## 1. Disjunction in classical logic



In classical logic, disjunction (\(\vee\)) is a binary sentential
operator whose interpretation is given by the following truth
table:



* (1) Disjunction in
classical logic




|  |  |  |
| --- | --- | --- |
| \(\phi\) | \(\psi\) | \((\phi \vee \psi)\) |
| 1 | 1 | 1 |
| 1 | 0 | 1 |
| 0 | 1 | 1 |
| 0 | 0 | 0 |




A disjunction \((\phi \vee \psi)\) is true iff at least one of the
disjuncts is true.



Adopting a natural deduction system, the proof-theoretical
contribution of disjunctive formulas can be defined by the following
two rules, which regulate (i) how disjunctions can be drawn as
conclusions (disjunction introduction rule, \(I\_{\vee}\), also known
as *addition*) and (ii) how conclusions can be drawn from
disjunctions (disjunction elimination rule, \(E\_{\vee}\), also known
as *reasoning by cases*):


* (2)
Disjunction introduction (\(I\_{\vee}\))



\[
\begin{prooftree}
 \AxiomC{\(\phi\)}
 \RightLabel{\(I\_{\vee}^R\)}
 \UnaryInfC{\((\phi \vee \psi)\)}
\end{prooftree}

\qquad

\begin{prooftree}
 \AxiomC{\(\psi\)}
 \RightLabel{\(I\_{\vee}^L\)}
 \UnaryInfC{\((\phi \vee \psi)\)}
\end{prooftree}
\]

* (3)
Disjunction elimination (\(E\_{\vee}\))



\[
\begin{prooftree}
 \AxiomC{\((\phi \vee \psi)\)}
 \alwaysNoLine
 \AxiomC{\([\phi]\)}
 \UnaryInfC{\(\vdots\)}
 \UnaryInfC{\(\theta\)}
 \AxiomC{\([\psi]\)}
 \UnaryInfC{\(\vdots\)}
 \UnaryInfC{\(\theta\)}
 \singleLine
 \RightLabel{\(E\_{\vee}\)}
 \TrinaryInfC{\(\theta\)}
\end{prooftree}
\]


Intuitively, the former tells us that we can conclude \((\phi \vee
\psi)\) on the basis of \(\phi\) (or of \(\psi\)), while the latter
states that \(\theta\) will follow from a disjunction \((\phi \vee
\psi)\) if it can be derived from \(\phi\) and also from \(\psi\).



One of the goals of a logical system is to arrive at a rigorous
characterization of the notion of validity. In a logical system, which
normally consists of a language, a proof-theory and a semantics,
validity can be defined proof-theoretically or semantically.
Proof-theoretically, validity is defined in terms of formal proofs. An
argument is proof-theoretically valid (\(\vdash\)) if there is a proof
from some or all of its premises to its conclusion. Semantically or
model-theoretically, validity is normally defined in terms of
truth-preservation. An argument is model-theoretically valid
(\(\models\)) if there is no interpretation (in the semantics) in
which its premises are all true and its conclusion false. In classical
logic, the proof-theoretical and the model-theoretical perspectives
have been proven to characterize exactly the same notion of validity
(*soundness and completeness theorems*): an argument is
proof-theoretically valid iff it is model-theoretically valid.



Below are some of the principles involving disjunction that classical
logic validates. In the list we adopt the model-theoretic notation
(\(\models\)) because such perspective will be more prominent in the
following section. \(\models \phi\) means \(\phi\) is true in all
interpretations. The last two principles in the list are
model-theoretic analogues of disjunction introduction and elimination,
respectively.



\[\begin{align}
& \models (\phi \vee \neg \phi) & \text{(law of excluded middle)} \\
& \models \neg (\phi \wedge \psi) \leftrightarrow (\neg \phi \vee \neg \psi) & \text{(De Morgan laws)} \\
& \models \neg (\phi \vee \psi) \leftrightarrow (\neg \phi \wedge \neg \psi) & \\
& \models \phi \wedge (\psi\_1 \vee \psi\_2) \leftrightarrow (\phi \wedge \psi\_1) \vee (\phi \wedge \psi\_2) & \text{(distributive laws)} \\
& \models \phi \vee (\psi\_1 \wedge \psi\_2) \leftrightarrow (\phi \vee \psi\_1) \wedge (\phi \vee \psi\_2) & \\
& \neg \phi, (\phi \vee \psi) \models \psi & \text{(disjunctive syllogism)} \\
& \phi \models (\phi \vee \psi), \quad \psi \models (\phi \vee \psi) & \text{(addition)} \\
& \phi \to \theta, \psi \to \theta \models (\phi \vee \psi) \to \theta & \text{(reasoning by cases)} 
\end{align}\]


These principles have been widely discussed and, at times, rejected in
the logical-philosophical literature. The following section briefly
summarizes parts of these discussions focusing on which alternative
interpretation of \(\vee\) these discussions have led to.


## 2. Non-classical variations


### 2.1 Law of excluded middle and the principle of bivalence



The law of excluded middle (LEM) states that any proposition of the
form \((\phi \vee \neg \phi)\) is logically valid. The semantic
principle of bivalence states that every proposition is either true or
false (and not both). Bivalence and LEM collapse in classical logic,
but are distinguished for example in supervaluationism (van Fraassen
1966), which rejects the former, while retaining the latter. Since
Aristotle, who was the first to state these principles, bivalence and
LEM have been criticized on various grounds. This section first
discusses intuitionistic logic, which invalidates LEM (and part of the
De Morgan laws); and then presents some canonical arguments against
bivalence and discuss how \(\vee\) is interpreted in a number of
non-bivalent systems including many-valued logic, dynamic semantics,
supervaluationism, as well as quantum logic, which besides bivalence
also invalidates the classical distributive laws.


#### 2.1.1 Disjunction in intuitionistic logic



LEM has been rejected in constructivism, in particular in
intuitionistic logic (see the entry on
 intuitionistic logic).
 The standard informal interpretation of logical operators in
intuitionistic logic is the so-called proof-interpretation or
Brouwer-Heyting-Kolmogorov (BHK). On such interpretation the meaning
of a statement \(\phi\) is given by explaining what constitutes a
proof of \(\phi\) rather than in terms of its truth. The BHK
interpretation of disjunction reads as follows:


* (4) A proof of
\((\phi\vee \psi)\) consists of a proof of \(\phi\) or a proof of
\(\psi\).



On such interpretation, the question concerning the validity of LEM is
then equivalent to the question of the possibility of unsolvable
(mathematical) problems (Brouwer 1908, translated in Heyting (ed)
1975): LEM
should then fail because it is doubtful that for any mathematical
statement \(\phi\), either there is a proof of \(\phi\) or a proof of
\(\neg
 \phi\).[1]




Intuitionistic logic can be described as classical logic without LEM
(or the principle of double negation \((\neg \neg \phi \to \phi)\)),
but with the classical law of contradiction \(((\phi \to \psi) \to
((\phi \to \neg \psi) \to \neg \phi))\) and *ex falso
quodlibet* \((\neg \phi \to (\phi \to \psi))\). Related to its
rejection of LEM, intuitionistic logic satisfies the *disjunction
property* (Godel 1932). A logic is said to have the
disjunction property if whenever \((\phi \vee \psi)\) is provable in
the logic, so is at least one of \(\phi\) and \(\psi\). Classical
logic does not have such property (\(\vdash\_{CL} (p \vee \neg p)\),
without \(\vdash\_{CL} p \) or \(\vdash\_{CL} \neg p\)) but has a weaker
property called *Hallden-completeness*: every provable
disjunction whose disjuncts don't share any propositional
variables has at least one of those disjuncts provable.



The rejection of LEM in intuitionistic logic also implies the
rejection of classical *reductio ad absurdum* as a legitimate
method of (mathematical) proof. Intuitionistically, *ad
absurdum* one can only prove negative statements (via negation
introduction \((\phi \to \bot) \to \neg \phi\)). This is because the
principle of double negation (\(\neg \neg \phi \to \phi\)) does not
hold in intuitionistic logic. If it did, LEM would follow by *modus
ponens* from the intuitionistically provable \(\neg \neg (\phi
\vee \neg \phi)\) (in general it holds that if \(\phi\) is classically
provable, then \(\neg\neg \phi\) is intuitionistically provable).



Finally note that only one of the De Morgan laws is intuitionistically
valid, of the other only one half still holds:


* \(\models\_{IT} (\neg \phi \wedge \neg \psi) \leftrightarrow \neg
(\phi \vee \psi)\) (De Morgan laws)
* \(\models\_{IT} (\neg \phi \vee \neg \psi) \to \neg (\phi \wedge
\psi) \), but \(\not\models\_{IT} \neg (\phi \wedge \psi) \to (\neg
\phi \vee \neg \psi)\)



Intuitively, \( \neg (\phi \wedge \psi) \) intuitionistically asserts
that an effective method has been given, which would convert any proof
of \((\phi \wedge \psi)\) into a proof of a contradiction. But this
does not imply that either a proof of \(\neg \phi\) (i.e., an
algorithm which would convert any proof of \(\phi\) into a proof of a
contradiction) or a proof of \(\neg \psi\) has been given, which is
what \((\neg \phi \vee \neg \psi)\) intuitionistically asserts.


#### 2.1.2 Disjunction in multi-valued logics



The semantic principle of bivalence states that every proposition is
either true or false (and not both). Early arguments against bivalence
were linked to the problem of determinism. In *De
Interpretatione* (part 9), Aristotle discusses the status of
statements about contingent future events, and seems to conclude that
for these statements the principle of bivalence should be rejected,
otherwise determinism would follow. His argument can be reconstructed
as follows. Consider the statement *A sea battle will be fought
tomorrow*. If the statement is true, then the sea battle will
necessarily take place. If the sentence is false, then it will be
impossible for the sea battle to take place. Assuming that statements
are either true or false, one concludes that the sea battle is either
necessary or impossible. To escape this fatalistic conclusion
Aristotle rejects the principle of bivalence (while possibly retaining
the law of excluded middle, see van Fraassen 1966: 493-495, and
the entry on
 fatalism).
 Aristotle's sea battle argument, although criticized by many,
constituted one of Lukasiewicz' original motivations for the
development of his trivalent logic (Lukasiewicz 1920,
translated in McCall 1967). Since then various multi-valued logics have been
developed where formal disjunction \(\vee\) receives a non-classical
interpretation. In Lukasiewicz' original system, but also
in strong Kleene three-valued logic (Kleene 1952), disjunction is
interpreted according to the following truth table, where # stands
for *indefinite* or *possible* in Lukasiewicz, and
for *undefined* in Kleene (Lukasiewicz's and
Kleene's system differ in their interpretation of implication,
we will disregard this issue here). Priest's (1979) Logic of
Paradox (LP) also adopts the following table for disjunction. There #
should be read as *both true and false*:



* (5)Disjunction in
strong Kleene logic




|  |  |  |  |
| --- | --- | --- | --- |
| \(\vee\_s \) | 1 | # | 0 |
| 1 | 1 | 1 | 1 |
| # | 1 | # | # |
| 0 | 1 | # | 0 |




On this interpretation, a disjunction is true if at least one of the
disjuncts is true, false if both disjuncts are false, undefined
otherwise.



In Bochvar's internal three-valued logic, also known as
Kleene's weak three-valued logic, disjunction receives a
different interpretation. The symbol # should be read here as
*meaningless*:



* (6) Disjunction in weak
Kleene logic




|  |  |  |  |
| --- | --- | --- | --- |
| \(\vee\_w\) | 1 | # | 0 |
| 1 | 1 | # | 1 |
| # | # | # | # |
| 0 | 1 | # | 0 |




While on a strong Kleene interpretation, a disjunction can be true
even if one of the disjuncts is undefined, on a weak Kleene
interpretation, if one of the disjuncts is meaningless, the whole
disjunction is meaningless as well.



Negation receives the following interpretation in these systems:



* (7) Negation in strong
and weak Kleene logic




|  |  |
| --- | --- |
| \(\neg \) |  |
| 1 | 0 |
| # | # |
| 0 | 1 |




In both strong and weak Kleene logics then, \((\phi \vee \neg \phi)\)
never has value \(0\), but it doesn't always have value \(1\)
either: if \(\phi\) is assigned value #, so is its negation, but then
\((\phi \vee \neg \phi)\) will also have value # on both the strong
and weak interpretations of \(\vee\). Thus \((\phi \vee \neg \phi)\)
fails to be logically valid in these systems (unless both \(1\) and #
are taken to be designated values, as in Priest's LP, which
validates LEM. Priest's notion of validity is defined later in
this section). Other many-valued logics, such as the Lukasiewicz
continuum-valued logic, also fail to validate LEM, for similar
reasons. In continuum-valued logic, the set of truth values is the set
of real numbers between \(0\) and \(1\), where \(1\) stands for
completely true, \(0\) for completely false, \(0,5\) for half true,
etc. Disjunction and negation are analyzed as \(F\_{\vee}(x,
y)=max(x,y) \) and \(F\_{\neg} (x)= 1-x\) respectively, where \(max\)
means "the maximum of". But then since \(max(x, 1-x)\)
need not be equal to \(1\), \((\phi \vee \neg \phi)\) can fail to be
completely true in this system.



The difference between the strong and weak Kleene's treatment of
disjunction can be better appreciated by looking at the phenomenon of
presupposition, which constitutes one of the most well-known
linguistic motivations for the rejection of bivalence (other common
motivating phenomena include the semantic paradoxes and vagueness, see
the entries on
 liar paradox
 and
 sorites paradox,
 and
 section 2.1.4
 below). Consider Russell's (1905) example: 


* (8) The king
of France is bald.



According to bivalence,
 (8)
 must be either true or false. Which is it? Russell's answer is
well known. According to his theory of descriptions
 (8)
 expresses the conjunction "there is a unique king of France and
he is bald" of which the first conjunct is false, and therefore
the whole sentence is false. Strawson (1950) however criticized
Russell's analysis and argued that the existence and uniqueness
of the king of France is not part of what is asserted by an utterance
of
 (8)
 rather it is part of what is taken for granted or
*presupposed*. If there is no king of France, the sentence is
not false, rather it is neither true nor false. Adopting a
three-valued logic we can assign to
 (8)
 value #. Consider now the following three examples where
 (8)
 occurs in a disjunction.


* (9)  Either
the king of France is bald or the king of France is not
bald.
* (10)  Either
there is no king of France or the king of France is bald.
* (11)  Either
Barack Obama is tall or the king of France is
bald.



Both strong and weak Kleene systems predict
 (9)
 to be undefined/meaningless, since both disjuncts are
undefined/meaningless. However, the predictions of the two systems
with respect to
 (10)
 and
 (11)
 diverge: the strong Kleene system predicts both
 (10)
 and
 (11)
 to be true since at least one of the disjuncts is true (assuming
Barack Obama is in fact tall), while the weak system predicts both
 (10)
 and
 (11)
 to be meaningless since at least one of the disjuncts is meaningless.
Intuitively, however,
 (10)
 is true while
 (11)
 is more readily judged as undefined/meaningless. Thus the predictions
of neither systems are in agreement with common sense judgments.
Linguists have observed that the intuitive difference between
 (10)
 and
 (11)
 is that in the former, but not in the latter the existence of a
unique king of France (the presupposition of one of the disjuncts) is
entailed by the negation of the other disjunct. Karttunen (1973), who
systematically studied the projection behavior of presuppositions
(i.e., how the presupposition of a constituent projects at the level
of the complex sentence), described disjunction as a *filter*.
In his taxonomy of embedding operators, *plugs* block all
presuppositions in their scope (an example is *told that*),
*holes* allow presupposition to project freely (e.g.,
negation), while *filters* allow only some presuppositions to
project. Various analyses have been proposed attempting to capture how
presuppositions project when embedded in disjunctions or other complex
sentences. A prominent example is Peters (1979), who showed how
Karttunen's observations can be accounted for within a
multi-valued logic with special non-symmetric connectives (see the
entry on
 presupposition
 for an overview).


#### 2.1.3 Disjunction in dynamic semantics



Another influential attempt to formalize Karttunen's
generalization with respect to presupposition projection, is the
dynamic account of Heim (1983), further developed in Beaver (2001). In
a dynamic semantics, the interpretation of sentences is given in terms
of their context change potentials rather than their truth conditions
(see the entry on
 dynamic semantics).
 A context (or information state) \(c\) is defined as a set of
possible worlds, the worlds compatible with the information available
to the conversationalists, along the lines of Stalnaker's (1978)
notion of a context set. The meaning of an atomic sentence \(p\) is
identified with its potential to update the input context \(c\), i.e.,
its potential to eliminate all non-\(p\) worlds from \(c\): \(c[p]=\{w
\in c \mid w(p) =1\}\). Here is the dynamic clause for
disjunction.


* (12)
Disjunction in dynamic semantics



\[
c[(\phi \vee \psi)] = c [\phi] \cup (c \setminus c [\phi])[\psi]
\]


The result of updating a context \(c\) with a disjunction \((\phi \vee
\psi)\) consists in the union of (i) the result of updating \(c\) with
\(\phi\) and (ii) the result of updating with \(\psi\) the context \(c
\setminus c [\phi]\) obtained by subtracting from \(c\) all worlds
which verify \(\phi\). By defining the presupposition of a sentence as
what has to be true (or supported) in a context \(c\) in order for an
update with the sentence to be defined in \(c\), the dynamic account
of disjunction captures the pattern in
 (9)-(11).
 Simplifying, suppose \(c\) does not support the information that
there is a king of France, then
 (9)
 and
 (11)
 will be undefined in \(c\), because \(c [\phi]\) and/or \((c
\setminus c [\phi])[\psi]\) will be undefined. But
 (10)
 will be defined. This is because the second disjunct in
 (10),
 which presupposes that there is a king of France, is interpreted here
with respect to the local context \((c \setminus c [\phi])\) which is
the context obtained by subtracting from \(c\) all worlds which verify
the first disjunct, i.e., all worlds in which there is no king of
France. Such local context will support the information that there is
a king of France. So the whole disjunction does not presuppose
anything. The ordering between disjuncts however matters in a dynamic
account and so the analysis predicts a difference in presupposition
between
 (10)
 and the following variant where the presupposition trigger occurs in
the first disjunct rather than the second:


* (13) Either
the king of France is bald or there is no king of France.



This prediction however does not seem to be borne out.



A related discussion concerns the behavior of anaphora in the context
of disjunction and in particular the challenge presented by the
so-called "bathroom" example
 (14)
 from Barbara Partee, where the anaphoric pronoun *it* in the
second disjunct refers back to the negative indefinite *no
bathroom* in the first disjunct, while negative indefinites are
normally inaccessible to subsequent pronouns as illustrated in
 (15):


* (14) Either
there is no bathroom in the house, or it is in a funny
place.
* (15) There is
no bathroom in the house. # It is in a funny place.



The contrast between
 (14)
 and
 (15)
 seems to provide evidence for the analysis of disjunction presented
in
 (12)
 according to which the second disjunct must be interpreted with
respect to a context supporting the negation of the first disjunct (so
a context supporting the information that there is a bathroom in the
house). Note however that the principle of double negation \((\neg
\neg \phi \to \phi)\) is invalidated in standard dynamic semantics for
anaphora (Groenendijk and Stokhof 1991) and so a full account of these
examples requires some adjustment. Various solutions have been
proposed (e.g., Simons 1996), but the discussion about anaphora (and
presupposition projection) in disjunctions has not been settled
yet.



It is easy to see that a dynamic semantics with presuppositions does
not validate LEM, because, if \(\phi\) contains a presupposition
trigger, \((\phi \vee \neg \phi)\) will be undefined in any context in
which the presuppositions of \(\phi\) are not satisfied. For the
fragment without presuppositions, however, LEM is validated, but there
is a non-trivial sense in which bivalence is still rejected: truth and
falsity in dynamic semantics are defined relative to a context, for
example, \(\phi\) is true (or supported) in \(c\) iff \(c[\phi]=c\);
and \(\phi\) is false in \(c\) iff \(c[\phi]=\emptyset\). But then
\(p\) might be neither true nor false in a context \(c\) (e.g., if
\(c\) contains both \(p\) worlds and non-\(p\) worlds), while, for all
\( c\), it holds that \((p \vee \neg p)\) is true in \(c\).


#### 2.1.4 Disjunction in supervaluationism



Another system which rejects bivalence while validating LEM is
supervaluationism (van Fraassen 1966). Let \(V\) be a set of classical
valuations \(v\), where a classical valuation is a function assigning
T(rue) or F(alse) to any statement in the relevant language. A
supervaluation \(s\_V\) is then a function assigning T (F) to exactly
those statements assigned T (F) by all valuations in \(V\). Since
\(V\) can contain classical valuations that assign different values to
one and the same statement (e.g., if \(v\_1(p)\)=T and \(v\_2(p)=\)F for
\(v\_1,v\_2 \in V\)), supervaluations have truth-value gaps (\(s\_V(p)\)
is undefined), so the law of bivalence fails. LEM, by contrast, is
validated, since \(v(\phi \vee \neg \phi)\)=T for all \(\phi\) and
classical \(v\), and thus \(s\_V(\phi \vee \neg \phi)\) = T, for any
\(s\_V\). Thus, in supervaluationism, a disjunction can be supertrue
(i.e., can be assigned T in some \(s\_V\)) without any of its disjuncts
being supertrue.



Supervaluational semantics has been largely applied to explain
phenomena of vagueness (Lewis 1970; Fine 1975b; Kamp 1975; see also the entry on
 vagueness). A predicate \(P\) is vague if it exhibits borderline
cases for which it is not clear whether \(P\) truly applies or not.
Prominent examples of vague predicates are relative adjectives like
*tall* or *clever*, while adjectives like *prime*
as in *3 is a prime number* are normally taken not to be vague.
The basic insight of any supervaluational account of vagueness is that
a vague language admits of several precisifications (formalized as a
set \(V\) of classical valuations) and the semantic value of a
statement is fixed only insofar as all those precisifications agree: a
statement \(\phi\) is true if it is supertrue (\(s\_V(\phi)=T\)), that
is, true on every admissible precisification (\(v(\phi)=T\), for all
\(v\in V\)), and it is false if it is superfalse (\(s\_V(\phi)=F\)),
that is, false on every admissible precisification (\(v(\phi)=F\), for
all \(v\in V\)); otherwise it has no semantic value. A common argument
in favor of supervaluationism relies on the observation that while a
predicate like *tall* may be vague, the compound predicates
*tall or not tall* and *tall and not tall* are not
vague, the former being true and the latter being false of all
entities in the predicate application domain. Thus, while *\(a\) is
tall* may fail to be true or false (and the same goes for its
negation), *\(a\) is either tall or not tall* and *\(a\) is
tall and not tall* are always true and false respectively, as
predicted by a supervaluational semantics. More recent experimental
work, however, has cast some doubts on these observations (e.g.,
Ripley 2011). Supervaluationism has further been criticized for giving
rise to counterexamples to familiar inference-patterns (Williamson
1994: 151-152 and 162-163), (see also Keefe 2000; Williams
2008 for further discussion). For example, it is easy to see that in
this framework, one loses the disjunction elimination rule, at least
in languages with a determinately/supertrue operator \(D\). Given a
class \(V\) of classical valuations, we can introduce the operator
\(D\) with the stipulation that \(v(D \phi)=1\) iff \(v\_1(\phi)=1 \),
for all \(v\_1\) in \(V\). Defining supervalidity, \(\models\_{sv}\), in
terms of preservation of supertruth, we have that \(p \models\_{sv} D
p\), \(\neg p \models\_{sv} D \neg p\) and \(\models\_{sv} p \vee \neg
p\), but \(\not\models\_{sv} Dp \vee D \neg p\). So the disjunction
elimination rule does not preserve supervalidity in this language (see
also Humberstone, chapter 6, pages 830-833). Whether
supervalidity is the right notion of validity from a supervaluational
perspective is however controversial (e.g., Varzi 2007).


#### 2.1.5 Disjunction in quantum logic



Another logic which lacks bivalence is quantum logic, which also
typically rejects the distributive laws of classical logic (Birkhoff
and von Neumann 1936; Putnam 1968). Quantum logic was started by
Birkoff and von Neumann for studying the relation among physical
observables in quantum physics. Quantum logic rejects bivalence
because a state in a quantum system typically assigns probability
values to experimental propositions rather than plain true or false.
To see why quantum physics can be taken to provide evidence against
the distributive laws of classical logic consider a particle moving on
a line. Suppose \(p\) asserts that the particle's momentum is
within a certain interval, while \(q\) and \(r\) assert that the
particle's position is in the intervals \(\alpha\) and \(\beta\)
respectively. Suppose the statement \(p \wedge (q \vee r)\) is
confirmed by our observation, so is true (or has value 1). By the
distributive laws we can classically conclude \((p \wedge q) \vee (p
\wedge r)\). But in quantum theory the latter might still be false
(have value 0), since both conjunctions \((p \wedge q)\) and \((p
\wedge r)\) might assert tighter restrictions on simultaneous values
of position and momentum than is allowed by Heisenberg's
uncertainty principle (see the entry on
 the uncertainty principle).
 So in the "logic of quantum physics" a disjunction \((q
\vee r)\) can have value 1 in a state without any of the disjuncts
having value 1: if \(q\) has value 1 or \(r\) has value 1, then \((q
\vee r)\) has value 1, but the other direction fails.



Various ways have been proposed to develop a quantum logic which
rejects the distributive laws while saving as much as possible of
classical logic. One way discussed by Dummett (1978) (who was not a
proponent of quantum logic) involves restricting the elimination rule
for disjunction so that the distributive laws are no longer derivable
(see Humberstone 2011: 298-302 and 918-922 for more
details). Arguably more natural characterizations of quantum logic use
algebraic semantics and probability theory (see the entry
 quantum logic and probability theory).


### 2.2 Disjunctive syllogism and addition



Disjunctive Syllogism (DS) states that we can infer \(\psi\) from \(
(\phi \vee \psi) \) and \(\neg \phi\). DS, which is valid in classical
logic, but also in intuitionistic logic and in the multi-valued logics
discussed in the previous section (with the exception of PL), has
however been rejected in relevance logics (Anderson and Belnap 1962,
1975). Relevance logics are non-classical logical systems developed to
avoid the paradoxes of material and strict implication (see the entry
on
 relevance logic).
 These are classically valid principles, like \(p \to (q \vee \neg
q)\), which however strike us as counterintuitive. This is so because,
according to the relevantist, they involve a fallacy of relevance: in
these cases the antecedent is typically irrelevant to the consequent.
Belnap's formal criteria of relevance (also known as variable
sharing principle) is that in any provable implication, the antecedent
and the consequent should share at least one propositional variable, and that no
inference can be shown valid if the premises and conclusion do not
share at least one propositional variable.



Historically, relevance logic rejected DS because of the role it plays
in C.I. Lewis "independent" proof of the fact that an
impossible proposition \((A \wedge \neg A)\) can lead to any
proposition whatsoever, which is one of the paradoxes of strict
implication:




|  |  |  |
| --- | --- | --- |
| 1. |  \((A \wedge \neg A)\)  |  [assumption] |
| 2. | \( A\)  |  [from 1, by \(E\_{\wedge}\)] |
| 3. |  \(\neg A\)  |  [from 1, by \(E\_{\wedge}\)] |
| 4. |  \((A \vee B)\)  | [from 2, by \(I\_{\vee}\), or addition] |
| 5. |  \(B\)  |  [from 3,4 by disjunctive syllogism] |



According to Anderson and Belnap "the inference from \(\neg A\)
and \(A\vee B\) to \(B\) is an error: [...] a fallacy of
relevance" (Anderson and Belnap 1962: 19). Formally, the
inference is rejected because according to Anderson and Belnap's
notion of tautological entailment (see their system E) \(\neg A \wedge
(A\vee B)\) entails \(B\) if and only if \((\neg A \wedge A) \vee
(\neg A \wedge B)\) entails \(B\), only if \((\neg A \wedge A)\)
entails \(B\), which fails since the premise \((\neg A \wedge A)\) is
not relevant to the conclusion \(B\) (no shared propositional
variables). Anderson and Belnap further argued that we can still have
a version of disjunctive syllogism involving an intensional notion of
disjunction \(\vee\_i\), for which addition wouldn't hold.
Intensional disjunction is defined as follows: \((A\vee\_i B):= (\neg A
\to\_i B) \), where \(\to\_i\) is an intensional implicational
connective, i.e., one satisfying the formal criterion of relevance
introduced above. The English *or* would then be ambiguous
between intensional and extensional (truth-functional) disjunction,
and only for the latter would addition hold. Intuitively, the
intensional variety of *or* would be one requiring
"relevance" between the disjuncts. On the intensional
reading, "\(A\) or \(B\)" would entail that \(A\)
and \(B\) are so related that we are entitled to say "If
\(A\) had not been true, \(B\) would have been true" or
"If \(B\) had not been true, \(A\) would have been
true" or the like. A disjunction like *Either Napoleon was
born in Corsica or else the number of the beast is perfect*
clearly fails to have this property and therefore is of the
truth-functional kind. Whereas *That is either Drosophilia
Melanogaster or D. virilis, I'm not sure which* appears to
entail that if it is not the one then it is the other, and thus is of
the intensional kind. The distinction between intensional and
extensional disjunction has been criticized by various scholars,
including Burgess (1981, 1983), but has been defended by Read (1981,
1983) who proposed the following example as a clear case of
truth-functional disjunction, for which addition is valid: *You
qualify for a grant if either you are over 65 or you can earn less
than 2000 pounds a year* (Read 1981: 68). Now if English
*or* had an intensional and an extensional sense, it should be
possible to find both in a large variety of linguistic contexts. On a
closer inspection however it appears that it is quite hard to find
clear cases of intensional disjunction in embedded positions like in
the antecedent of a conditional (as in Read's example), or under
negation. For example, *That is not Drosophilia Melanogaster or D.
virilisit*, on its most natural reading, simply means that that is
not either, but then *or* must be extensional here because to
derive "not \(A\) and not \(B\)" from "not
(\(A\) or \(B\))" addition is needed. This seems to
suggest that phenomena of relevance should be captured on a pragmatic
level rather than on the level of logic (Grice 1989). An explicit
Gricean account of relevance of disjunctive sentences has been indeed
proposed by Simons (2001), who observes that felicitous utterances of
"\(A\) or \(B\)" require the disjuncts to be
relevant alternatives, and argues that such condition, which she calls
the relatedness condition, can be derived from general principles of
conversation interacting with truth functional interpretation of
*or*. In "Further Notes on Logic and Conversation"
Grice had suggested that to felicitously use a disjunctive sentence
"\(A\) or \(B\)", a speaker must be in possession of
a reasonable argument with "\(A\) or \(B\)" as
conclusion [by Quality] which does not contain one of the disjuncts as
a step [by Quantity] (Grice 1989: 44). According to Simons to arrive
at such a kind of evidence is almost certainly impossible in case the
disjuncts are not (contextually) related to each other (as they were
for example in Belnap and Anderson's Napoleon example, which
indeed requires a special context to be acceptable). Simons (2001)
also discusses a second requirement that a disjunction needs to
satisfy in order to constitute a felicitous contribution to an
ordinary conversation, namely what she calls the distinctness
condition (aka known as Hurford's constraint, from Hurford
1974), which states that the disjuncts in a clausal disjunction must
be distinct alternatives. The following, which does not satisfy such a
constraint since one of the disjunct entails the other, is indeed
highly anomalous (Simons 2001: example (2)):


* (16) Either
there is dirt in the fuel line or there is something in the fuel
line.



Hurford's constraint has more recently received some attention
in the semantic/pragmatic literature because of its role in the debate
between localist and globalist analysis of scalar implicatures (e.g.,
Chierchia, Fox, and Spector 2012).




Disjunctive syllogism is also invalid (or better quasi-valid) in
Priest's Logic of Paradox (LP). In PL both \(1\) and \(\#\) are
designated values (i.e., preserved in valid inferences), where #
stands for both true and false. More formally, validity is defined as
follows: \(\Sigma \models \phi\) iff there is no valuation \(v\) such
that for all \(\psi \in \Sigma: v(\psi) = 1\) or # and \(v(\phi) =0\).
Adopting the strong Kleene tables for disjunction and negation
presented above, we obtain that \(\phi, (\neg \phi \vee \psi)
\not\models \psi\), because \(\phi\) and thus also \((\neg \phi \vee
\psi)\) might be both true and false, while \(\psi\) is false. The
schema \(\phi , (\neg \phi \vee \psi) \models \psi\) however only
fails in case \(\phi\) is a *dialetheia* (both true and false),
in this sense is the scheme quasi-valid. Priest (2006: ch. 8) then argues that applications
of disjunctive syllogism can be justified/are reasonable in case
\((\phi \wedge \neg \phi)\) is rationally rejectable.
  




We conclude this section with a final remark on addition, which
according to Anderson and Belnap does not hold for intensional
disjunction. The validity of addition has also been disputed in
relation to imperative logic. We don't seem to be able to
conclude
 (18)
 from
 (17)
 (Ross' (1941) paradox):


* (17) Post
this letter!
* (18) Post
this letter or burn it!



One way to tackle this would be to treat *or* in
 (18)
 as a case of intensional disjunction. This solution however would
fail to account for a characteristic aspect of the interpretation of
disjunctive imperatives which arguably explains the failure of
addition in these cases, namely their choice offering potential. The
most natural interpretation of disjunctive imperatives is as one
presenting a choice between different actions:
 (18)
 implies that you may post the letter and you may burn it (a free
choice inference). Imperative
 (17)
 then cannot imply
 (18)
 otherwise when told the former one would be justified in burning the
letter rather than posting it (e.g., Mastop 2005; Aloni 2007; Aloni
and Ciardelli 2013). More on free choice in
 section 6.


## 3. Disjunction in language



From a linguistic point of view disjunction is a kind of coordination,
where coordination refers to syntactic constructions in which two or
more units of the same type are combined into a larger unit and still
have the same semantic relations with other surrounding elements
(Haspelmath 2007). An open question is whether disjunctive
coordination is a universal that can be found in all languages. All
languages appear to possess coordination constructions of some kind,
but not all languages seem to have explicit coordinators like
*and* and *or*. For example, Maricopa (a Yuman language
of Arizona described by Gill 1991) and Dyribal (an Australian
Aboriginal language described by Dixon 1972) seem to lack explicit
coordination structures, so in these languages there is no word
corresponding to *or* (see also Winter 1995). This does not
necessarily mean however that these languages lack a way to express
disjunctive meanings. Maricopa and Dyribal appear to be able to convey
"\(A\) or \(B\)", without explicit disjunctive
coordinator, by adding a suffix/particle expressing uncertainty to the
main verb.
 (19)
 illustrates this strategy for Maricopa (from Gil 1991: 102):


* (19) 
Johns John-nom
Bills Bill-nom
v[?]aawuumsaa. 3-come-pl-fut-infer
'John or Bill will come'



That it is the "uncertainty" suffix *saa*
which is responsible for the disjunctive interpretation is evidenced
by the fact that if omitted the interpretation of the sentence becomes
conjunctive (Gil 1991). The close connection between linguistic
disjunction and uncertainty (or ignorance) will be further scrutinized
in the following sections. Let us now turn to languages which do have
specialized disjunctive words like *or*. A first difference
between logical disjunction (\(\vee\)) and disjunctive words in human
languages is that while the former is a binary operator, there is no
theoretical finite limit to the number of units that the latter can
join. Another striking difference is that while logical disjunction is
a sentential operator, linguistic disjunction is typically
cross-categorial. For example, English *or* can coordinate
expressions of different syntactic category as illustrated below,
where the units are noun phrases in
 (20a)
 and verb phrases in
 (20b):


* (20) John or
Mary sang. (\(\Leftrightarrow\) John sang or Mary sang)
* (21) Every
man sang or danced. (\(\not\Leftrightarrow\) every man sang or every
man danced)



Adopting an algebraic perspective Keenan and Faltz (1985) showed that
we can capture all these uses identifying disjunction with the join
operator in a Boolean algebra (or, simplifying a bit, set union). In
the special case of disjunction at the sentence level, the Boolean
operator boils down to the classical propositional operator on truth
values (see
 section 7.1
 for a recent account which identifies disjunction with the join
operator in a Heyting algebra, which at the sentence level yields a
non-classical (inquisitive) propositional operator). As an
illustration consider the following interpretation of generalized,
cross-categorial *or* in terms of generalized union, adapted
from Gazdar (1980) (see also Winter 2001). The crucial assumption
behind this definition is that noun phrases like *John* or
*every man* denote sets or functions rather than individuals
(as in Montague 1973).


* (22)
Generalized, cross-categorial disjunction



\[\lbrack\!\lbrack or\rbrack\!\rbrack = \sqcup\_{\langle\tau,\tau \tau\rangle} =\left\{ 
 \begin{array}{l l}
 \vee\_{\langle t,tt\rangle} & \quad \mathrm{if } \ \tau=t \\
 \lambda X\_{\tau}\lambda Y\_{\tau} \lambda Z\_{\sigma\_1}. X(Z) \sqcup\_{\langle\sigma\_2,\sigma\_2 \sigma\_2\rangle} Y(Z) & \quad \mathrm{if } \ \tau=\sigma\_1 \sigma\_2 
 \end{array} \right.\]


Assume that (i) verb phrases (VPs) denote functions from individuals
into truth values (type \(\langle e,t\rangle\)), (ii) noun phrases
denote functions from VP denotations into truth values (type
\(\langle\langle e,t\rangle, t\rangle\)), and (iii) sentences denote
truth values (type \(t\)). Then, talking about characteristic
functions as if they were sets, we have that *John* denotes the
set of all John's properties, *Mary* denotes the set of
all Mary's properties, and *sang* denotes the set of all
individuals that sang. Given
 (22),
 *John or Mary* denotes the union of John's properties
with Mary's properties, i.e., the set consisting of exactly
those properties that pertain to either John or Mary. So *John or
Mary sang* will come out true just in case this union has as a
member the property of singing. This is the case only if either John
sang or Mary sang (see
 23a).
 On the other hand, given
 (22),
 *sang or danced* denotes the union of the set of individuals
that sang with the set of individuals that danced. So every man sang
or danced will come out true just in case this union is a member of
the property-set denoted by every man. From this we cannot infer that
either every man sang or every man danced (see 23b).


* (23)  a.
 John or Mary sang.
  

**sing**\('\) \(\in \{ P \mid j\in P \} \cup \{ P \mid
m\in P\}\)
  

\(\Leftrightarrow\) **sing**\('\) \(\in \{ P \mid j\in P
\) or \( m\in P\}\)
  

\(\Leftrightarrow\) \( j\in \) **sing**\('\) \(\vee\) \(
m\in\) **sing**\('\)
* b.
 Every man sang or
danced.
  

**sing**\(' \cup\) **dance**\('\) \(\in \{ P
\mid \) **man**\(' \subseteq P\}\)
  

\(\Leftrightarrow\) **man**\(' \subseteq \)
**sing**\(' \cup\) **dance**\('\)
  

\(\not\Leftrightarrow\) **man**\(' \subseteq \)
**sing**\('\) \(\vee\) **man**\(' \subseteq
\) **dance**\('\)



One of the assumptions behind this cross-categorial analysis is that
the input to the semantic component of grammar involves the units as
coordinated at "surface structure" (i.e., no syntactic
*conjunction reduction*, mapping non-sentential coordination to
sentential coordination at "deep structure"). This is
crucial for example to capture example (23b) where non-sentential
disjunction could not be syntactically reduced to sentential
disjunction without change of meaning. Rooth and Partee (1982) however
discussed a number of counterexamples to such an analysis involving
cases of wide-scope *or* in opaque contexts. Their famous
example is repeated below (Rooth and Partee 1982: example (13)):


* (24)  Mary is
looking for a maid or a cook.



As Rooth and Partee observed the sentence has three readings: (i) the
normal *de dicto* reading, according to which Mary would be
satisfied if she found any maid and she would also be satisfied if she
found any cook (this reading can be generated by combining the verb
directly with the disjunctive noun phrase *a maid or a cook*);
(ii) the normal *de re* reading according to which Mary is
looking for a specific person and this person is either a maid or a
cook (this reading can be generated by quantifying in the disjunctive
noun phrase into the translation of the sentence *Mary is looking
for him\(\_1\)*); but also (iii) a wide scope *or* *de
dicto* reading in which either Mary is looking for a maid, any
maid, or she is looking for a cook, any cook, we don't know
which of the two. The latter reading cannot be generated by the
standard Montagovian techniques. To capture these readings Rooth and
Partee (1982) proposed a dynamic analysis of disjunction, while Winter
(2000), building on Larson (1985), proposed a syntactic account where
*or* can move to various positions correlated to the positions
where overt *either* can occur.


## 4. Disjunction in conversation



There are various conclusions one normally draws from the assertion of
a disjunctive sentence like
 (25):


* (25) Mary is
patriotic or quixotic.
* a.
at least one of the two
is true
* b.
at most one of the two
is true [exclusive inference]
* c.
speaker doesn't
know which is true [ignorance inference]



Since only (25a) follows from the classical truth-functional account
of disjunction, there seem to be a divergence in meaning between
\(\vee\) from classical logic and its natural language counterpart.
Grice influentially argued that the assumption that such divergence
does in fact exist is a mistake originating "from inadequate
attention to the nature and importance of the conditions governing
conversation" (Grice 1989: 24).



Expanding on parts of Grice's celebrated argument (Grice 1989:
44-46), suppose one would propose to analyze *or* in such
a way that not only (25a) would logically follow from
 (25)
 but also (25b) and/or (25c). A major problem for such a strong
analysis of *or* is that it would fail to account for the fact
that both the exclusive and the ignorance inferences are easily
cancellable. One can say *Mary invited John or Bill or both*
(cancellation of exclusive inference), or *The prize is either in
the attic or in the garden. I know that because I know where I put it,
but I am not going to tell you* (cancellation of ignorance-modal
inference, from Grice 1989: 45). A strong theorist could then respond
that there are two senses of *or*, a strong one and a weak
(truth-functional) one, with the latter employed in the previous cases
of cancelation. But as Grice replied, 



> 
> 
> if *or* is supposed to possess a strong sense, then it should
> be possible for it (*or*) to bear this sense in a reasonably
> wide range of linguistic settings, for example it should be possible
> to say *It is not the case that A or B* where we are denying
> that \(A\) or \(B\) (in the strong sense of *or*). (Grice
> 1989: 45)
> 
> 
> 



That this is not possible is illustrated by the oddity of the
following two cases: *It is not the case that Mary invited John or
Bill, because she invited both* or *It is not the case that the
prize is either in the attic or in the garden, because I know that it
is in the garden*. Since strong senses of *or* seem to be
restricted to "unenclosed" uses for which an alternative
explanation is available, Grice's conclusion is that only
 (25a)
 should be taken as part of the semantic contribution of the sentence
(what is said). The exclusive and ignorance inferences in
 (25b)
 and
 (25c)
 are merely pragmatic effects (conversational implicatures) which
derive from interactions between a weak (truth-functional)
interpretation of *or* with general principles of
conversation.



On Grice's account, conversation is a purposeful and cooperative
enterprise governed by what he calls the Cooperative Principle:


* (CP)  Make your
conversational contribution such as is required, at the stage at which
it occurs, by the accepted purpose or direction of the talk exchange
in which you are engaged.



Subsumed under this general principle, Grice distinguishes four
categories of more specific maxims, including the maxim of Quantity
(here simplified).


* Quantity
* (i)
 Make your contribution
as informative as is required (for the current purposes of the
exchange);
* (ii)
 Do not make your
contribution more informative than is required.



Conversational implicatures are pragmatic inferences arising from the
interplay between basic (weak) semantic content and these principles
of social interaction. A speaker conversationally implicates what she
must be assumed to believe in order to preserve the assumption that
she is adhering to the CP and maxims.



Gricean reasonings leading from the assertion of a disjunction to the
ignorance and exclusive inference can be summarized as follows:


* (26) Mary is
patriotic or quixotic \(\Rightarrow\) speaker doesn't know
which
* If the speaker had
known that Mary is patriotic, she should have said so (by Quantity).
Assuming that the speaker made the most
informative relevant statement she could, the hearer can infer that
the speaker doesn't know that Mary is patriotic. Similar
reasoning for the second disjunct.
* (27) Mary is
patriotic or quixotic \(\Rightarrow\) not both
* If the speaker had
known that Mary is patriotic and quixotic, she should not have used
*or*, but *and* (by Quantity).
Assuming that the speaker made the most informative relevant statement
she could, the hearer can infer that the speaker doesn't know
that Mary is patriotic *and* quixotic. Assuming that the
speaker is opinionated (either believes (A and B) or believes not (A
and B)) one can conclude that Mary is not patriotic *and*
quixotic.



The assumption in
 (27)
 that the speaker must be opinionated about the conjunctive statement
is however problematic. Given that the use of *or* implicates
that the speaker doesn't know which, why should there be any
pressure to think that the speaker would know both if it were true? A
number of authors have indeed argued that Gricean implicatures are
always epistemically modalized: so in
 (27)
 only the proposition that the speaker *doesn't know*
that Mary is patriotic and quixotic is derivable by Gricean means
*contra* the predictions of the classical formalisation of
Gricean implicatures of Gazdar (1979) (see Soames 1982: 521 and Horn
1989: 543). Many other attempts to formalization of (variations) of
Gricean reasonings have been proposed in the literature, including
Horn (1984) but also the more recent Sauerland (2004), van Rooij and
Schulz (2004) and Franke (2011). All these formalizations assume a
truth-functional analysis of disjunction and other connectives, as the
baseline for the pragmatic reasoning. In what follows we will discuss
recent developments in linguistic analysis of disjunction, including
also challenges to a Gricean pragmatic view.


## 5. Inclusive and exclusive uses of *or*



While the Gricean argument in the previous section quite conclusively
excludes that English *or* is ambiguous between an inclusive
and an exclusive interpretation (*contra*, for example, Tarski
1939:
 21),[2]
 recently some linguists have observed that some disjunctive
constructions in languages other than English seem to force exclusive
uses. Szabolcsi (2002, 2015) discusses the case of Hungarian
*vagy*-*vagy* and Spector (2014) the case of
French *soit-soit*. Russian *ili-ili*,
Italian *o-o*, French *ou -ou* and German
*entweder*-*oder* seem to behave in a similar
fashion. These are all cases of fully-iterated disjunctions, with a
disjunctive particle preceding each disjunct. Note however that not
all fully-iterated disjunctive constructions are of these kinds, for
example English *either-or* constructions are not always
exclusive (*Nobody ate either rice or beans* simply means
nobody ate either), and iterated disjunctive constructions in Sinhala
and Malayalam are not exclusive at all (see Szabolcsi 2015). Finally
also Latin *aut*, which is often taken as a paradigmatic
example of exclusive disjunction (e.g., Copi
 1971),[3]
 has been shown to have inclusive uses (at least in its not iterated
variant), for example *Nemo timebat tribunos aut plebes (No one
feared the magistrates or the mob)* just means no one feared
either. See Dik (1968: 274-276) and Jennings (1994:
239-251) for discussion and more examples.



To show that French *soit*-*soit* constructions tend to
force exclusivity inferences, while French plain disjunction
*ou* doesn't, Spector discusses the following examples.
The reply in both
 (28)
 and
 (29)
 contradicts the exclusive inference, but at the same time asserts that
the first sentence is true. According to Spector, the fact that such a
reply is infelicitous in
 (29)
 shows that the exclusive inference is obligatory in this case.


* (28)  a.
Marie ira au
cinema lundi ou mardi.
  

'Marie will go to the movies on Monday or Tuesday.'
* b.
Absolument! Et elle ira
meme a la fois lundi ET mardi.
  

'Absolutely! She will even go both days.'
* (29)  a.
Marie ira au
cinema soit lundi soit mardi.
  

'Marie will go to the movies SOIT on Monday SOIT on
Tuesday.'
* (23)  b.
 # Absolument! Et elle
ira meme a la fois lundi ET mardi.
  

'Absolutely! She will even go both days.'



It is important to notice at this point that logic textbook exclusive
disjunction, represented as \(\infty\) in the following truth table,
would not deliver the correct results for these constructions.




|  |  |  |  |
| --- | --- | --- | --- |
| \(\phi\) | \(\psi\) | \(\phi \vee \psi\) | \(\phi \infty \psi\) |
| 1 | 1 | 1 | 0 |
| 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 1 |
| 0 | 0 | 0 | 0 |



First of all, it is well known that using \(\infty\) would give wrong
predictions for the cases with more than two disjuncts. For example,
\(\alpha \infty (\beta \infty \gamma)\) is true, if all disjuncts are
true, but *Marie ira au cinema soit lundi soit mardi soit
mercredi* is not (see Reichenbach 1947; Simons 2000). Secondly,
the use of exclusive disjunction would also fail to explain certain
embedded uses of *soit*-*soit* constructions. While
these constructions are ungrammatical under negation, making therefore
hard to apply the Gricean argument of the previous section to these
cases, they are licensed in the scope of a universal quantifier, but
the inferences corresponding with the exclusive reading are no longer
present in such cases:


* (30)  a.
 Tous mes
etudiants etudient soit l'allemand soit
l'anglais.
  

'Every student of mine studies SOIT German SOIT
English.'
* b.
 Absolument! Et certains
etudient meme les deux.
  

'Absolutely! And some even study both.'



On the other hand, Spector notices that the following would be deviant
as a reply to (30a):


* (31) 
#Absolument! Et ils etudient meme les deux.
  

'Absolutely! And they even study both.'



Notice that
 (31)
 negates the exclusive (scalar) implicature derived from
 (30a)
 by Gricean reasoning using
 (32)
 as a relevant alternative:


* (32)  Tous
mes etudiants etudient l'allemand et
l'anglais.
  

'Every student of mine studies German and
English.'



The generalization suggested by Spector is then that
*soit-soit* and other iterated disjunctions
*obligatorily* trigger the Gricean exclusive inferences which
are normally optionally triggered by plain disjunction.



As mentioned above, besides conveying obligatory exclusive effects,
these iterated disjunctions are ungrammatical under negation. Let me
conclude this section with a note on the interaction between
disjunction and negation. Interactions between disjunction,
conjunction and negation in classical logic are regulated by the de
Morgan laws. English *or* seems to validate the second de
Morgan law \(\neg (\phi \vee \phi)\) \(\leftrightarrow\) \((\neg \phi
\wedge \neg \psi)\). On its preferred reading,
 (33)
 means that we didn't close the door and we didn't close
the window.


* (33)  We
didn't close the door or the window.



Sentence
 (33)
 however has also a second reading, on which it means we did not close
the door or did not close the window, but I am not sure which. As
Szabolcsi (2002) observed, speakers in many languages including
Hungarian, Russian, Italian and French find that their counterpart of
 (33)
 only has this second reading. On this reading disjunction scopes over
negation, and so Szabolcsi proposes to treat disjunction words in
these languages as positive polarity items, i.e., roughly, expressions
that cannot be interpreted (anti-licensed) in the immediate scope of a
negation, unless the negation is itself in a negative or more
generally downward entailing (DE) context. Spector introduces a
distinction between local and global positive polarity items and
argues that single disjunction in French and other languages are local
positive polarity items (anti-licensed in the *immediate scope of
negation*, unless certain constraints are met) as Szabolcsi had
proposed, but the iterated disjunctive constructions discussed above
are global polarity items (anti-licensed under the scope of negation,
*however distant the negation is*, unless certain constraints
are met).


## 6. Modal accounts of disjunction and free choice



We can think of disjunction as a means of entertaining different
alternatives. If I say *either it is raining or it is snowing*,
I normally convey that both alternatives are open options for me.
Grice, as we just saw, argued against a semantic account of such
effects, which he labeled as the non truth-functional ground of
disjunction (Grice 1989). Zimmermann (2000), by contrast, proposes a
modal analysis of linguistic disjunction which identifies the semantic
contribution of *or* with precisely these epistemic effects
(see also Geurts 2005 for a further development of this idea). On
Zimmermann's account linguistic disjunctions should be analyzed
as conjunctive lists of epistemic possibilities:


* (34)  \(S\_1\)
or...or \(S\_n\) \(\mapsto\) \(\Diamond S\_1 \wedge \dots \wedge
\Diamond S\_n\),
  

where \(\Diamond\) is an epistemic possibility operator



Someone uttering a sentence of the form "\(S\_1\) or...or
\(S\_n\)" generally conveys that (i) any of the \(S\_1,\dots,
S\_n\) may be true (genuineness) and (ii) that between them, the
\(S\_1,\dots, S\_n\) covers all the relevant possibilities
(exhaustivity). While on a standard truth-functional analysis only
(ii) is part of the meaning of *or*, Zimmermann identifies (i)
as the essential contribution of linguistic disjunction with
exhaustivity being conveyed by intonation or other devices, for
example *either ... or*. Evidence for this division of labor
comes from the cases of so-called "open" disjunctions,
i.e., disjunction with a terminal rise. In the following examples
\(\uparrow\) and \(\downarrow\) indicate rising and falling pitch
contours respectively.


* (35) Where
shall we go?
* a.
 London\(\uparrow\) or
Berlin\(\uparrow\) or Paris\(\downarrow\)
* b.
 London\(\uparrow\) or
Berlin\(\uparrow\) or Paris\(\uparrow\)



While the terminal fall in the closed disjunction
 (35a)
 indicates that the speaker consider her list of options to be
exhaustive, the terminal rise in
 (35b)
 conveys the opposite effect. On Zimmermann's analysis is then
the terminal fall which contributes exhaustivity and not
*or*.



The main motivation behind Zimmermann's modal analysis however
comes from phenomena of free choice. Sentences of the form "You
may \(A\) or \(B\)" are normally understood as implying
"You may A and you may B". The following, however, is not
a valid principle in standard deontic logic, e.g., von Wright
(1968).


* (36) \(P(\alpha
\vee \beta) \to P \alpha \)   [Free Choice
Principle]



As Kamp (1973) pointed out, plainly making the Free Choice principle
valid, for example by adding it as an axiom, would not do because it
would allow us to derive \(P q\) from \(P p\) as shown in
 (37),
 which is clearly unacceptable:


* (37)  1.
\(P p\)  
[assumption]
* 2.
\(P (p \vee q)\)  
[from 1, by principle (38)]
* 3.
\(P q \)   [from 2,
by free choice principle]



The step leading to 2 in
 (37)
 uses the following principle which holds in standard deontic
logic:


* (38)  \(P
\alpha \to P(\alpha \vee \beta)\)



Intuitively, however,
 (38)
 seems invalid (*You may go to the beach* doesn't seem to
imply *You may go to the beach or the cinema*), while
 (36)
 seems to hold, in direct opposition to the principles of deontic
logic. Von Wright (1968) labeled this the paradox of free choice
permissions. Similar paradoxes arise also for imperatives (see
Ross' paradox, introduced in
 section 2),
 epistemic modals (Zimmermann 2000), and other modal
constructions.



Various solutions have been proposed to the paradox of free choice.
Many have argued that what we called the Free Choice Principle is
merely a pragmatic inference and therefore the step leading to 3 in
 (37)
 is unjustified. One argument in favor of such a pragmatic account
comes from the observation that free choice effects disappear in
negative contexts. For example, *No one is allowed to eat the cake
or the ice-cream* cannot merely mean that no one is allowed to eat
the cake and the ice-cream, as would be expected if free choice
effects were semantic entailments rather than pragmatic implicatures
(Alonso-Ovalle 2006). Various ways of deriving free choice inferences
as implicatures have been proposed (e.g., Gazdar 1979; Kratzer and Shimoyama 2002; Schulz
2005; Fox 2007 and Franke 2011; see however Fusco 2014 for a critical
discussion of pragmatic accounts to free choice).



Others have proposed modal systems where the step leading to 3 in
 (37)
 is justified while the step leading to 2 is no longer valid, e.g.,
Aloni 2007, which proposes a uniform account of free choice effects of
disjunctions and indefinites under both modals and imperatives. Simons
(2005) and Barker
(2010) also proposed
semantic accounts of free choice inferences, the latter crucially
employing an analysis of *or* in terms of linear logic additive
disjunction combined with a representation of strong permission using
the deontic reduction strategy (as in Lokhorst 2006).



Finally Zimmermann (2000) distinguishes between
 (36),
 which, according to him, is an unjustified logical principle, from
the following intuitively valid principle:


* (39) 
\(X\) may \(A\) or may \(B\) \(\models\) \(X\) may
\(A\) and \(X\) may \(B\)



By analyzing disjunctions as conjunctions of epistemic possibilities,
the correct logical rendering of
 (39)
 seems to be the following:


* (40)  \(
\Diamond P \alpha \wedge \Diamond P \beta \models P \alpha \wedge P
\beta \)



Zimmermann, however, actually derives only the weaker principle in
 (41)
 (under certain assumptions including his Authority principle). \(\Box
\alpha\) should be read here as "it is certain that
\(\alpha\)":


* (41)  \(
\Diamond P \alpha \wedge \Diamond P \beta \models \Box P \alpha \wedge
\Box P \beta \)



Although Kratzer and Shimoyama, Alonso-Ovalle, Aloni, Simons and
Zimmermann differ in their solution of the free choice paradox, they
all agree in endorsing an "alternative-based" analysis of
*or* according to which a disjunctive sentence "\(A\)
or \(B\)" contributes the set of propositional alternatives
\(\{A, B\}\). Alonso-Ovalle (2006), for example, uses this alternative
set in his pragmatic derivation of the free choice implicatures. Aloni
(2007) instead assumes that modals and imperatives explicitly operate
on the alternative sets introduced in their scope. For example, on her
analysis, *may*\((\phi)\) is true only if the ordinary meaning
of *may* is true of all alternative propositions generated by
\(\phi\). Thus, *You may go to the beach or to the cinema* is
true only if *You may go to the beach* and *You may go to
the cinema* are both true. In the following section we have a
closer look at this alternative-based view.


## 7. Alternative-based accounts of disjunction



In the previous section, we saw that recent semantic work has argued
that disjunctions like "\(A\) or \(B\)" should be
treated as generating a set of propositions \(\{A, B\}\) rather than
yielding the classical join of the two disjuncts. The intuition behind
these accounts is that the main function of a disjunctive sentence is
to present a set of alternatives. A disjunction offers
characterizations of (at least) two different possible states of
affairs, and (under certain circumstances) asserts that at least one
of these holds in the actual world. Besides phenomena of free choice
discussed in the previous section, other linguistic facts have been
argued to justify an alternative-based analysis of disjunction. In
this final section, we first introduce the recent inquisitive
semantics as an example of an "alternative-based" system
with a fully developed logic, and then briefly discuss these further
linguistic motivations.




![[a square with a number on a gray circle in each quadrant. The numbers, left to right then top to bottom, are 11, 10, 01, and 00. The first three of these numbers have a curve enclosing them]](fig-a.png)

(a) classical




![[a square with a number on a gray circle in each quadrant. The numbers, left to right then top to bottom, are 11, 10, 01, and 00. The first two of these numbers have a curve enclosing them as do the first and third numbers.]](fig-b.png)

(b) inquisitive




Figure 1. \((p \vee q)\) in classical
logic and in inquisitive logic.



### 7.1 Inquisitive semantics



In standard logic-based analyses of linguistic meanings, the
*semantic content* of a sentence \(\phi\) is defined as the set
of evaluation points which verify \(\phi\). Normally, evaluation
points are identified with possible worlds and so the semantic content
of a sentence is identified with a set of possible worlds (a
proposition), those worlds in which the sentence is true. In
inquisitive semantics, instead, evaluation points are not worlds, but
information states, which are defined as sets of possible worlds
(those worlds compatible with the information encoded in the state).
The semantic content of \(\phi\) is then inquisitively identified with
the set of states which support \(\phi\), thus a *set of sets*
of possible worlds, rather than a set of possible worlds. The original
motivation for this move was to arrive at a uniform notion of semantic
content which would work for both declarative and interrogative
meaning (see Ciardelli, Groenendijk and Roelofsen 2015), since the
latter are standardly characterized as sets of propositions, rather
than as sets of possible worlds (see the entry on
 questions).
 The inquisitive move of taking states as evaluation points rather
than worlds has important consequences for disjunction. The
inquisitive clause of disjunction reads as follows:


* (42)  \(s
\models (\phi\vee \psi)\) iff \(s \models \phi\) or \(s \models
\psi\).



The interpretation of the connective is given in terms of support in
an information state (rather than truth in a world). For a disjunction
to be supported in a state, at least one of the disjuncts should be
supported, where an atomic sentence is supported in a state \(s\) iff
it is true in all worlds in \(s\). Fig. 1a and b from (Ciardelli and
Roelofsen 2011) illustrates how the resulting characterization of the
semantic content of disjunctive sentences crucially differs from the
classical characterization. These figures assume a propositional
language with only two atoms \(p\) and \(q\); world \(11\) makes both
\(p\) and \(q\) true, world \(10\) makes \(p \) true and \(q\) false,
*et cetera*.
 Figure 1a
 depicts the classical meaning of \((p \vee q)\): the set of all
worlds that make either \(p\) or \(q\), or both, true.
 Figure 1b
 depicts the semantic content associated with \((p \vee q)\) in
inquisitive semantics. Only the maximal states supporting the
disjunction, the so-called *alternatives*, are depicted. A
state \(s\) supports the disjunction if and only if it is included in
either of these alternatives. The semantic content of \((p \vee q)\)
contains then two alternatives. One alternative is made up of all
worlds that make \(p\) true, and the other of all worlds that make
\(q\) true.



From a logical point of view, inquisitive logic can be axiomatised by
expanding intuitionistic logic with the Kreisel-Putnam axiom scheme
\((\neg \phi \to (\psi\_1\vee \psi\_2)) \to (\neg \phi \to \psi\_1) \vee
(\neg \phi \to \psi\_2)\) and the double negation axioms restricted to
atoms (Ciardelli and Roelofsen 2011), and so is one of the
intermediate logics between intuitionistic and classical logic (see
the entry on
 intuitionistic logic, section 6.1).
 As in intuitionistic logic, in inquisitive logic LEM is not provable
and the disjunction property holds (\((\phi \vee \psi)\) is
inquisitively provable iff \(\phi\) is inquisitively provable or
\(\psi\) is). From a model-theoretic point of view, LEM is not
inquisitively valid because both \(\phi\) and \(\neg \phi\) may fail
to be supported in a state that represents ignorance with respect to
\(\phi\). Recall that the main motivation behind inquisitive logic is
to capture both informational and inquisitive content of sentences.
Intuitively, \((\phi \vee\neg \phi)\) is not valid because while
informationally vacuous, it can still raise an issue, namely the issue
whether \(\phi\) is the case or not and, therefore, it is not trivial
from an inquisitive point of view.



From an algebraic perspective, as we saw in
 section 3,
 there is a long tradition in natural language semantics that analyzes
disjunction words as expressing a join operator in a Boolean algebra,
which, at the sentential level, delivers the least upper bound of the
two disjuncts with respect to classic entailment. Recent work showed
that alternative-based systems don't need to abandon the elegant
uniform algebraic perspective of the classical analysis: while
classical entailment gives rise to a Boolean algebra, inquisitive
entailment gives rise to a complete Heyting algebra, with meet, join,
and relative pseudo-complement operators (Roelofsen 2013). Thus if we
identify disjunction with the join operator in such a Heyting algebra,
this automatically generates the desired cross-categorial,
alternative-based notion (Ciardelli and Roelofsen 2015).


### 7.2 Examples of linguistic applications



A number of linguistic phenomena have been argued to justify an
alternative-based analysis of disjunction. In the previous section we
discussed the case of free choice. The remaining part of this section
briefly reviews two additional cases: conditionals and questions.


#### 7.2.1 Disjunction in the antecedent of a conditional



The first phenomenon concerns the interpretation of disjunction in the
antecedent of a counterfactual conditional. According to the classical
treatment due to Stalnaker (1968) and Lewis (1973), a counterfactual
\(\phi\leadsto\psi\) is true in a world \(w\) just in case, among all
worlds that make \(\phi\) true, those that *differ minimally*
from \(w\) also make \(\psi\) true. It is well known however that a
Lewis/Stalnaker analysis invalidates the following inference pattern
(Fine 1975a; Nute 1975):


* (43) 
\((\phi\_1 \vee \phi\_2)\leadsto\psi\) \(\models\) \(\phi\_1
\leadsto\psi\)



> 
> 
> for indefinitely close \(\phi\_1\)-worlds may be \(\neg \psi\)-worlds
> while all of the much closer \(\phi\_2\)-worlds are \(\psi\)-worlds.
> However, the counterfactual *if Thorpe or Wilson were to win the
> next General Election, Britain would prosper* does seem to imply
> *if Thorpe were to win the next General Election, Britain would
> prosper*. (Fine 1975a:
> 453)
> 
> 
> 



One possible reaction would be to abandon a Lewis/Stalnaker treatment
of counterfactuals. But another solution is to adopt an alternative
treatment of disjunction (Alonso-Ovalle 2006, 2009; van Rooij 2006).
Indeed, if the disjunctive antecedent is taken to generate two
alternatives and if verifying the counterfactual involves separately
checking every alternative generated by the antecedent, the problem is
avoided.


#### 7.2.2 Disjunction in questions



An alternative analysis of disjunction further allows a perspicuous
representation of the ambiguity of disjunctive questions like
 (44),
 between a polar reading (expected answers: *yes/no*) and an
alternative reading (expected answers: *coffee/tea*) (e.g., von
Stechow 1991 and Aloni, Egre, & Jager 2013, for the
embedded case).


* (44)  Do you
want coffee or tea?



The alternative reading, which was problematic for standard analyses
of questions (Groenendijk and Stokhof 1984), can be easily represented
by adopting alternative/inquisitive disjunction (see
 figure 1).
 The polar interpretation naturally follows because the classical
notion of disjunction can be easily recovered from the
alternative-based interpretation by adding a closure operator.



In English, alternative and polar readings of questions can be
distinguished by intonation or by using the contrastive marker
*either...or*, the question *Do you want either coffee
or tea?* only has a polar interpretation. Many languages including
Mandarin Chinese, Finnish and Basque can use different disjunctive
coordinators to disambiguate in these cases (Haspelmath 2007). These
languages have two words for interrogative disjunction and standard
disjunction. Interrogative disjunction (e.g., Basque *ala*) can
only occur in interrogative clauses, where it forces an alternative
reading, standard disjunction (e.g., Basque *edo*) can occur in
both declarative and interrogative clauses, in the latter case it
forces a polar interpretation. The following Basque example from
Saltarelli (1988: 84) illustrates:


* (45)  Te-atea-ART
**ala**
or-INT
kafe-a
coffee-ART
nahi
want
duzu?you.it
'Do
you want coffee or tea?' (expected answers:
coffee/tea)
* (46)  Te-a
tea-ART
**edo**
or-STA
kafe-a
coffee-ART
nahi
want
duzu?you.it
'Do
you want either coffee or tea?' (expected answers:
yes/no)



The interaction between disjunctive words, questions and intonation is
much more complex than has been exposed here, see Pruitt and Roelofsen
(2013) for a description of the data and an illustration on how
alternative-based systems can be employed to clarify these
phenomena.