the
polls as a Republican, Carter second as a Democrat, and then Anderson
as a distant third, but also a Republican. In this context, McGee
argues that the following instance of MP is problematic:



(3)
A Republican will win. If a Republican wins, then if Reagan does
not win, Anderson will win. Therefore if Reagan does not win, Anderson
will win.


Here the conclusion is problematic, because intuitively, if Reagan
does not win, Carter appears as the more likely winner. Recently
Mandelkern (2020) has argued that IE should not be viewed as
unrestrictedly valid. Mandelkern's argument is in part based on
logical considerations and in part based on natural language examples.
Mandelkern points out that in a situation (adapted from McGee's
counterexample) in which Reagan is ahead of Carter and Anderson, but
the relative situation of Anderson and Carter is not known, there is a
contrast between:



(4)
If a Republican will win the election, and Anderson will win if
Reagan doesn't win, then both Republicans are currently in a
stronger position to win than Carter.
(5)
If a Republican will win the election, then if Anderson will win
if Reagan doesn't, then both Republicans are currently in a
stronger position to win than Carter.


According to Mandelkern, the latter is less acceptable than the
former, because the second conditional antecedent "Anderson will
win if Reagan doesn't" is contextually equivalent to
"a Republican will win". Mandelkern does find IE
acceptable if the nested antecedent is restricted to a non-conditional
sentence, however.



Another argument against IE, put forward in Adams (1975), concerns the
equivalence of \(A>(B >A)\) and \((A\wedge B) >A\). Whereas
the latter is a clear logical truth, the former is not. S. Kaufmann
(2005) thus argues that the validity or invalidity of IE depends on
specific causal assumptions linking the consequent and the
antecedents.



The case of SDA is also a bone of contention between theorists.
Chellas (1975) and Fine (1975) first pointed out that SDA seems
intuitively valid, against Lewis's and Stalnaker's
theories. However, as pointed out by Ellis, Jackson, and Pargetter
(1977), SDA and LLE together imply Monotonicity: from \(A>C\), it
follows that \(((A\wedge B) \vee (A\wedge \neg B)) >C\) by LLE, and
thus by SDA, \((A\wedge B) >C\).



Since then, various theorists have proposed to abandon LLE (viz., Nute
1980; Fine 2012; Ciardelli, Zhang, & Champollion 2018; Santorio
2018) in order to endorse SDA without Monotonicity. Santorio (2018) in
particular provides a systematic comparison between pragmatic accounts
based on the idea that SDA might be based on a mechanism akin to
scalar implictures (see Klinedinst 2009), and semantic accounts based
on the idea that logically equivalent antecedents can directly
generate different alternatives. Santorio's proposal, in a
nutshell, is that \(A>C\) is true iff the alternative propositions
\(A\_1,\dots,A\_n\) that are ways for \(A\) to be true are such that the
closest worlds in them are \(C\)-worlds.



Importantly, Santorio's account drops LLE but nevertheless makes
SDA only optionally valid. Lassiter (2018) recently suggested that
some classic counterexamples put forward against SDA by McKay and
Inwagen (1977), and originally judged inconclusive, can be
strengthened. For example, assuming Jim likes odd numbers but is
indifferent between them, and actually bet on 5, one can accept:



(6)
If Jim had bet on 1 or 3, there is an exactly 50% probability that
he would have bet on 3.


SDA would predict:



(7)


a.
If Jim had bet on 1 there is an exactly 50% probability that he
would have bet on 3.
b.
If Jim had bet on 3 there is an exactly 50% probability that he
would have bet on 3.
 


Both conclusions are problematic under at least one reading. However,
we note that there is also a reading that would make them fine, if the
evaluation of chance is prior to Jim's betting.



In summary, we see that although OI, IE and SDA appear intuitively
valid, to accept them involves substantive trade-offs regarding other
basic principles.


## 4. Premise Semantics


### 4.1 Motivations



Some early theories of conditionals said that a counterfactual is
true, or assertable, if and only if its antecedent, together with
further "co-tenable" premises, implies its consequent
(Ramsey 1931 commenting on Mill; Chisholm 1946; Goodman 1947, 1955;
Mackie 1962). D. Lewis called such theories metalinguistic, the idea
being that "if \(A\), \(C\)" is true provided there is an
argument leading from the sentence \(A\) and additional premises to
\(C\). On these accounts, the counterfactual is either a sentence
meaning that some suitable argument backing it exists, or--as in
Mackie's version--it is itself an elliptical presentation
of such an argument. The principal problem of any such theory is to
specify which "additional premises" are suitable to be
conjoined to a given antecedent and which are not. Goodman (1955)
views the additional premises as general laws, supplemented by
relevant factual conditions, but the problem is to determine which
facts are to be assumed, and which facts are to be retracted. Rescher
(1964) treats additional potential premises as ordered according to
their position in a system of "modal categories".



An account of the co-tenability idea based on possible worlds was
further elaborated in the work of Veltman (1976) and Kratzer (1979),
in what has come to be known as *premise semantics* (an
expression coined by D. Lewis 1981). Premise semantics can be
presented either as providing acceptance conditions or as providing
truth conditions. It is compatible with an epistemic as well as with
an ontic interpretation of conditionals. The former builds on the idea
that the premises represent essentially an agent's basic
beliefs; this is the approach of Veltman (1976) who took inspiration
from Ramsey. The latter takes the premises to represent the basic
facts of a possible world; this is the approach of Kratzer (1979,
1981), who was inspired by Rescher and Lewis (D. Lewis 1981) and has
functions assigning a premise set to every possible world. In any
case, the idea of premise semantics is this. Relative to a given set
\(\Gamma\) of premises (assigned to a possible world \(w\)), a
conditional \(A>B\) is accepted (or true at \(w\)) if and only if
for every maximal subset of \(\Gamma'\) of \(\Gamma\) that is
consistent with \(A\) it holds that \(\Gamma' \cup \{A\}\) implies
\(B\).


### 4.2 Premise frames



More formally, a *premise frame* is a pair \(\langle W,
\Gamma\rangle\) where \(W\) is a non-empty set of possible worlds and
\(\Gamma: W \to \cP(\cP(W))\) is as a function assigning to any world
\(w\) a set of propositions--the premises for \(w\). One may, but
need not stipulate that \(w\) does, and all other worlds do not,
belong to every proposition in \(\Gamma(w)\) (Centering). For the sake
of simplicity, we assume that \(W\) is finite. A *premise
model* is a triple \(M = \langle W, \Gamma, V\rangle\) such that
\(\langle W, \Gamma\rangle\) is a premise frame and \(V\) a valuation
function. For any proposition \(F\), let \(\Gamma(w)\bot F\) denote
the set of all subsets of \(\Gamma(w)\) that are consistent with \(F\)
but have no proper supersets in \(\Gamma(w)\) that are also consistent
with \(F\). That is, \(\Gamma(w)\bot F\) is the set of subsets of
\(\Gamma(w)\) that are maximally consistent with \(F\). The truth
conditions for conditionals are:

 

\[\tag{PS} M, w \Vdash A >C \text{ iff for each } X \in \Gamma(w)\bot\lvert A\rvert^M, \\ \bigcap (X\cup \lvert A\rvert^M) \subseteq \lvert C\rvert^M. \]


This means that a (counterfactual) conditional is true at a world
\(w\) if each set of basic beliefs or facts associated with this world
that is maximally consistent with \(A\), taken together with \(A\),
implies \(C\).



Kratzer shows how this definition gives the flexibility to either
"lump" some facts together, or to keep facts separate from
each other relative to a conditional antecedent. One of her
illustrative examples is the
 following.[4]
 Angelika and Regina must pass a bridge one after the other. Angelika
takes one minute to pass the bridge, and Regina is waiting one minute
before passing. Assuming Angelika had passed the bridge in 40 seconds,
would Regina still be waiting for one minute? Let \(B\) represent
"Angelika passes in one minute", \(C\) "Regina waits
one minute" and \(A\) represent "Angelika passes in 40
seconds", with the proviso that \(A\) and \(B\) are
inconsistent. The issue is whether the conditional \(A>C\),
"if Angelika had passed in 40 seconds, Regina would have waited
one minute", is true or false. One option is to let
 \(\Gamma(w)=\{B,C\}\).[5]
 \(\Gamma(w)\cup \{A\}\) is inconsistent, but contains as maximal
\(A\)-consistent set only \(\{A,C\}\) and the intersection of this set
entails \(C\). This predicts that Regina would have waited one minute
indeed under the assumption that Angelika had passed faster. Another
option is to let \(\Gamma(w)=\{B\cap C\}\). This time, \(\Gamma(w)\cup
\{A\}\) contains \(\{A\}\) as sole maximal \(A\)-consistent set, and
this set fails to entail \(C\). Although the two premise sets have the
same intersection, the lumping of premises in the second set leads one
to retract more facts when making the counterfactual assumption. This
flexibility, as emphasized by Kratzer, accounts for the
context-sensitivity of conditionals.



It is easy to see how premise semantics invalidates Transitivity. For
example, if

 

\[\Gamma(w) = \{A\supset B,\, A\supset\neg C,\, \neg A\wedge(B\supset C)\},\]


then we get that both \(A>B\) and \(B>C\) are true, but
\(A>C\) is false at \(w\) (in fact, \(A>\neg C\) is true; see
 Fig. 2).
 There are similar counterexamples to Monotonicity and
Contraposition.



![a diagram: link to extended description below](figure2.svg)

Figure 2: Counterexample against
Transitivity in a premise model [An
 extended description of figure 2
 is in the supplement.]



### 4.3 Correspondence with ordering semantics



David Lewis (1981) proved that the truth conditions of premise
semantics and ordering semantics fundamentally correspond to each
other. One can start with a premise model and define a strict partial
ordering for an o-model that gives the same results: define \(u
\prec\_w v\) iff the set of all premises in \(\Gamma(w)\) true at world
\(u\) is a proper superset of the set of premises true at world \(v\).
And conversely, one can start with an o-model based on a strict
partial ordering and define a premise model that gives the same
results: define \(\Gamma(w)\) as the set of all propositions of the
form

 

\[X\_v := \{u\in W: u\prec\_w v \text{ or } u=v\},\]


for every \(v\in W\).



Lewis in his paper discusses several refinements of the
correspondence, depending on which conditions are imposed on ordering
frames, and further results can be found in Chemla (2011). From a
philosophical and foundational perspective, the correspondence may be
taken to suggest that the choice between an ordering-first approach
and a premise-first approach doesn't matter. Arguably, however,
a premise semantics may be judged more explanatory than one in which
the ordering comes first, for two reasons. First of all, from a
cognitive perspective, it may be argued that premise semantics is
closer to implementing the Ramsey test idea, since an ordering
semantics only implicitly refers to the idea of making adjustments to
one's belief set. Secondly, in an o-semantics the question is
indeed moot what determines similarity between worlds. Of course,
which facts to admit or not in premise semantics is also problematic,
but using propositions as an ordering source makes clearer that
similarity between worlds is a relative rather than an absolute
matter. In recent years, several authors have proposed to further
clarify the problem of premise selection in counterfactuals by looking
at causal models of reasoning (K. Schulz 2007; Briggs 2012; S.
Kaufmann 2013; Santorio 2019).



Based on the correspondence between ordering semantics and premise
semantics, Kratzer (1979) mentions that Burgess's system
**B** is sound and complete for the latter. Veltman
(1985: 108-132) proved that flat conditionals interpreted by
premise semantics can be characterized by the axioms of system
**P**. Results obtained in the 1990s show that such
conditionals have an additional special property that cannot, however,
be expressed as an axiom of conditional
 logic:[6]



\((\*)\)
If \((A\vee B) >C\) is accepted (or true at \(w\)), then there
are \(D\) and \(E\) such that \(D\wedge E\) entails \(C\), and \(A
>D\) and \(B >E\) are both accepted (or true at \(w\)).


Among the recent modifications of premise semantics with causal
models, Santorio's (2019) filtering semantics ends up giving up
OR.


## 5. Probabilistic Logics



The idea that the right semantics for conditionals may be
probabilistic is appealing for at least three reasons. First of all,
as pointed out by Adams (1965), it is unclear whether it makes sense
to declare either true or false conditionals with false antecedents.
Adams thus prefers to think of conditionals as having assertability
conditions, rather than truth conditions, and justified assertability
is naturally handled in terms of probability. Adams's scepticism
about truth conditions for conditionals is shared by several authors,
in particular Gibbard (1980) and Edgington (1995), and it was further
fueled by Lewis's (1976) triviality results. Secondly, many
conditional sentences appear to express only a relation of probable
inference between antecedent and consequent. Thus, whereas "if
\(a=b\), then \(a+1=b+1\)" expresses a purely deductive
relation, "if you boil this egg, it will become hard"
expresses an inductive relation, and "if the light does not turn
on, then the lamp must be broken" expresses an abductive
relation (Douven & Verbrugge 2010; Krzyzanowska, Wenmackers,
& Douven 2013; Douven 2016). But in the same way in which
inductive and abductive inferences express that a conclusion may be
inferred from given premises with only a reasonable probability,
putting these inferences into a conditional appears to demand the
assignment of a degree of probability to the corresponding sentences.
Thirdly, as stressed by Stalnaker (1970),



> 
> 
> [a]lthough the interpretation of probability is controversial, the
> abstract calculus is a relatively well defined and well established
> mathematical theory.
> 
> 
> 



Indeed, based on its mathematical simplicity, the theory offers a
universal framework in the psychology of reasoning, where conditionals
occupy central stage (Over, Hadjichristidis, Evans, Handley, &
Sloman 2007). We open this section with Adams's logic for
conditionals. We then explain Lewis's triviality results and the
problem they raise for the assignment of probabilities to compounds of
conditionals.


### 5.1 Adams's logic



Central to probabilistic treatments of conditionals is the notion of
conditional probability. Given a Boolean formula \(A\) of \(L\_0\), and
a set \(W\) of possible worlds over which they are interpreted, we let
\(\Pr(A)\) denote \(\Pr(\lvert A\rvert)\), namely the probability
assigned to the worlds in which \(A\) is true. For \(A\) and \(C\) two
Boolean formulae, the conditional probability
\(\Pr(C\mkern2mu|\mkern2muA)\) is definable by the Ratio Formula:


* **(Ratio Formula)**
\(\Pr(C\mkern2mu|\mkern2muA)=\displaystyle\frac{\Pr(A\wedge
C)}{\Pr(A)}\), provided \(\Pr(A)>0\).



Now consider the problem of determining the probability that if a fair
die lands even, it will land on six. The probability of the material
conditional "either it is not even, or it is a six" is



\[\Pr(1\vee 3\vee 5\vee 6)=\nicefrac{2}{3}.\]


This clearly exceeds the probability sought, which is instead captured
in terms of the conditional probability of getting a six assuming it
is an even number,

 

\[\Pr(6 \mid 2\vee 4 \vee 6)=\nicefrac{1}{3}.\]


More generally, the probability \(\Pr(A\supset C)\) of any material
conditional is never less than the conditional probability
\(\Pr(C\mkern2mu|\mkern2muA)\), a point apparently first stressed by
Reichenbach (1949, p. 437).



Adams's central assumption is that for every conditional
sentence \(A>C\) in which \(A\) and \(C\) are Boolean sentences,
and for every probability function \(\Pr\), the probability of the
conditional equals the corresponding conditional probability:


* **(Adams's Thesis)** For every Boolean
sentences \(A\), \(C\), and every probability function \(\Pr\),
\(\Pr(A>C)=\Pr(C\mkern2mu|\mkern2muA)\), provided
\(\Pr(A)>0\).



Adams's Thesis has two sides. From the inner standpoint of
Adams' logic of conditionals, it may be viewed as a
*definition*: the conditional probability defines the degree of
probability assigned to a simple conditional. From the outer
standpoint of ordinary English, Adams's Thesis makes an
empirical claim: namely, that the *assertability* of a simple
conditional is a function of the conditional probability of the
consequent given the antecedent.



Stipulating that \(\Pr(A>C)=1\) when \(\Pr(A)=0\), Adams
investigates several notions of validity for the language \(L\_1\) of
flat conditionals. His original definition is the following:


* **(Probabilistic Validity)** An inference is
*probabilistically valid* if and only if, for any positive
\(\varepsilon\), there exists a positive \(\delta\) such that under
any probability assignment under which each of the premises has a
probability greater than \(1 - \delta\), the conclusion will have a
probability of at least \(1 - \varepsilon\).



One way of understanding this definition is by contraposing it: an
argument is invalid when an arbitrarily high probability can be
assigned to the premises, but such that the conclusion falls short of
a high probability.



Adams contrasts this definition with an alternative criterion for
validity that is perhaps more intuitive:


* **(Strict Validity)** An inference is *strictly
valid* if and only if its conclusion has probability 1 under any
probability assignment under which its premises each have probability
1.



Every probabilistically valid inference is strictly valid, but not
vice versa. Strict validity coincides with classical validity. In
later work, Adams uses an alternative definition of probabilistic
validity, based on the notion of uncertainty. Given a probability
function \(\Pr\), the uncertainty U\((A)\) of a sentence \(A\) is
defined as \(1-\Pr(A)\):


* **(p-Validity)** An inference is *p-valid* if
and only if for every probability assignment, the uncertainty of the
conclusion is no greater than the sum of the uncertainty of the
premises.



Probabilistic validity and \(p\)-validity can be shown to coincide
(Adams 1975: Thm. 3.1). For Boolean formulae, moreover, Adams shows
that the three definitions of validity coincide with classical
validity. For conditional sentences, however, the resulting logic is
weaker, and corresponds in fact to system **P**.



As in the semantics considered in
 sections 3
 and 4, Monotonicity, Transitivity, and Contraposition fail
in Adams' semantics. Consider Transitivity: it is easy to
construct models in which \(\Pr(B\mkern2mu|\mkern2muA)=1\),
\(\Pr(C\mkern2mu|\mkern2muB)>1-\delta\) for positive, but
arbitrarily small \(\delta\), and yet \(\Pr(C\mkern2mu|\mkern2muA)=0\)
(see
 Fig. 3).



![a diagram: link to extended description below](figure3.svg)

Figure 3: Counterexample against
Transitivity in an Adams model [An
 extended description of Figure 3
 is in the supplement.]




This yields a counterexample according to either definition of
probabilistic validity. Similar examples can be constructed for
Monotonicity, Contraposition, and the paradoxes of the material
conditional.


### 5.2 Lewis's triviality theorem



Viewed as logics that tell us which conditionals are consequences of a
given set of conditionals, Adams's logic coincides, over their
common language, with Stalnaker's and Lewis's logics. That
is, for every argument with a finite set of premises \(\Gamma\) and
conclusion \(A\) of \(L\_1\), \(A\) is a \(p\)-consequence of
\(\Gamma\) iff \(A\) is a consequence of \(\Gamma\) according to
Stalnaker's semantics (see Gibbard 1980 for a
 proof).[7]
 However, the main limit of Adams's logic is that it does not
account for embeddings of conditionals (negated conditionals,
disjunctions of conditionals, right-nested and left-nested
conditionals). Nevertheless, the coincidence of the logics generated
by possible worlds and probabilistic semantics suggests that
Adams' Thesis could be extended to the language \(L\_2\) of
arbitrarily complex conditionals. This hypothesis is known as
Stalnaker's Hypothesis, who formulates it in his 1970
 article.[8]


* **(Stalnaker's Hypothesis)** For every
probability function \(\Pr\) and for every conditional \(A>C\),
possibly complex: \(\Pr(A>C)=\Pr(C\mkern2mu|\mkern2muA)\), provided
that \(\Pr(A)>0\).



The empirical adequacy of Stalnaker's Hypothesis depends on the
treatment of nested conditionals. However, Lewis (1976) shows that
under rather uncontroversial assumptions on the probability of nested
conditionals, the thesis leads to triviality. Lewis assumes the
following Factorization Hypothesis (Fitelson 2015):


* **(Factorization Hypothesis)** For every probability
function \(\Pr\) and for all sentences \(A\) and \(B\) such that
\(\Pr(A\wedge B)>0\):
* \[\Pr(B >C\mid A)=\Pr(C\mid A\wedge B).\]



Given Stalnaker's Hypothesis, the Factorization Hypothesis is
equivalent to the Law of Import-Export, namely \(\Pr(A > (B >
C))=\Pr((A\wedge B) > C).\)



From Factorization, Lewis derives the following result, which
basically expresses that the probability of the conditional collapses
to the unconditional probability of its consequent:


* **Triviality Theorem (Lewis)** If \(A\) is
probabilistically compatible with both \(C\) and \(\neg C\), that is,
if \(\Pr(A\wedge C)>0\) and \(\Pr(A\wedge \neg C)>0\), then
\(\Pr(A>C)=\Pr(C)\).



The proof goes as follows:

 

\[\begin{align} 
\Pr(A> C)&=\Pr(A>C\mid C)\Pr(C) + \Pr(A>C\mid \neg C)\Pr(\neg C) \\
 & \hspace{4cm} {\scriptsize\text{(Expansion)}}\\ 
\Pr(A>C\mid C)&=\Pr(C\mid A\wedge C) = 1 \\
 & \hspace{4cm} {\scriptsize\text{(by Factorization)}}\\ 
\Pr(A>C\mid \neg C)&=\Pr(C\mid A\wedge \neg C) = 0 \\
 & \hspace{4cm} {\scriptsize\text{(by Factorization)}}\\ 
\Pr(A>C)&=1\cdot \Pr(C)+0\cdot \Pr(\neg C) = \Pr(C) 
\end{align}\]


Lewis himself states further variants, and several generalizations
have been proposed since (Hajek & Hall 1994; Bradley 2000;
Milne 2003; Fitelson 2015). Bradley (2000) in particular shows that
the following Preservation Condition inspired by Belief Revision
theory also leads to triviality.


* **Preservation Condition (Bradley):** For every
probability function \(\Pr\) and all sentences \(A\) and \(C\), if
\(\Pr(A)>0\) and \(\Pr(C)=0\), then \(\Pr(A>C)=0\).



The condition is weaker than Stalnaker's Thesis, and therefore
it extends triviality to an even wider class of conditionals.


### 5.3 Cardinality and imaging



Lewis's triviality results expose a mismatch between
\(\Pr(C\mkern2mu|\mkern2muA)\) and \(\Pr(A>C)\). To clarify the
phenomenon, we ask what each quantity represents in light of
Lewis's result, already for simple conditionals.


#### 5.3.1 Cardinality



Let us start with \(\Pr(C\mkern2mu|\mkern2muA)\). Even with a fixed
probability assignment over a finite set of worlds, the conditional
probability may not correspond to the probability of any set of
worlds. Let \(W=\{a,b,c\}\) and let

 

\[\Pr(a)=\Pr(b)=\Pr(c)=&frac13;.\]


Clearly,

 

\[\Pr(a\mid a\vee b)=1/2.\]


However, in the Boolean algebra generated from \(W\), no
possible-worlds proposition gets a probability of 1/2 under
\(\Pr\). Hajek (1989) in fact showed that there are more
distinct conditional probability values than distinct unconditional
probability values, for any finite set of possible worlds \(W\) that
has at least three worlds with non-zero probability. As pointed out in
Egre and Cozic (2011), this result offers a structural
analogy with undefinability results in generalized quantifier theory.
"Most \(A\)s are \(C\)s" is not definable by applying the
unrestricted "most" to first-order definable propositions.
Similarly here, we may view the result as showing that in a sentence
like "there is a probability of 1/2 that if \(A\) then
\(C\)", the embedded conditional does not express any
proposition.



This result, however, is compatible with the idea that if-clauses
fundamentally act as restrictors of overt or covert operators (see
Lewis 1975; Kratzer 2012; and
 section 8.1
 below). Egre and Cozic (2011) use the result to
vindicate that view, though Charlow (2016) argues that the restrictor
view is not in itself immune to triviality. Another take on it
concerns three-valued approaches. In the three-valued case, the
assertability of a proposition is naturally viewed as the probability
of that proposition being true given it has a determinate truth value
(McDermott 1996). Assuming De Finetti's scheme for the
conditional, the assertability of \(A>C\) is then equal to the
probability that \(A\wedge C\) is true, given that \(A\) is true, and
so to the conditional probability. This derives Adams' Thesis
for simple conditionals. That fact notwithstanding, note that with
\(a, b,c\) as atoms and \(\Pr\) as equiprobable on them, again there
is no three-valued proposition whose probability is 1/2 for the
conditional \(a\vee b > a\), if we define a possible world by an
assignment of one of the three values to atomic propositions (and
handle disjunction Strong Kleene style). Rather, if three-valued
conditionals do express propositions, those propositions then need to
be viewed as relations between those worlds where the sentence is true
and those where it is defined, so again as more complex objects than
set of worlds (cf. Dubois & Prade 1994 who represent them as
pairs).


#### 5.3.2 Imaging



Consider now \(\Pr(A>C)\). Assume Stalnaker's semantics for
\(A>C\); what does the probability of that conditional correspond
to, if not the conditional probability? D. Lewis (1976) identifies a
revision rule he calls *imaging*. Given \(\Pr\), he defines
\(\Pr\mkern.1mu'\_A\) to be the imaging under the antecedent, where
\(\Pr\mkern.1mu'\_A(w)=0\) if \(w\) is not an \(A\)-world; and if \(w\)
is an \(A\)-world, then \(\Pr\mkern.1mu'\_A(w)\) equals \(\Pr(w)\)
augmented by the sum of the non-\(A\) worlds to which \(w\) comes
closest according to Stalnaker's selection function. Lewis shows
that the probability of Stalnaker's conditional is the
probability of the consequent under imaging by the antecedent. Whereas
conditioning upon \(A\) kills off the non-\(A\) worlds and spreads
their mass uniformly over the \(A\)-worlds, imaging upon \(A\) is a
different operation, which shifts the weight of non-\(A\) worlds
non-uniformly to the \(A\)-worlds depending on the selection function
(see
 Fig. 4).




![a diagram: link to extended description below](figure4a.svg)

(a) Conditionalization on \(A\) results in posterior probabilities
\(\Pr'(u)=\frac{p}{p+q}\), \(\Pr'(v)=\frac{q}{p+q}\) and
\(\Pr'(w)=0\).




![a diagram: link to extended description below](figure4b.svg)

(b) Imaging on \(A\) results in posterior probabilities
\(\Pr'(u)=p+r\), \(\Pr'(v)=q\) and \(\Pr'(w)=0\).




Figure 4: Example with \(W=\{u,v,w\}\)
and prior probabilities \(\Pr(u)=p\), \(\Pr(v)=q\) and \(\Pr(w)=r\).
In the figure a red dashed arrow from \(x\) to \(y\) represents that
\(y\) is the closest \(A\)-world to \(x\) [An
 extended description of Figure 4
 is in the supplement.]




An example from Edgington (2020) can help to figure out the
difference. Assume a stack of 100 straws, out of which 90 are 10 cm
long, 1 is 11 cm long, and 9 are 20 cm long. Consider the conditional
"if the straw is longer than 10 cm, then it is shorter than 15
cm". Assume similarity between worlds is set by the closeness in
lengths between straws. The probability of the set of worlds in which
Stalnaker's conditional is true is \(\nicefrac{91}{100}\), since
the 10-cm-straw worlds and the 11-cm-straw world are such that the
closest world in which the straw is longer than 10 cm is indeed a
world where it is less than 15 cm. But the conditional probability for
the conditional is \(\nicefrac{1}{10}\), since only one of the straws
that are 11 cm or 20 cm long is less than 15 cm.



The upshot of these considerations is that starting from
Stalnaker's conditional, we can see that its probability is not
the conditional probability, but the imaging probability (for more on
imaging, we refer to Gunther 2018). Conversely, starting from the
conditional probability of \(C\) given \(A\), we have seen that it is
not in general equal to the probability of any possible world
proposition. Importantly, Adams's Thesis remains coherent for
simple conditionals *qua* definition, because it stipulates
that the probability of a conditional sentence is a certain
conditional probability. However, as an empirical thesis,
Stalnaker's Hypothesis states a substantive identity between
quantities defined separately, and we now see better why it should
fail. Whether Adams' Thesis holds as a matter of empirical fact
and even for simple conditionals is still debated. Recent findings, to
which we return in
 section 7,
 suggest that relevance considerations also come into play (Douven
2016; Skovgaard-Olsen, Singmann, & Klauer 2016).


### 5.4 Probabilities for compound conditionals



Stalnaker's Thesis is consistent for simple conditionals, but
fails for conditionals of arbitrary complexity. A challenge is to
extend Stalnaker's Thesis to more or less extended classes of
compound conditionals. Several proposals have been made toward that
goal. The associated literature is quite technical, however, and we
give only a few landmarks here. Important is that all the resulting
systems mentioned here conservatively extend Adams's basic
logic, namely system **P** (i.e., they preserve the same
axioms for simple conditionals, though not necessarily for compound
conditionals).



Van Fraassen (1976) extends Stalnaker's Thesis to right-nested
conditionals of the form \(A> (B>C)\) in which \(A, B,C\) are
Boolean, and to left-nested conditionals \((A>B)>C\) with the
same restriction, and he can also deal with specific conjunctions and
disjunctions of conditionals. Van Fraassen's proposal involves a
product space construction, relying on so-called Stalnaker-Bernoulli
models, allowing him to assign probabilities to infinite sequences of
worlds based on standard Stalnaker models. The construction is used in
Stalnaker and Jeffrey (1994) and extended in S. Kaufmann (2009). One
noteworthy aspect of the construction is that the probability of
right-nested conditionals of the form \(A>(B>C)\) is not in all
cases identical to the probability of \((A\wedge B) > C\). McGee
(1989) on the other hand proposes a distinct construction that does
not permit left-nested conditionals, but which is intended to secure
the law of import-export for right-nested conditionals (famously, by
giving up Modus Ponens for nested conditionals). Both approaches have
some probabilistic principles in common, however, for instance



\[\Pr(A\wedge (A>B))=\Pr(A\wedge B)\]


holds in both theories. For both approaches, moreover, the associated
logics are semantically interpretable starting from Stalnaker's
possible world semantics. Bradley (2012) builds on the
Jeffrey-Stalnaker and McGee approaches to show that for simple
conditionals Adams Thesis can be derived by assigning probabilities to
pairs of worlds, instead of just worlds. The probability of a pair
\((w\_0,w\_1)\) represents the probability that \(w\_0\) is the actual
world, and that \(w\_1\) is its selected alternative given the
antecedent under discussion (note that one needs such probabilities
for all potential antecedents). A generalization of this property to
nested conditionals is outlined, with philosophical emphasis on the
idea that the probability of a conditional does not depend on a single
world, but also on the nonfactual relations of counterfactual
dependence between worlds.



A related approach to iterated conditionals is based on the so-called
coherence approach to probability, coming from de Finetti's
betting view of probability (see Coletti & Scozzafava 1999; Gilio
2002; Gilio & Sanfilippo 2014; and Pfeifer 2014 for an
introduction). The idea is that probability degrees can be attached to
conditional events so long as they do not violate certain coherence
constraints on those bets (no sure losses or "Dutch
books"). In this framework the probability of an event is
defined in terms of the technical notion of prevision. The prevision
\(x\) attached to \(A>C\) corresponds to the fact that



> 
> 
> for every real number \(s\), you are willing to pay (resp., to
> receive) an amount \(sx\) and to receive (resp., to pay) \(s\), or
> \(0\), or \(sx\), according to whether \(A\wedge C\) is true, or
> \(A\wedge\neg C\) is true, or \(\neg A\) is true (bet called off),
> respectively. (Sanfilippo, Gilio, Over, & Pfeifer (2020 notation
> adapted))
> 
> 
> 



Depending on independence assumptions concerning basic propositions,
previsions can be defined not only for conjunctions of conditionals,
but also for various nested conditionals. A recent overview of results
in that domain appears in Sanfilippo et al. (2020).


## 6. Belief Revision Approaches



While Stalnaker rendered Ramsey's suggestion about the role of
"if \(A\)" in terms of closest antecedent world, and Adams
focused on the probabilistic interpretation of it, Peter
Gardenfors (1978) introduced a semantics of conditionals that
uses the Ramsey idea in a direct and purely qualitative way. His
approach is resolutely epistemic--or more precisely,
doxastic-- and it is based on the notion of belief or acceptance.
Precursors of this approach were William Harper (1976) and Isaac Levi
(1977).


### 6.1 Belief revision models and the Ramsey test



A *belief set* over a language \(L\) is a set of sentences that
is closed under a given background logic \(\iCn\) (i.e., a
*theory* in the logician's sense). A *belief revision
model* (BRM) is a pair \(M = \langle \cK, \*\rangle\), where
\(\cK\) is a set of belief sets over \(L\) and \*: \(\cK \times L \to
\cK\) is a belief revision function assigning to every belief set
\(K\) and every sentence \(A\) a revised belief set \(K\*A\). The
acceptance condition for conditionals at a belief set in a belief
revision model is a straightforward formalization of Ramsey's
test:

 

\[\tag{RT} M, K \Vdash\_{acc} A >B \text{ if and only if } B \in K\*A.\]


Gardenfors's (1978, 1988: sects. 7.1-7.2) initial
idea was to take conditionals that are accepted at a belief set as
*beliefs*, i.e., as elements of this very belief set. Let us
call this Gardenfors's Ramsey test:

 

\[\tag{GRT} A >B \in K (\text{in } M) \text{ if and only if } B \in K\*A.\]


Principles for the logic of conditionals then fall out directly of the
following rationality postulates for belief revision introduced by
Gardenfors (1978) and Alchourron, Gardenfors, and
Makinson ("AGM", 1985). Using the abbreviation \(K+A\) for
\(\iCn(K\cup\{A\})\), the eight AGM-postulates read as follows: For
all belief sets \(K\) (in all BRMs \(M\)) and all sentences \(A\) and
\(B\),




|  |  |  |
| --- | --- | --- |
| (\(\*\)1) | \(K\*A\) is a belief set. | (Closure) |
| (\(\*\)2) | \(A\in K\*A.\) | (Success) |
| (\(\*\)3) | \(K\*A\subseteq K+A.\) | (Inclusion) |
| (\(\*\)4) | If \(\neg A\not\in K\), then \(K+A \subseteq K\*A\). | (Vacuity) |
| (\(\*\)5) | \(K\*A = K\_{\bot}\) only if \(\vdash \neg A\). | (Consistency) |
| (\(\*\)6) | If \(\vdash A\equiv B\), then \(K\*A = K\*B\). | (Extensionality) |
| (\(\*\)7) | \(K\*(A\wedge B) \subseteq(K\*A)+B\). | (Superexpansion) |
| (\(\*\)8) | If \(\neg B \notin K\*A\), then \((K\*A)+B \subseteq K\*(A\wedge
B)\) | (Subexpansion, Rational Monotonicity) |



The numbering is AGM's, the names are due to Hansson (1999).
Postulates (\(\*\)3) and (\(\*\)4) taken together say that if the input
\(A\) is consistent with the belief set \(K\), then the background
logic alone is sufficient to guide belief revision. When (\(\*\)1) and
(\(\*\)2) are present and \(\iCn\) is a monotonic consequence
operation, (\(\*\)4) is equivalent to the *Preservation
Principle* stating that if \(A\) is consistent with \(K\), then
all elements of \(K\) are preserved in \(K\*A\):



(Pres)
If \(\neg A\not\in K\), then \(K \subseteq K\*A\). (Preservation)


AGM called (\(\*\)1)-(\(\*\)6) the *basic* set of
postulates. The *supplementary* postulates (\(\*\)7) and
(\(\*\)8) concern composite belief revisions of the form \(K\*(A\wedge
B)\). The idea is that, if \(K\) is to be changed minimally so as to
include two sentences \(A\) and \(B\), such a change should be
possible by first revising \(K\) with respect to \(A\) and then
expanding \(K\*A\) by \(B\)--provided that \(B\) does not
contradict the beliefs in \(K\*A\).



The logic of conditionals then mirrors the logic of belief revision:
(\(\*\)1) corresponds to RCK, (\(\*\)2) to ID, (\(\*\)3) to SM, a
weakened version of (\(\*\)4) (viz., if \(A \in K\) and \(K\) is
consistent, then \(K \subseteq K\*A\)) to CS, (\(\*\)6) corresponds to
LLE, (\(\*\)7) to OR and (\(\*\)8) to RMon. In establishing this logic,
Gardenfors used the following notion of validity: a sentence
\(A\) is valid in a BRM \(M\) if there is no consistent \(K\in \cK\)
with \(\neg A\in K\). A sentence \(A\) is valid iff it is valid in
every BRM. Arlo-Costa and Levi (1996) call this notion
*negative validity* and contrast it with a notion of
*positive validity* that they argue is preferable: \(A\) is
positively valid in a BRM \(M\) if \(A\in K\) for every \(K\in
\cK\).


### 6.2 Gardenfors's triviality theorem



Gardenfors (1986; 1988 sects. 7.4-7.7) soon noticed that
coupling the Ramsey test with the AGM theory of belief revision leads
to a trivialization, which is in many respects similar to
Lewis's trivialization (see
 section 5.2).


* **Triviality Theorem (Gardenfors):**A belief
revision model satisfies the AGM postulates
 (\(\*\)4)
 and
 (\(\*\)5)
 and Gardenfors's Ramsey Test only if it is trivial in the
sense that it does not contain any belief set that is logically
compatible with three sentences \(A\), \(B\) and \(C\) that are
pairwise inconsistent with each other.



A quick and simple proof goes as follows: Suppose for reductio that we
have a BRM \(M = \langle \cK, \*\rangle\) with a belief set \(K\in
\cK\) that is logically compatible with three pairwise inconsistent
sentences \(A\), \(B\) and \(C\). First notice that

 

\[B\in K+(\neg A\wedge B) = (K+(A\vee B))+\neg A \subseteq(K+(A\vee B))\*\neg A,\]


by (\(\*\)4), and similarly \(C\in (K+(A\vee C))\*\neg A\). So by
GRT,

 

\[\neg A > B \in K+(A\vee B)\]


and

 

\[\neg A > C \in K+(A\vee C).\]


Since \(A\) is logically stronger than both \(A\vee B\) and \(A\vee
C\), we have

 

\[K+(A\vee B)\subseteq K+A\]


and

 

\[K+(A\vee C)\subseteq K+A.\]


But since \(A\) is compatible with \(K\), \(K+A \subseteq K\*A\), by
(\(\*\)4). So both \(\neg A > B\) and \(\neg A > C\) are in
\(K\*A\). This implies, by GRT again, that both \(B\) and \(C\) are in
\((K\*A)\*\neg A\). Since \(B\) and \(C\) are inconsistent with each
other, the latter belief set is inconsistent, contradicting (\(\*\)5).
So the supposition is impossible.



Gardenfors's triviality theorem was widely perceived as an
unexpected and somewhat shocking result. Postulate
 (\(\*\)5)
 can hardly been seen as the culprit, so the triviality theorem was
interpreted by Gardenfors basically as a conflict between
 (\(\*\)4)
 or the Preservation Principle on the one hand, and the Ramsey Test on
the other. But the reactions to his result were mixed, and no
consensus has emerged so far.



A first position, taken by Levi and very much in line with those
denying that conditionals have truth values, was that conditionals do
not express propositions, but epistemic appraisals, and thus should
not be objects of belief and elements of belief sets, but rather
sentences that can only be accepted. This amounts to rejecting
GRT.



Another group of researchers suggested to modify or restrict the
Ramsey test. Solutions of this kind were discussed, amongst others, by
Rott (1986), Gardenfors (1987, 1988), Levi (1996: Chapter 2),
Lindstrom and Rabinowicz (1998) and Nute and Cross (2001).
Arguably, none of these adaptations of the Ramsey test has retained
the intuitive appeal of the original idea.



Others think that the Ramsey test is perfectly all right. Some authors
argue that the right reaction is to say that conditionals should not
be analyzed by AGM-style *revisions*, but rather by
*updates* in the style of Katsuno and Mendelzon (1992). Members
of this party include Ryan and Schobbens (1997), Grahne (1998) and
Crocco and Herzig (2002). Updates in this sense are closely related to
Lewis's (1976) imaging (see
 section 5.3.2
 above). These operations violate the AGM-postulate (\(\*\)4) and the
Principle of Preservation, but satisfy the following Monotonicity
condition which is immediately entailed by GRT:

 

\[\tag{MON} \text{If } K\_1\subseteq K\_2 \text{ then } K\_1\*A \subseteq K\_2\*A.\]


Rott (2011) defends Gardenfors's Ramsey test by showing
that it does not create a specific problem for the Preservation
Principle. Its consequence is rather that both preservation of factual
beliefs and preservation of flat conditionals entail that there is no
preservation of right-nested conditionals of depth 2. GRT allows for
right-nested conditionals if a model for iterated belief revision is
available (other compounds of conditionals are more difficult to make
sense of). It can be shown, for instance, that supposing there is an
atom \(p\) on which the believer suspends judgment, the
conditional

 

\[(\neg A\vee p)>((\neg A\vee \neg p)>\neg A)\]


functions exactly like the epistemic modal "might \(\neg
A\)": it is believed (or accepted) if and only if \(A\) is not
believed. Thus, if our language includes right-nested conditionals,
there are no two belief sets \(K\) and \(K'\) such that \(K\) is a
strict subset of \(K'\). Rott argues that it has simply been a mistake
to take it for granted that Preservation can and should be transferred
from factual language to languages containing nested conditionals.



Detailed analyses of the dialectics of the situation emerging from
Gardenfors's triviality theorem can be found in
Lindstrom and Rabinowicz (1998), Nute and Cross (2001), and
Arlo-Costa (2007). An up-to-date discussion of related problems
is Boylan & Schultheis (forthcoming).


## 7. Relevance and Difference-Making



Since MacColl's and C. I. Lewis's strictures against
material implication, there has been an increasing number of voices
arguing that some notion of relevant connection between antecedent and
consequent is part of the meaning of conditionals. In the area of
conditionals, the notion of relevance is claimed by distinct
traditions, however, and means slightly different
 things.[9]



One tradition concerns so-called relevant logics, designed to
formulate a more demanding notion of implication than Lewis's
strict implication (cf. Meyer & Routley 1973). The idea of
relevant implication is based in large part on the idea that the
antecedent and the consequent of an acceptable conditional \(A>B\)
must be connected *in terms of topics*. For example, a
conditional of the form \((p \wedge \neg p) > q\) need not be
valid, despite having an impossible antecedent, since the consequent
shares no content with the antecedent. Various constraints can be
imposed on the idea of content-sharing, leading to various
non-classical logics of so-called relevant implication (see
Mares's 2020 SEP entry on relevant logics for an overview, and
Weiss 2019 on the use of analytic entailment to deal with
Sextus' so-called fourth conditional).



Another approach to the notion of relevance is in terms of the idea
that the antecedent of a conditional should make a difference to the
truth, assertability or probability of the consequent. Consider a
conditional like "if London is in England, then Aristotle was a
Greek philosopher". The conditional sounds odd because the truth
of the antecedent does not matter to the truth of the consequent.
While relevant logics capture the idea of relevance in terms of
topic-sharing, here the idea of relevance is captured in terms of the
difference that the antecedent makes for the consequent.



The idea of *difference-making* itself can be formulated in
several ways. One way is proposed by Douven, in what he calls the
"Evidential support thesis" ("EST", Douven
[2008, 2016: 108]). Basically, the EST says that a conditional
\(A>C\) is relevantly assertable provided \(\Pr(C|A)\) exceeds a
sufficiently high threshold and \(\Pr(C|A)>\Pr(C)\). Douven's
main argument in support of the latter condition--which is a
relevance criterion--is that high probability is not enough for
assertion. He gives the example of the following minimal pair:



(8)


a.
If there is at least 1 Head in the first 10 tosses of this coin,
then there is at least 1 Head in the first 100,000 tosses.
b.
If Chelsea wins the Champion's league, then there is at
least 1 Head in the first 100,000 tosses.
 


Both statements are such that the probability of the consequent given
the antecedent is high. However, in (8)-b the antecedent does not
increase the probability of the consequent, it is irrelevant in that
sense, and the conditional may be called a "non-interference
conditional". Douven (2016) defines validity in terms of
acceptability-preservation according to the EST criterion, for all
thresholds \(t\in [0.5, 1)\). The resulting logic is weak, in
particular it violates Modus Ponens, RW, AND, CMon, Cut, OR and
Contraposition. Douven establishes the validity and invalidity of
specific principles (see Douven 2016: Theorem 5.2.1), but a sound and
complete axiomatization of his logic is still an open problem.



More recently, an alternative approach was proposed by Crupi and
Iacona (forthcoming-b), yielding a stronger logic. Crupi and Iacona
use the Rips measure of confirmation and associate to a conditional
\(A>C\) a score defined as

 

\[\frac{P(C|A)-P(C)}{1-P(C)}\]


when \(P(C|A)>P(C)\), and defined to be equal to \(0\) otherwise.
This yields a modified score of "uncertainty" for a
conditional, which they substitute for the standard notion of
uncertainty in Adams's definition of \(p\)-validity. The logic
they identify violates RW and Cut, but it validates AND, CMon, OR,
Modus Ponens, restricted forms of Aristotle's and
Abelard's theses (principles of connexive logic), and even
Contraposition. Crupi and Iacona (forthcoming-a) present a
possible-worlds semantics for what appears to be the same logic, in
which contraposition plays a central role.



Ahead of this recent wave of work, Rott (1986) pioneered another
measure of relevance in a belief revision framework, a measure picked
up by Spohn (2013), and proposed independently in various accounts of
causality. Arguing against Conjunctive Sufficiency (see
 sections 2.2),
 Rott' suggests that a "difference-making"
conditional \(A>C\) is accepted provided \(C\in K\*A\) and \(C\notin
K\*\neg A\). This is a relevantized variant of the Ramsey Test, ruling
that the conditional is accepted provided the revision by \(A\)
results in the belief that \(C\), but the revision by \(\neg A\) does
not result in the belief that \(C\). Using the terminology of
"difference-making conditionals", Rott (forthcoming) calls
the violation of RW the hallmark of conditionals that encode the idea
that the antecedent is positively relevant to the consequent. This
yields a logic that also invalidates CMon, Cut, Or and Contraposition,
but it validates And, Aristotle's theses, a restricted form of
Abelard's thesis as well as conditions such as



(\(>\)1)
If \(A >B\wedge C\), then \(A>B\) or \(A>C\).
(\(>\)2a)
\(A>C \tiff (A >A\wedge C \text{ and }A >A\vee C)\).


In general conditionals with a relevance reading tend to behave in
rather unusual and unruly ways. Raidl (2021) develops a general
technique of transferring completeness results for more
"well-behaved" suppositional conditionals \(>\) to
relevance-expressing conditionals definable in terms of \(>\). He
is thus able to cover the proposal of Crupi and Iacona as well as that
of Rott in a unified manner.



Rott's proposal turns out to be related to what is known as the
"difference measure" or as the "score of
contingency"

 

\[\Delta(p)=P(C\mkern2mu|\mkern2muA)-P(C|\neg A)\]


in the psychology of causality (Shanks 1995; Cheng 1997). In
particular, (Skovgaard-Olsen et al. 2016) found that depending on
whether this measure is positive, negative, or null, the probability
of a conditional may or may not equal the conditional probability. In
other words, relevance considerations appear to interact with
assignments of probabilities to conditionals, the main finding being
that Adams' Thesis appears to hold good only when \(\Delta(p)\)
is positive. The equation breaks down, on the other hand, when
\(\Delta(p)\) is null, so precisely in the case of non-interference
conditionals. Furthermore, Rooij and Schulz (2019) argue that an even
better predictor of the acceptability of conditionals is given by a
normalized measure of contingency, that is, by the measure



\[\Delta^\*(p) = \frac{P(C|A)-P(C|\neg A)}{1-P(C|\neg A)}.\]


Whether in logic or in the psychology of reasoning, the investigation
of relevance measures and of the resulting inferences is currently a
very active field in the study of conditionals.


## 8. Modals and Speech Acts



The previous sections have looked at propositional logics of
conditionals, which extend Boolean logic with a distinguished
conditional operator. To wrap up, we look briefly at some richer
languages, in particular at the interplay of conditionals with
modalities, and at the interplay of conditionals with other speech
acts than assertions.


### 8.1 Modalities



Possible-worlds analyses of conditionals make it possible to define
modalities from conditionals as follows (Stalnaker 1968; Nute
1980):

 

\[\begin{align} \Diamond A & \equiv \neg (A > \neg A)\\ \Box A &\equiv (\neg A > A)\end{align}\]


Conversely, we saw how strict conditional analyses treat the
conditional as definable from modalities. Those facts may seem to
suggest that modals and conditionals are interdefinable. However, the
interaction between them is actually more complex.



Linguistic facts about conditionals suggest that across a wide range
of languages, modal operators commute with conditional
antecedents.



(9)
If \(A\) then necessarily \(C \equiv\) Necessarily if \(A\) then
\(C\).
(10)
If \(A\) then possibly \(C \equiv\) Possibly if \(A\) then
\(C\).


Such facts centrally motivate Kratzer's account of conditionals
as restrictors of modal operators (Kratzer (2012), inspired by D.
Lewis 1975). As Kratzer (1991, pp. 468-469) puts it,
"there is a very close relationship between *if*-clauses
and operators like *must*", and more specifically
"for each world, the *if*-clause has the function of
restricting the set of worlds which are accessible from that
world".



Kratzer formulates the restrictor view of conditionals by relativizing
the semantics of modals to two premise sets, a so-called modal base
("hard" facts known or believed by the speaker) and an
ordering source ("soft" facts taken to be revisable).
Consider a model \(\langle W, \Gamma, \Delta\rangle\) where \(\Gamma\)
is a premise function for the ordering source, and \(\Delta\) the
premise function for the modal base. Let us assume the Limit
assumption and let \(\max\_{\Gamma(w)}(F)\) be the set of worlds in a
proposition \(F\) that satisfy a maximal set of propositions in
\(\Gamma(w)\). Kratzer's semantics for \(\Box\) is:



**(Kratzer's *Must*)**

 

\[ M, w,\Gamma, \Delta\Vdash \Box C \tiff \max\_{\Gamma(w)}\left(\bigcap \Delta(w)\right)\subseteq \lvert C\rvert^{M,w,\Gamma, \Delta}\]


The conditional "if \(A\) then \(C\)" can be treated as a
binary modality \(\Box(A,C)\), with the following truth conditions,
where \((\Delta+A)(w)\) equals \(\Delta(w) \cup \{\lvert
A\rvert^{M,w,\Gamma, \Delta}\}\):


**(Restricted *Must*)** \[M, w,\Gamma, \Delta\Vdash \Box (A,C) \tiff M, w, \Gamma, (\Delta+A) \Vdash \Box C\]



For bare conditionals, Kratzer postulates that the antecedent also
restricts a covert modal. This particular assumption is a source of
controversy. Several authors point out that while *if*-clauses
may restrict operators, modal operators and conditional operators can
also have separate contributions (von Fintel 1994; M. Kaufmann &
S. Kaufmann 2015).



An important issue is whether the interaction of modals and
conditionals can be accounted for when both types of operators are
logically represented in the language. We highlight two recent
accounts along those lines. Both take inspiration from Yalcin's
(2007) conception of modal operators as expressive devices, and also
from Veltman's (1996) and Gillies's (2004) emphasis on the
idea that utterances involving epistemic modals update informational
states.



Ciardelli (forthcoming) states truth conditions for a language with
modal operators and a conditional operator. He defines the semantic
value of sentences relative to two parameters, a state parameter \(s\)
(set of worlds), and an attitude parameter \(a\). In Ciardelli's
approach, *if*-clauses restrict the state \(s\), as in
Kratzer's approach, but modals basically act on the attitude
parameter \(a\), which can express various modal forces (universal,
existential, or probabilistic). The framework allows him to derive as
equivalences the facts
 (9)
 and
 (10),
 in the following guise:

 

\[\begin{align} A > \Box C & \equiv \Box(A > C)\\ A > \Diamond C & \equiv \Diamond (A > C) \end{align}\]


Bare conditionals of the form \(A>B\) turn out equivalent to
conditionals \(A>\Box B\), but only under the default attitude
coming with universal force. Ciardelli (forthcoming) also derives the
incompatibility of the conditional \(A>C\) with the
might-conditional \(A>{\Diamond \neg C}\) (If-Might Contradiction,
henceforth IMC). The same incompatibility is at the heart of
Santorio's (forthcoming) so-called path semantics for
conditionals. Santorio exposes a contradiction between this principle
and two other principles of conditional logic, namely CEM and the
Duality principle whereby \(A>\Diamond C\,\equiv\, \neg (A >\neg
C)\) (endorsed by D. Lewis 1973, though rejected by Stalnaker 1968).
His semantics validates IMC and preserves CEM, but drops Duality. A
number of further predictions follow from Santorio's semantics,
in particular OI becomes valid, as well as restricted versions of IE.
Like Ciardelli, Santorio does not have to treat bare conditionals as
implicitly modalized, and like Ciardelli he can derive commutation
facts about modalities.



Beside those proposals, Omori (2019) focuses also on the interplay of
negation, modals, and conditionals in the framework of connexive
logics. Omori uses Odintsov and Wansing's (2010) four-valued
framework for modal logic, to show that conditional negations of
varying strength for \(A>B\) can be derived from the outer negation
of a conditional (building on distinctions made in Egre
& Politzer 2013), in particular \(A>{\Diamond\neg B}\) instead
of \(A >\neg B\). He gives sound and complete axiomatizations for
the resulting conditional logics. An earlier connexive system is
Olkhovikov (2002)'s logic **LImp**,
effectively combining Cooper's conditional with a trivalent
necessity operator of determinate truth (mapping 1 to 1
and 1/2 and 0 to 0).


### 8.2 Beyond assertion



Conditionals do not only appear in declarative sentences, but also in
questions and in commands, as in:



(11)
If John visits, will Mary visit too?
(12)
If John visits, make sure to bring cheese!


The semantics of imperatives and interrogatives lies beyond the scope
of this entry. However, we mention such constructions because they too
may be seen as supporting Kratzer's restrictor analysis. In
particular, S. Kaufmann and Schwager (2011) treat imperatives as
involving a covert modal operator, which can have a deontic, or
bouletic meaning, but basically acting like "must". In the
case of questions, Isaacs and Rawlins (2008) treat questions as
prefixed with a covert speech act operator, coming with a modal domain
restricted by the *if*-clause. In a related way, Ciardelli,
Groenendijk, and Roelofsen (2018) point out that the semantics of a
question can be specified by resolution conditions relative to an
information state. Again, conditional questions can be viewed as
appropriately restricting the information state to the antecedent
worlds, so as to resolve the question relative to those (see
Ciardelli, forthcoming). For a polar question of form \(?M\)
("will Mary visit?") the information state \(s\) resolves
it if either \(s\) supports \(M\) or \(s\) supports \(\neg M\). In the
case of a conditional polar question like
 (11),
 the semantics of Ciardelli, Groenendijk, and Roelofsen (2018)
predicts that a state \(s\) will resolve a question of the form \(J
>{?M}\) provided either \(s\) supports \(J >M\) or \(s\)
supports \(J >\neg M\).



Further connections have been explored between the semantics of
interrogatives and the very structure of conditional sentences. Starr
(2014b) in particular argues that all conditionals have a
topic-comment structure, allowing to treat the antecedent as a covert
question itself. Finally, more aspects of the interaction of
conditional phrases with speech acts deserve to be mentioned,
including so-called *biscuit*-conditionals ("if you want
some biscuits, there are some in the cupboard", Austin 1956,
Lycan 2001), as well as concessives ("even if A, B"
sentences, see Lycan 2001, Guerzoni and Lim 2007). For both kinds,
permitted inferences differ significantly from those found in regular
conditionals (in particular, both entail the truth of the consequent).
This is evidence that the diversity found in logics of conditionals
also depends on the variety of uses and linguistic environments in
which *if*-clauses occur.