## 1. The Basics


### 1.1 Definition of Free Logic



Free logic is formal logic whose quantifiers are interpreted in the
usual way--that is, objectually over a specified domain
\(\bD\)--but whose singular terms may denote objects outside of
\(\bD\), or not denote at all. Singular terms include proper names
(individual constants),
 definite descriptions,
 and such functional expressions as '\(2 + 2\)'. Since
 classical (i.e., Fregean) predicate logic
 requires that singular terms denote members of \(\bD\), free logic is
a "nonclassical" logic. Where \(\bD\) is, as usual, taken
to be the class of existing things, free logic may be characterized as
logic the referents of whose singular terms need not exist.


### 1.2 How Free Logic Differs from Classical Predicate Logic



Karel Lambert (1960) coined the term 'free logic' as an
abbreviation for 'logic free of existence assumptions with
respect to its terms, singular and general'. General terms are
predicates. Lambert was suggesting that just as classical predicate
logic generalized Aristotelian logic by, *inter alia*,
admitting predicates that are satisfied by no existing thing
('is a Martian', 'is non-self-identical',
'travels faster than light'), so free logic generalizes
classical predicate logic by admitting singular terms that denote no
existing thing ('Aphrodite', 'the greatest
integer', 'the present king of France').



Because classical logic's singular terms must denote existing
things (when, as usual, '\(\exists\)' is read as
"there exists"), classical logic is unreliable in
application to statements containing singular terms whose referents
either do not exist or are not known to. Consider, for example, the
true statement:



(S)
We detect no motion of the earth relative to the ether,


using 'the ether' as a singular term for the light-bearing
medium posited by nineteenth century physicists. The reason why (S) is
true is that, as we now know, the ether does not exist. According to
classical logic, however, (S) is false, because it implies the
existence of the ether. Free logic allows such statements to be true
despite the non-referring singular term. Indeed, it allows even
statements of the form \({\sim}\exists x\) \(x=t\) (e.g., "the
ether does not exist") to be true, though in classical logic,
which presumes that \(t\) refers to an object in the quantificational
domain, they are self-contradictory.



Free logic accommodates *empty* singular terms (those that
denote no member of the quantificational domain \(\bD)\) by rejecting
inferences whose validity depends on the classical presumption that
they must denote members of \(\bD\). Consider, for example, the rule
of universal instantiation (specification): from the premise
"Every \(x\) (in \(\bD)\) satisfies \(A\)" we may infer
"\(t\) satisfies \(A\)." This rule, whose formal
expression is:

 

\[ \forall xA \vdash A(t/x), \]


is invalid in free logic; for even if every object in \(\bD\)
satisfies \(A\), if \(t\) does not denote a member of \(\bD\), then
\(A(t/x)\) may be false. (Here and elsewhere \(A(t/x)\) is the result
of replacing all occurrences of \(x\) in \(A\) by individual constant
\(t\); if there are no such occurrences, then \(A(t/x)\) is just
\(A\).) Likewise invalid is existential generalization: the principle
that from a formula \(A\) containing a singular term \(t\) we may
infer that there is something in \(\bD\) that satisfies \(A(x//t)\),
the result of replacing *one or more* occurrences of \(t\) in
\(A\) by \(x\):

 

\[ A \vdash \exists xA(x//t), \]


If \(t\) does not denote an object in \(\bD\) then the truth of \(A\)
does not guarantee that there exists in \(\bD\) an object that
satisfies \(A(x//t)\). Though free logic rejects such classical
inferences, it accepts no classically invalid inferences; hence it is
strictly weaker than classical logic for a language with the same
vocabulary.



To distinguish terms that denote members of \(\bD\) from those that do
not, free logic often employs the one-place "existence"
predicate, '\(\rE!\)' (sometimes written simply as
'\(\rE\)'). For any singular term \(t\), \(\rE!t\) is true
if \(t\) denotes a member of \(\bD\), false otherwise.
'\(\rE!\)' may be either taken as primitive or (in
bivalent free logic with identity) defined as follows:

 

\[ \rE !t \eqdf \exists x(x=t). \]


Using 'E!' we can express classical logic's blanket
presumption that singular terms denote members of \(\bD\) as an
explicit premise, E\(!t\), for selected terms \(t\). Thus we can
formulate the following weaker analogs of universal instantiation:



\[ \forall xA, \rE!t \vdash A(t/x) \]


and existential generalization:

 

\[ A, \rE!t \vdash \exists xA(x//t), \]


which are valid in free logic.


### 1.3 Relation of Free Logic to Inclusive Logic



Classical predicate logic presumes not only that all singular terms
refer to members of the quantificational domain \(\bD\), but also that
\(\bD\) is nonempty. Free logic rejects the first of these
presumptions. *Inclusive logic* (sometimes also called
*empty* or *universally free* logic) rejects them both.
Thus while inclusive logic for a language containing singular terms
must be free, free logics need not be inclusive.



Many existential assertions--e.g., \(\exists x(x=x),\) \(\exists
x(Px \rightarrow Px),\) \(\exists x(Px \rightarrow \forall
yPy)\)--are true in all nonempty domains and hence are valid in
both classical logic and non-inclusive free logic. But since all
existentially quantified formulas are false in the empty domain, none
are valid in inclusive logic. Correlatively, since all universally
quantified formulas are true in the empty domain, none are
self-contradictory in inclusive logic. Even vacuously universally
quantified formulas (formulas of the form \(\forall xA\), where \(x\)
is not free in \(A)\) are true in the empty domain. Hence the
schema:

 

\[ \forall xA \rightarrow A, \text{ where } x \text{ is not free in } A, \]


which is valid in both classical logic and non-inclusive free logic,
is invalid in inclusive logic. Inclusive logic also invalidates some
of the laws of confinement--e.g.,

 

\[ \forall x(P \amp A) \leftrightarrow(P \amp \forall xA), \text{ where } x \text{ is not free in } P, \]


that are used for prenexing formulas (giving quantifiers the widest
possible scope) or purifying them (giving quantifiers the narrowest
possible scope). And in inclusive logic the formula:

 

\[ \forall x(A \leftrightarrow x=t), \]


widely used in the theory of definite descriptions, is not equivalent,
as it otherwise is, to:

 

\[ \forall x(A \rightarrow x=t) \amp A(t/x), \]


since with \(\bD\) empty and \(A(t/x)\) false, the first but not the
second is true. Where there is need for such regularities, a
non-inclusive free logic may be preferable to an inclusive one. Yet
because inclusivity frees logic from one more existential presumption,
many free logicians favor it.


## 2. Formal Systems



Logics may be represented in various ways. Axiom systems, natural
deduction systems and trees (or, equivalently, tableaux) are among the
most common. This section presents all three for the bivalent
inclusive form of free logic known as Positive Free Logic (PFL) and
mentions some variants. (For the meaning of the term
"positive" in this context see
 Section 3.2).
 PFL is formulated in a first-order language \(\bL\) without sentence
letters or function symbols, whose primitive logical operators are
negation (not) '\({\sim}\)', the conditional (if-then)
'\(\rightarrow\)', the universal quantifier (for all)
'\(\forall\)', identity '=' and
'E!', the others being defined as usual. We assume for the
sake of definiteness that the formulas of \(\bL\) are closed (contain
no unquantified variables) and that they may be vacuously quantified
(have the form \(\forall xA\) or \(\exists xA\), where \(x\) does not
occur free in \(A)\). An occurrence of a variable is
*quantified* if it lies within the scope of an operator such as
'\(\forall\)' or '\(\exists\)' that binds that
variable; otherwise it is *free*.


### 2.1 Axiom Systems



PFL may be axiomatized, with modus ponens as the sole inference rule,
by adding the following schemas to the tautologies of classical
propositional logic:

 

\[\begin{align} \tag{A1} &A \rightarrow \forall xA \\ \tag{A2} &\forall x(A \rightarrow B) \rightarrow (\forall xA \rightarrow \forall xB) \\ \tag{A3} &\forall xA(x/t), \text{ if } A \text{ is an axiom} \\ \tag{A4} &\forall xA \rightarrow (\rE!t \rightarrow A(t/x)) \\ \tag{A5} &\forall x \rE!x. \end{align}\]


Once again, \(A(x/t)\) is the result of replacing all occurrences of
individual constant \(t\) in \(A\) by the variable \(x\). If there are
no such occurrences, then \(A(x/t)\) is just \(A\). In (A1) the
variable \(x\) is not free in \(A\) (since otherwise \(A\) would be an
open formula and formulas of \(\bL\) are closed). However, \(x\) may
be free in \(A\) or \(B\) in (A2) and in \(A\) in (A3) and (A4).



(A4) and (A5) are special axioms for free logic. The others are
classical. (A4) modifies the classical principle:

 

\[\tag{A4c} \forall xA \rightarrow A(t/x) \]


by using '\(\rE!\)' to restrict specification. (A4)
stipulates in effect that the quantifiers range over *all*
objects that satisfy '\(\rE!\)', (A5) that they range
*only* over objects that satisfy 'E!'. Omitting
(A5) and replacing (A4) with (A4c) yields classical logic. To obtain a
non-inclusive free logic, we may add to (A1)-(A5) the axiom
\(\exists x\rE!x\)--or any axiom of the form \(\exists xT\) such
that for any term \(t, T(t/x)\) is a tautology.



For languages containing the identity predicate, we also need:



\[\tag{A6} s=t \rightarrow(A \rightarrow A(t//s)), \]


where, as above, \(A(t//s)\) is the result of replacing one or more
occurrences of \(s\) in \(A\) by \(t\), and either

 

\[\tag{A7} t=t \]


if all self-identity statements, including those whose singular term
is empty, are to be true or

 

\[\tag{A7-} \forall x(x=x) \]


if not (see Sections
 3.1
 and
 3.2
 below). If 'E!' is defined in terms of the identity
predicate as indicated in
 Section 1.2,
 then (A4) takes the form:

 

\[ \forall xA \rightarrow(\exists y(y=t) \rightarrow A(t/x)). \]


'\(\rE!\)' cannot be defined without the identity
predicate (Meyer, Bencivenga and Lambert, 1982).



Free logic can be formalized without either '=' or
'\(\rE!\)'. (A1)-(A3) remain unchanged, but (A4) and
(A5) are replaced respectively by:

 

\[\begin{align} \tag{A\(4'\)} & \forall y(\forall xA \rightarrow A(y/x)) \\ \tag{A\(5'\)} & \forall x\forall yA \rightarrow \forall y\forall xA. \end{align}\]


(A\(4')\), like (A4), restricts specification to objects within
\(\bD\), but it uses a quantifier instead of '\(\rE!\)' to
do so. The quantifier permutation axiom (A\(5')\) is redundant in the
presence of the identity axioms but, as Fine proved in (1983), is
independent of the other axioms.



The formulas used in the axiom systems discussed so far are closed,
but some free logics allow open formulas--i.e., formulas that
contain free variables. These logics follow one of two conventions for
variable assignments. Those that assign to each free variable a member
of \(\bD\) are called *E\(^+\)-logics*; those that do not are
called *E-logics*. The following specification rule is valid in
E\(^+\)-logics but not in E-logics:

 

\[ \forall xA \vdash A(v/x). \]


(Here \(A(v/x)\) is the result of replacing every occurrence of the
variable \(x\) in \(A\) by a variable \(v\) that is free for \(x\) in
\(A\).) Conversely, the following substitution rule is valid in
E-logics but not in E\(^+\)-logics:

 

\[ A \vdash A(t/x). \]


But since this article employs closed formulas, the distinction
between E- and E\(^+\)-logics may here be ignored. (See Williamson
(1999) for an illuminating discussion of problems engendered by
permitting open formulas in inclusive logics.)


### 2.2 Natural Deduction Rules



PFL can also equivalently be formulated in a natural deduction system.
The introduction and elimination rules for the operators of
propositional logic and identity are as usual. The quantifier
introduction and elimination rules are restricted by use of the
predicate '\(\rE!\)', as follows:




|  |  |
| --- | --- |
| \((\forall\rI)\) | Given a derivation of \(A(t/x)\) from \(\rE!t\), where \(t\) is
new and does not occur in \(A\), discharge \(\rE!t\) and infer
\(\forall xA\). |
| \((\forall \rE)\) | From \(\forall xA\) and E\(!t\) infer \(A(t/x)\). |
| \((\exists \rI)\) | From \(A\) and \(\rE!t\) infer \(\exists xA(x//t)\). |
| \((\exists \rE)\) | Given \(\exists xA\) and a derivation of a formula \(B\) from
\(A(t/x) \amp \rE!t\), where \(t\) is new and does not occur in either
\(A\) or \(B\), discharge \(A(t/x) \amp \rE!t\) and infer \(B\) from
\(\exists xA\). |



The variable \(x\) need not be free in \(A\), in which case \(A(t/x)\)
is just \(A\). 'E!' may either be taken as primitive (in
which case it requires no additional rules) or defined in terms of the
identity predicate as in
 Section 1.2.
 For non-inclusive logic, we may add a rule that introduces \(\exists
x\rE!x\).


### 2.3 Tree Rules



Jeffrey-style tree rules (Jeffrey 1991) for PFL can be obtained by
replacing the classical rules for existentially and universally
quantified formulas with the following:




***Existential Rule***: If \(\exists xA\) appears
unchecked on an open path, check it, and


1. if \(x\) is free in \(A\), choose a new individual constant \(t\)
and list both \(\rE!t\) and \(A(t/x)\) at the bottom of every open
path beneath \(\exists xA\), and
2. if \(x\) is not free in \(A\), write \(A\) at the bottom of every
open path beneath \(\exists xA\).



***Universal Rule***: If \(\forall xA\) appears on
an open path, then


1. if \(x\) is free in \(A\), then where \(t\) is an individual
constant that occurs in a formula on that path, or a new individual
constant if there are none on the path, split the bottom of every open
path beneath \(\forall xA\) into two branches, writing \({\sim}\rE!t\)
at the bottom of the first branch and \(A(t/x)\) at the bottom of the
second, and
2. if \(x\) is not free in \(A\), write \(A\) at the bottom of every
open path beneath \(\forall xA\).




For languages that do not allow vacuous quantification, clause (ii)
can in each case be omitted. Non-inclusive free logic needs an
additional rule that introduces \(\rE!t\) for some new individual
constant \(t\) if a path does not already contain a formula of this
form.


## 3. Semantics



Semantics for free logics differ in how they assign truth-values to
atomic formulas that are *empty-termed*--i.e., contain at
least one empty singular term. There are three general approaches:


1. *Negative* semantics require all empty-termed atomic
formulas to be false,
2. *Positive* semantics allow some empty-termed atomic
formulas not of the form *E!t* to be true, and
3. *Neutral* (or *nonvalent*) semantics require all
empty-termed atomic formulas not of the form *E!t* to be
truth-valueless.


### 3.1 Negative Semantics



A negative semantics is a bivalent semantics on which all empty-termed
atomic formulas (including identity statements) are false. The
inclusive version presented here makes only minimal adjustments to
classical semantics to allow for non-denoting terms.



Let the language \(\bL\) be defined as in
 Section 2.
 Then a negative inclusive model for \(\bL\) is a pair \(\langle
\bD,\bI\rangle\), where \(\bD\) is a possibly empty set (the domain)
and \(\bI\) is an interpretation function that assigns referents to
individual constants and extensions to predicates such that:


1. for each individual constant \(t\) of \(\bL\), either \(\bI(t) \in
\bD\) or \(\bI(t)\) is undefined, and
2. for each \(n\)-place predicate \(P\) of \(\bL, \bI(P) \subseteq
\bD^n\).



\((\bD^n\) is the set of \(n\)-tuples of members of \(\bD\), a 1-tuple
of an object \(d\) being just \(d\) itself.) Given a model \(\langle
\bD,\bI\rangle\), we recursively define a valuation function \(\bV\)
that assigns truth values to formulas as follows:

 

\[\begin{aligned} \bV(Pt\_1 \ldots t\_n) &= \left\{\begin{array}{l} \rT \Leftrightarrow \bI(t\_1),\ldots,\bI(t\_n) \text{ are all defined and} \\ \quad\langle \bI(t\_1),\ldots, \bI(t\_n)\rangle \in \bI(P); \\ \rF \text{ otherwise.} \end{array}\right. \\ \bV(s=t) &= \left\{\begin{array}{l} \rT \Leftrightarrow \bI(s),\bI(t) \text{ are defined and } \bI(s)=\bI(t); \\ \rF \text{ otherwise.} \end{array}\right. \\ \bV(\rE!t) &= \left\{\begin{array}{l} \rT \Leftrightarrow \bI(t) \text{ is defined;} \\ \rF \text{ otherwise.} \end{array}\right. \\ \bV({\sim}A) &= \left\{\begin{array}{l} \rT \Leftrightarrow \bV(A) = \rF\\ \rF \text{ otherwise.} \end{array}\right. \\ \bV(A\rightarrow B) &= \left\{\begin{array}{l} \rT \Leftrightarrow \bV(A) = F \text{ or } \bV(B) = T; \\ \rF \text{ otherwise.} \end{array}\right. \\ \bV(\forall xA) &= \left\{\begin{array}{l} \rT \Leftrightarrow \text{ for all } d\in \bD, \bV\_{(t,d)}(A(t/x)) = \rT \\ \quad(\text{where } t \text{ is any individual constant not in } A \\ \quad\text{and } \bV\_{(t,d)} \text{ is the valuation function on the} \\ \quad\text{model } \langle \bD,\bI^\*\rangle \text{ such that } \bI^\* \text{ is just like } \bI\\ \quad\text{except that } \bI^\*(t)= d); \\ \rF \text{ otherwise.} \end{array}\right. \end{aligned}\]


(The metalinguistic symbol '\(\Leftrightarrow\)' means
"if and only if.") A logic adequate to this semantics may
be axiomatized by making three changes to the axioms of PFL. The first
is to add the axiom:

 

\[\begin{align} \tag{A-} &Pt\_1 \ldots t\_n \rightarrow \rE!t\_i, \text{ where } 1\le i\le n \text{ and} \\ & P \text{ is any primitive } n\text{-place predicate, including \(`\)='.} \end{align}\]


This expresses the convention that an atomic formula cannot be true
unless its terms refer. Second, because all empty-termed identity
statements are false on a negative semantics, (A7) is invalid and must
be replaced by (A7-). Third, since (A2), (A3), (A-) and (A7-) together
imply (A5), (A5) may be omitted. The resulting logic is known as NFL
(Negative Free Logic). For languages with function symbols, negative
free logic requires in addition this *axiom of strictness*:



\[ \rE!f(t\_1 ,\ldots ,t\_n) \rightarrow \rE!t\_i, \text{ where } 1\le i\le n, \]


which assures that a function has a value only if each of its
arguments does. Because of its unusual treatment of identity, negative
free logic validates the equivalence:

 

\[ t=t \leftrightarrow \rE!t. \]


(This equivalence is sometimes taken as a definition of E\(!t\).)
Identity statements in negative free logic thus have existential
implications. This may be problematic in certain contexts. According
to Shapiro and Weir (2000), for example, use of such an
"existential" notion of identity sullies the
"epistemic innocence" of some recent efforts to base
neo-logicist philosophies of mathematics on free logic.



Negative free logic is also peculiar in that it validates the
principle of *indiscernibility of nonexistents*:



\[ ({\sim}\rE!s \amp{\sim}\rE!t) \rightarrow(A \rightarrow A(t//s)), \]


where \(A(t//s)\) is the result of replacing one or more occurrences
of \(s\) in \(A\) by \(t\).


### 3.2 Positive Semantics



Positive semantics allow some empty-termed atomic formulas not of the
form \(\rE!t\) to be true. They are typically bivalent, though there
are variants that allow truth-value gaps or extra truth values. Only
bivalent semantics are considered in this section.



Positive semantics treat formulas of the form \(t=t\) as true, whether
or not \(t\) is empty. Hence they validate (A7), which affirms all
self-identity statements, not merely the weaker (A(7-), which affirms
only self-identities between nonempty terms.



Like negative semantics, some positive semantics require each singular
term to denote either a member of \(\bD\) or nothing at all. But then
when a term fails to denote, the truth value of an atomic formula
containing it cannot as usual be a function of its denotation, and the
formula must be evaluated in some nonstandard way. To avoid such
irregularity and yet permit empty-termed formulas to be true, other
positive semantics allow singular terms to denote, and predicates to
be satisfied by, nonmembers of \(\bD\). These nonmembers are collected
into a second or *outer* domain \(\bD\_o\), in contrast to which
\(\bD\) is described as the *inner* domain. The result is a
*dual-domain* semantics.



Positive semantics with dual domains are generally the simplest. The
members of the outer domain \(\bD\_o\) typically represent
"non-existing" things. Depending on the application, these
may be theoretical or ideal entities, error objects (in computer
science), fictional objects, merely possible (or even impossible)
objects, and so on. Some authors make \(\bD\) a subset of \(\bD\_o\),
which is the convention throughout this article; others make the two
disjoint. In a bivalent dual-domain semantics each singular term
denotes an object in \(\bD\_o\) though possibly not in \(\bD\). Thus
\(\bD\), though not \(\bD\_o\), may be empty. Predicates are assigned
extensions from \(\bD\_o\), and the truth-values of atomic formulas
(whether empty-termed or not) are computed in the usual Tarskian
fashion: an atomic formula is true if and only if the \(n\)-tuple of
objects denoted by its singular terms, taken in order, is a member of
the predicate's extension. Identity statements are no exception.
Statements of the form \(s=t\) are true if and only if \(s\) and \(t\)
denote the same object. Hence, even if empty-termed, they may be
true.



More formally, a dual-domain model for a language \(\bL\) of the sort
defined in
 Section 2
 is a triple \(\langle \bD,\bD\_o,\bI\rangle\), where \(\bD\) is a
possibly empty inner domain, \(\bD\_o\) is a nonempty outer domain such
that \(\bD \subseteq \bD\_o\), and \(\bI\) is an interpretation
function such that for every individual constant \(t\) of \(\bL,
\bI(t) \in \bD\_o\), and for every \(n\)-place predicate \(P\) of
\(\bL, \bI(P) \subseteq \bD\_o^n\). Given a model \(\langle
\bD,\bD\_o,\bI\rangle\), the valuation function \(\bV\) assigns truth
values to atomic and quantified formulas as follows:

 

\[\begin{aligned} \bV(Pt\_1 \ldots t\_n) &= \left\{\begin{array}{l} \rT \Leftrightarrow \langle\bI(t\_1),\ldots,\bI(t\_n)\rangle \in \bI(P); \\ \rF \text{ otherwise.} \end{array}\right. \\ \bV(s=t) &= \left\{\begin{array}{l} \rT \Leftrightarrow \bI(s)=\bI(t); \\ \rF \text{ otherwise.} \end{array}\right. \\ \bV(\rE!t) &= \left\{\begin{array}{l} \rT \Leftrightarrow \bI(t) \in \bD; \\ \rF \text{ otherwise.} \end{array}\right. \\ \bV(\forall xA) &= \left\{\begin{array}{l} \rT \Leftrightarrow \text{ for all } d\in \bD, \bV\_{(t,d)}(A(t/x)) = \rT \\ \quad(\text{where } t \text{ is not in } A \text{ and } \bV\_{(t,d)} \text{ is the}\\ \quad\text{valuation function on the model } \langle \bD,\bD\_o,\bI^\*\rangle \\ \quad\text{such that } \bI^\* \text{ is just like } \bI \text{ except that}\\ \quad\bI^\*(t)= d); \\ \rF \text{ otherwise.} \end{array}\right. \end{aligned}\]


The clauses for '\({\sim}\)' and ' \(\rightarrow\)
' are the same as in negative free logic. PFL with classical
identity -- that is, the logic axiomatized by (A1)-(A7)
-- is sound and complete with respect to this semantics (Leblanc
and Thomason 1968).



Dual-domain semantics have been criticized as ontologically
extravagant. In response, some authors have advocated single-domain
positive semantics, which assign no denotation to empty singular
terms. In such semantics empty-termed atomic formulas require
unconventional treatment. Typically such semantics determine the
truth-values of atomic formulas in two different ways: a Tarksi-style
calculation for formulas whose terms all refer, and a separate
truth-value assignment for empty-termed atomic formulas. The details,
however, tend to get complicated. Antonelli (2000), for example,
advocated such a single-domain free logic, which he called
proto-semantics, but more recently (2007, p. 72) he has characterized
all semantics for positive free logic as "somewhat
artificial" and has questioned the logical character of free
quantification in general.


### 3.3 Neutral Semantics



Neutral semantics make all empty-termed atomic formulas not of the
form *E!t* truth-valueless. Truth-valueless formulas are often
said to have "truth-value gaps." Neutral semantics are of
two types: ordinary neutral semantics, which provide conventions for
calculating the truth values of complex formulas directly from their
components, even when there are empty terms, and supervaluational
semantics, which calculate the truth values of complex formulas by
considering all the values that their components could have if their
empty terms had referents. Ordinary neutral semantics will be
considered in this section, supervaluations in
 Section 3.4.



The uniform policy of making all empty-termed atomic formulas
truth-valueless has the advantages of plausibility and simplicity at
the atomic level, but it complicates the evaluation of complex
formulas. How are the logical operators to function when some of the
values on which they usually operate are absent? Some cases are fairly
clear. The negation of a truth-valueless formula, for example, is
generally taken to be truth-valueless. But:


* If \(A\) is true and \(B\) truth-valueless, is \(A \rightarrow B\)
false or truth-valueless?
* If \(A\) is false and \(B\) truth-valueless, is \(A \rightarrow
B\) true or truth-valueless?
* Let \(A = (B \amp C)\), where \(x\) is free in \(B, B\) be true of
some but not all members of \(\bD\), and \(C\) be closed and
truth-valueless. Clearly this open formula is either truth-valueless
of every object in \(\bD\) or truth-valueless of some and false of
others. In either case, is \(\exists xA\) truth-valueless or
false?



At one extreme, we might want the operators to generate as many
plausible truth values as possible in order to validate as many
classically valid formulas as we can. At the other, one might arrange
things so that *all* empty-termed formulas are truth-valueless,
which would produce a very weak logic (Lehman 2001). But however we
choose, many formulas that are valid in both classical predicate logic
and the usual forms of free logic--indeed, even in propositional
logic--will become invalid. The law of noncontradiction, for
example:

 

\[ {\sim}(A \amp{\sim}A) \]


is truth-valueless whenever \(A\) is (unless we make negations of
truth-valueless statements true) and hence becomes invalid. Of course
this law and many other standard logical principles remain *weakly
valid*--i.e., not false on any model--and it is possible
to construct a logic based on weak validity rather than ordinary
validity. But because any such logic will still be weaker than
classical logic and because its theorems need not even be true, most
logicians reject this strategy. For more on neutral free logic, see
Lehman 1994, 2001, and 2002, pp. 233-237.


### 3.4 Supervaluations



Neutral semantics can be made to validate all the theorems of standard
free logics by augmenting them with supervaluations. Supervaluations
were first formalized by van Fraassen (1966). The version presented
here is a variant of Bencivenga's approach (1981 and 1986).



The fundamental idea is this: when empty terms deprive a formula of
truth-value, supervaluational semantics nevertheless accounts it true
(or false) if all possible ways of assigning referents to those terms
agree in making it true (or false). This strategy restores validity to
many principles that would lose it in an ordinary neutral semantics.
The following instance of the law of noncontradiction:



\[ {\sim}(Pt \amp{\sim}Pt), \]


for example, is truth-valueless when \(t\) is nondenoting (assuming an
ordinary neutral semantics that makes the negation of a
truth-valueless formula truth-valueless). Hence in such a semantics
the law itself is invalid. Yet were we to assign a referent to \(t\),
that referent would either be in the extension of \(P\) or not. If it
were, then \(Pt\) would be true. If it were not, then \(Pt\) would be
false. In either case \({\sim}(Pt \amp{\sim}Pt)\) would be true. Thus,
since all possible ways of assigning referents to \(t\) agree in
making \({\sim}(Pt \amp{\sim}Pt)\) true, we should count \({\sim}(Pt
\amp{\sim}Pt)\) itself as true. In this way the law of
noncontradiction can be preserved.



More explicitly, a supervaluation begins with a neutral model \(\bM\)
with a single, possibly empty domain. We then construct the set of
*completions* of \(\bM\). These may be regarded as bivalent
dual-domain positive models whose inner domain is the domain of
\(\bM\), but which also have an outer domain \(\bD\_o\) to provide
referents for the empty terms. In each completion, singular terms that
are *non*empty in \(\bM\) retain their referents, and those
that are empty in \(\bM\) denote a member of \(\bD\_o - \bD\). For each
\(n\)-place predicate \(P\), the extension of \(P\) is a subset of
\(\bD\_o^n\) and a superset of \(P\text{'s}\) extension in \(\bM\).



From these completions we now construct a supervaluation. A
*supervaluation* of \(\bM\) is a partial assignment of
truth-values to formulas that makes a formula true if all completions
of \(\bM\) make it true, false if they all make it false, and
truth-valueless if they disagree. A formula is valid on a
supervaluational semantics if and only if it is true on all
supervaluations. This semantics validates all and only the theorems of
PFL (Bencivenga 1981, Morscher & Simons 2001, pp.
14-18).



Supervaluations employ what Bencivenga (1986) calls a
"counterfactual theory" of truth: an empty-termed
statement *is* true if it *would be* true on any
assignment of referents to its empty terms. This has struck many
critics as simply false. Moreover, the logic itself leaves much to be
desired. For one thing, supervaluational consequence is too strong.
Thus, for example, although the formula \(Pt \rightarrow \rE!t\) is
(quite properly) not valid on a supervaluational semantics,
nevertheless since \(\rE!t\) is true on every supervaluation on which
\(Pt\) is true, the sequent (derivability statement) \(Pt \vdash
\rE!t\) is *im*properly semantically valid. Therefore, although
PFL is sound on supervaluational semantics and every semantically
valid formula is a theorem of PFL, not all semantically valid sequents
are provable in PFL. In fact, supervaluational consequence is not
axiomatizable by any extension of free logic. This follows from a
result of Woodruff (1984), who has shown that supervaluational
semantics has many of the undesirable properties of second-order
semantics. Jerry A. Fodor and Ernest Lapore (1996) argue, furthermore,
that the completions needed to construct supervaluations are not
meaning-preserving. Hence, they conclude, two alleged advantages of
supervaluations--that they explain the meaningfulness of
sentences with truth value gaps and that they allow us to preserve
classical logic--are illusory. Finally, since supervaluations are
built from completions that are in effect positive dual-domain models,
we may wonder whether the detour through supervaluations is worth the
trouble, since positive dual-domain models alone are simpler and more
adequate to PFL.


## 4. Generic Anomalies



While problems noted above are specific to particular forms of free
logic, there are anomalies that infect all, or nearly all, forms. This
section considers three: (1) a cluster of problems related to the
application of primitive predicates to empty terms, (2) the failure of
substitutivity *salva veritate* of co-referential expressions,
and (3) the inability of free logic to express sufficient conditions
for existence.


### 4.1 Problems with Primitive Predicates



In classical logic and in positive free logic any substitution
instance of a valid formula (or form of inference) is itself a valid
formula (or form of inference). But in negative or neutral free logic
this is not the case. A *substitution instance* is the result
of replacing primitive non-logical symbols by possibly more complex
ones of the same semantic type--\(n\)-place predicates with open
formulas in \(n\) variables, and individual constants with singular
terms--each occurrence of the same primitive symbol being
replaced by the same possibly complex symbol. The replacement of an
occurrence of a primitive \(n\)-place predicate \(P\) in some formula
\(B\) by an open formula \(A\) with free variables \(x\_1 ,\ldots
,x\_n\) is performed as follows: where \(t\_1 ,\ldots ,t\_n\) are the
individual constants or variables immediately following \(P\) in that
occurrence, replace P\(t\_1 \ldots t\_n\) in \(B\) by \(A(t\_i
/x\_i)\)--the result of replacing \(x\_i\) by \(t\_i\) in \(A\), for
each \(i\), \(1\le i\le n\).



Let \(P\), for example, be a primitive one-place predicate. Then if
the semantics is negative, \(Pt \rightarrow \rE!t\) is valid. But now
consider the substitution instance \({\sim}Pt \rightarrow \rE!t\), in
which the open formula \({\sim}Px\) is substituted for \(P\). This
substitution instance is false when \(t\) is empty. Hence valid
formulas may have invalid substitution instances. The same holds for
ordinary neutral semantics that make conditionals true whenever their
consequents are true.



In a negative semantics, moreover, the truth value of an empty-termed
statement depends arbitrarily on our choice of primitive predicates.
Consider, for example, a negative free logic interpreted over a domain
of people that takes as primitive the one-place predicate
'\(A\)', meaning "is an adult," and defines
"is a minor" by this schema:

 

\[ Mt \eqdf {\sim}At. \]


For any non-denoting name \(t, At\) is false in this theory; hence
\(Mt\) is true. If we take 'is a minor' as primitive
instead, the truth-values of \(At\) and \(Mt\) are reversed. But why
should truth-values depend on primitiveness in this way?



Positive semantics avoid these anomalies. But, if bivalent, in
application they force us to assign truth values to empty-termed
formulas in some other way, often without sufficient reason. Consider,
for example, these three formulas, all of which contain the empty
singular term '\(1/0\)' (where '\(/\)' is the
division sign):

 

\[\begin{aligned} 1/0 &= 1/0 \\ 1/0 &\gt 1/0 \\ 1/0 &\le 1/0 \end{aligned}\]


Assuming a bivalent positive semantics, which ones should we make true
and which false? Since the semantics is positive, '\(1/0 =
1/0\)' is automatically true. One might argue further that since
'\(\le\)' expresses a relationship weaker than
'=' and since '\(1/0 = 1/0\)' is true,
'\(1/0 \le 1/0\)' should be true as well. But that is
merely to mimic with empty terms an inference pattern that holds for
denoting terms. To what extent is such mimicry justified? Suppose we
do decide to make '\(1/0 \le 1/0\)' true; should we
therefore make '\(1/0 \gt 1/0\)' false? There are no
non-arbitrary criteria for answering such questions. To a large
extent, of course, the answers don't matter. There are no facts
here; any consistent convention will do. But that's just the
problem. Some convention is needed, and establishing one can be a lot
of bother for nothing.


### 4.2 Substitutivity Failures



Classical predicate logic has the desirable feature that co-extensive
open formulas may be substituted for one another in any formula
*salva veritate*--i.e., without changing that
formula's truth value. (Open formulas \(A\) and \(B\) in \(n\)
free variables \(x\_1 ,\ldots ,x\_n\) are *coextensive* if and
only if \(\forall x\_1 \ldots \forall x\_n (A \leftrightarrow B)\) is
true.) But, as Lambert noted in 1974, this principle fails for nearly
all free logics with identity. Consider, for example, the formula
\(t=t\), where \(t\) is empty, which is an instance of the open
formula \(x=x\). Now \(x=x\) is coextensive with both \((x=x \amp
\rE!x)\) and \((\rE!x \rightarrow x=x)\), since all three formulas are
satisfied by all members of \(\bD\). Hence if co-extensive open
formulas could be exchanged *salva veritate*, \((t=t \amp
\rE!t)\) and \((\rE!t \rightarrow t=t)\) would have the same truth
value as \(t=t\). But on nearly all free logics this is not the case.
Positive free logic and the supervaluations described in
 Section 3.4
 make \(t=t\) true and \((t=t \amp \rE!t)\) false; negative free logic
makes \(t=t\) false and \((\rE!t \rightarrow t=t)\) true; and any
ordinary neutral free logic whose conditionals are true whenever their
antecedents are false makes \(t=t\) truth-valueless and \((\rE!t
\rightarrow t=t)\) true. Many find this troubling because, since
Frege, it has been widely held that (1) extensions of complex
linguistic expressions should be functions of the extensions of their
components (so that co-extensive components should be exchangeable
without affecting the extension of the whole) and (2) the extension of
a formula (or statement) is a truth value.



One possible response is to reject (2). Leeb (2006) develops for a
version of PFL a dual-domain semantics in which the extensions of
formulas are abstract states of affairs. In this semantics,
co-referential open sentences are exchangeable not *salve
veritate*, but (as he puts it) *salve extensione*; that is,
the exchange does not alter the state of affairs designated by the
statement in which it occurs. But Leeb's state-of-affairs
semantics is so complex that it may discourage application.



Those who wish to retain (2) may be consoled by the following
observation: though substitutivity *salve veritate* of
co-extensive open formulas fails for nearly all free logics, a related
but weaker principle, the substitutivity *salve veritate* of
co-*comprehensive* open formulas, is valid for positive free
logics. Open formulas \(A\) and \(B\) in \(n\) free variables \(x\_1
,\ldots ,x\_n\) are *co-comprehensive* if every assignment of
denotations in the outer domain \(\bD\_o\) to \(x\_1 ,\ldots ,x\_n\)
satisfies \(A\) if and only if it satisfies \(B\). Among the open
formulas mentioned in the previous paragraph, for example, \(x=x\) and
\((\rE!x \rightarrow x=x)\) are co-comprehensive in a dual-domain
positive free logic, being satisfied by all members of \(\bD\_o\), but
\((x=x \amp \rE!x)\) is not co-comprehensive with them, since it is
satisfied only by the members of \(\bD\). Unlike co-extensiveness,
however, co-comprehensiveness is not expressible in the language of
PFL. But it becomes expressible with the introduction of quantifiers
over the outer domain--a strategy considered in
 Section 5.5.


### 4.3 Inexpressibility of Existence Conditions



'Whatever thinks exists,' 'Any necessary being
exists', 'That which is immediately known exists':
such statements of sufficient conditions for existence are prominent
in metaphysical debates. But, somewhat surprisingly, they are not
expressible in free logic. Their apparent form is \(\forall x(A
\rightarrow \rE!x)\). But because the universal quantifier ranges just
over \(\bD\), which is also the extension of E!, this form is valid in
free logic--as it is in classical logic with \(\rE!x\) expressed
as \(\exists y\,y=x\). No statement of this form--not even
'all impossible things exist'--can be false. Hence on
free logic all such statements are equally devoid of content. Argument
evaluation suffers as a result. Consider, for example, the obviously
valid inference:




|  |  |
| --- | --- |
|  | I think. |
|  | Whatever thinks exists. |
| \(\therefore\) | I exist. |



Its natural formalization in free logic is \(Ti, \forall x(Tx
\rightarrow \rE!x) \vdash \rE!i\). But this form is invalid. To obtain
the conclusion, we must first deduce \(Ti \rightarrow \rE!i\) by
specification from the second premise and then use modus ponens with
the first. But since the logic is free, specification requires the
question-begging premise \(\rE!i\).



Unsatisfactory entailments involving "existence
hedges"--predications that entail the existence of their
objects--have recently come up in discussions of neutral free
logic in particular. Daniel Yeakel (2016, p. 379) argues that
"on any neutral free logic either some existence hedges will
entail some undesired existence claims, or they will not entail some
desired existence claims." But the example of the previous
paragraph works for negative and positive free logics as well. A
remedy is not to be found in free logic alone, but again
quantification over the outer domain of a dual-domain semantics may
help (see
 Section 5.5).


## 5. Some Applications



This section considers applications of free logic in theories of
definite descriptions, languages that allow partial or non-strict
functions, logics with Kripke semantics, logics of fiction and logics
that are in a certain sense "Meinongian." Free logic has
also found application elsewhere--most prominently in theories of
predication, programming languages, set theory, logics of
presupposition (with neutral semantics), and definedness logics. For
more on these and other applications, see Lambert 1991 and 2001b;
Lehman 2002, pp. 250-253; and Nolt 2006, pp.
1039-1053.


### 5.1 Theories of Definite Descriptions



The earliest and most extensive applications of free logic have been
to the theory of definite descriptions. A definite description is a
phrase that may be expressed in the form "the \(x\) such that
\(A\)," where \(A\) is an open formula with only \(x\) free.
Formally, this is written using a special logical operator, the
definite description operator '\(\iota\)', as \(\iota
xA\). *Contra* Russell, free logic treats definite descriptions
not as merely apparent singular terms in formulas whose logical form
is obtainable only by elaborate contextual definitions, but as genuine
singular terms. Thus, like an individual constant, \(\iota xA\) may be
attached to predicates and (under appropriate conditions) substituted
for variables. For any object \(d\) in the domain \(\bD, \iota xA\)
denotes \(d\) if and only if among all objects in \(\bD\), \(d\) and
only \(d\) satisfies \(A\). If in \(\bD\) there is more than one
object satisfying \(A\), or none, \(\iota xA\) is empty. The
description operator therefore satisfies:

 

\[\tag{LL} \forall y(y=\iota xA \leftrightarrow \forall x(A \leftrightarrow x=y)), x \text{ free in } A. \]


(This formula is sometimes called "Lambert's Law,"
though the earliest published mention of it was apparently in Hintikka
(1959), p. 83. In the context of Hintikka's system,
however, it had various unwelcome consquences (see Lambert
1962; van Fraassen 1991, pp. 8-10).) Adding (LL) to
the free logic defined by (A1)-(A6) and (A7-) gives the minimal
free definite description theory MFD. MFD is the core of virtually all
free description theories, which therefore differ only in the
additional principles they endorse.



There is plenty of room for variation, for MFD fails to specify truth
conditions for atomic formulas (including identities) when they
contain empty descriptions, and there are many ways to do it. Making
all atomic formulas containing empty descriptions false yields a
negative free description theory axiomatizable by adding (LL) to NFL
(Burge 1974, Lambert 2001h). The result is essentially Bertrand
Russell's theory of definite descriptions, but with the
description operator taken as primitive rather than contextually
defined.



The simplest *positive* free description theory makes all
identities between empty terms true. Known as FD2, it may be
axiomatized by adding (LL) and:

 

\[ ({\sim}\rE!s \amp{\sim}\rE!t) \rightarrow s=t \]


to PFL. FD2 is akin to Gottlob Frege's theory of definite
descriptions; but whereas Frege chose a single arbitrary existing
object to serve as the conventional referent for empty singular terms,
FD2 makes this object non-existent. FD2 is readily modeled in a
dual-domain positive semantics with just one object in the outer
domain.



On FD2 all empty descriptions are intersubstitutable *salve
veritate*. But this result is subject to counterexamples in
ordinary language. This statement:



The golden mountain is a possible object,



for instance, is true, while this one:



The set of all non-self-membered sets is a possible object,



is false--though each applies the same predicate phrase 'is
a possible object' to an empty description. Thus we may prefer a
more flexible positive free description theory on which identities
between empty terms may be false. The literature presents a surprising
diversity of these (Lambert 2001a, 2003c, 2003d, 2003h; Bencivenga
2002, pp. 188-193; Lehman 2002, pp. 237-250).


### 5.2 Logics with Partial or Non-Strict Functions



Some logics employ primitive *n-place function
symbols*--symbols that combine with \(n\) singular terms to
form a complex singular term. Thus, for example, the plus sign
'+' is a two-place function symbol that, when placed
between, say, '2' and '3', forms a complex
singular term, '\(2 + 3\)' that denotes the number five.
Similarly, '\(^2\)' is a one-place function symbol that,
when placed after a term denoting a number, forms a complex singular
term that denotes that number's square. Semantically, the
extension of a function symbol is a function whose arguments are
members of the quantificational domain \(\bD\), and the resulting
complex term denotes the result of applying that function to the
referents of the \(n\) component singular terms, taken in the order
listed. Since classical logic requires every singular term (including
those formed by function symbols) to refer to an object in \(\bD\),
for each such function symbol \(f\), it requires that:



\[ \forall x\_1 \ldots \forall x\_n\exists y(y = f(x\_1, \ldots, x\_n)). \]


Hence classical logic prohibits primitive function symbols whose
extensions are partial functions--functions whose value is for
some arguments undefined. Such, for example, is the binary division
sign '/', since when placed between two numerals the
second of which is '0', it forms an empty singular term.
Similarly, the limit function symbol 'lim' yields an empty
singular term when applied to the name of a non-coverging sequence.
Classical logic can accommodate function symbols for partial functions
via elaborate contextual definitions. But then (as with Russellian
definite descriptions) the form in which these function symbols are
usually written is not their logical form. Free logic provides a more
elegant solution. Because it allows empty singular terms, symbols for
partial functions may simply be taken as primitive.



In applications of free logic involving partial functions, the
existence predicate '\(\rE!\)' is often replaced by the
postfix definedness predicate '\(\downarrow\)'. For any
singular term \(t, t\downarrow\) is true if and only if \(t\) has some
definite value in \(\bD\). Thus, for example, the formula
'\((1/0)\downarrow\)' is false. While some writers (e.g.,
Feferman (1995)) distinguish '\(\downarrow\)' from
'\(\rE!\)', the literature as a whole does not, and
'\(\downarrow\)' is often merely a syntactic variant of
'\(\rE!\)'.



In addition to partial functions, *positive* free logics can
also readily handle *non-strict* functions. A non-strict
function is a function that may yield a value even if not all of its
arguments are defined. The binary function \(f\) such that \(f(x,y) =
x\), for instance, can yield a value even if the \(y\)-term is empty.
So, for example, the formula \(f(1, 1/0) = 1\) can be regarded as
true. Logics for non-strict functions must be positive because in a
negative or neutral logic empty-termed atomic formulas, such as \(f(1,
1/0) = 1\), cannot be true. Free logics involving non-strict functions
find application in some programming languages (Gumb 2001, Gumb and
Lambert 1991). Such logics may employ a dual-domain semantics in which
the referents of empty functional expressions such as
'1/0' are regarded as *error objects*--objects
that correspond in the running of a program to error messages. Thus,
for example, an instruction to calculate \(f(1, 1/0)\) might return
the value 1, but an instruction to calculate \(f(1/0, 1)\) would
return an error message.


### 5.3 Logics with Kripke Semantics



Kripke semantics for quantified
 modal logics,
 tense logics,
 deontic logics,
 intuitionistic logics, 
and so on, are often free. This is because they index
truth to certain objects that we shall call "worlds," and
usually some things that we have names for do not exist in some of
these worlds. Worlds may be conceived in various ways: they may, for
example, be understood as possible universes in alethic modal logic,
times or moments in tense logic, permissible conditions in deontic
logic, or epistemically possible states of knowledge in intuitionistic
logic. Associated with each world \(w\) is a domain \(\bD\_w\), of
objects (intuitively, the set of objects that exist at \(w)\). An
object may exist in (or "at") more than one world but need
not exist in all. Thus, for example, Kripke semantics for tense logic
represents the fact that Bertrand Russell existed at one time but
exists no longer by Russell's being a member of the domains of
certain "worlds"--that is, times (specifically,
portions of the last two centuries)--but not others (the present,
for example, or all future times). Two natural assumptions are made
here: that the same object may exist in more than one world (this is
the assumption of *transworld identity*), and that some
singular terms--proper names, in particular--refer not only
to an object at a given world, but to that same object at every world.
Such terms are called *rigid* *designators*. Any logic
that combines rigid designators with quantifiers over the domains of
worlds in which their referents do not exist must be free.



Kripke semantics gives predicates different extensions in different
worlds. Thus, for example, the extension of the predicate 'is a
philosopher' was empty in all worlds (times) before the dawn of
civilization and more recently has varied. For rigidly designating
terms, this raises the question of how to evaluate atomic formulas at
worlds in which their referents do not exist. Is the predicate
'is a philosopher' satisfied, for example, by Russell in
worlds (times) in which he does not exist--times such as the
present? The general answers given to such questions determine whether
a Kripke semantics is positive, negative or neutral.



For negative or neutral semantics, the extension at \(w\) of an
\(n\)-place predicate \(P\) is a subset of \(\bD\_w^n\). An atomic
formula can be true at \(w\) only if all its singular terms have
referents in \(\bD\_w\); if not, it is false (in negative semantics) or
truth-valueless (in neutral semantics). In a positive semantics,
atomic formulas that are empty-termed at \(w\) may nevertheless be
true at \(w\). Predicates are usually interpreted over the union
\(\bU\) of domains of all the worlds, which functions as a kind of
outer domain for each world, so that the extension of an \(n\)-place
predicate \(P\) at a world \(w\) is a subset of \(\bU^n\). Some
applications, however, require predicates to be true of--and
singular terms to be capable of denoting--objects that exist in
no world. If so, we may collect these objects into an outer domain
that is a superset of \(\bU\). (They might be fictional objects,
timeless Platonic objects, impossible objects, or the like.)



Quantified formulas, like all formulas, are true or false only
relative to a world. Thus \(\exists xA\), for example, is true at a
world \(w\) if and only if some object in \(\bD\_w\) satisfies \(A\).
Except in intuitionistic logic, where it has a specialized
interpretation, the universal quantifier is interpreted similarly:
\(\forall xA\) is true at \(w\) if and only if all objects in
\(\bD\_w\) satisfy \(A\). Kripke semantics often specify that for each
\(w, \bD\_w\) is nonempty, so that the resulting free logic is
non-inclusive--but we shall not do so.



Any of various free modal or tense logics can be formalized by adding
to a language \(\bL\) of the sort defined in
 Section 2
 the sentential operator '\(\Box\)'. If \(A\) is a
formula, so is \(\Box A\). In alethic modal logic, this operator is
read "it is necessarily the case that." More generally, it
means "it is true in all accessible worlds that," where
accessibililty from a given world is a different relation for
different modalities: possibility for alethic logics, permissibility
for deontic logics, various temporal relations for tense logics, and
so on. A typical bivalent Kripke model \(\bM\) for such a language
consists of a set of worlds, a binary accessibility relation \(\bR\)
defined on that set; an assignment to each world \(w\) of a domain
\(\bD\_w\); an "outer" domain \(\bD\_o\) of objects (which
typically is either \(\bU\) or a superset thereof); and a two-place
interpretation function \(\bI\) that assigns denotations at worlds to
individual constants and extensions at worlds to predicates. For each
individual constant \(t\) and world \(w, \bI(t,w)\in \bD\_o\). In such
a model, a singular term is a rigid designator if and only if for all
worlds \(w\_1\) and \(w\_2\), \(\bI(t,w\_1) = \bI(t,w\_2)\). For every
\(n\)-place predicate \(P, \bI(P,w) \subseteq \bD\_w^n\) if the
semantics is negative or neutral; if it is positive, \(\bI(P,w)
\subseteq \bD\_o^n\). Truth values at the worlds of a model \(\bM\) are
assigned by a two-place valuation function \(\bV\) (where \(\bV(A,w)\)
is read "the truth value \(\bV\) assigns to formula \(A\) at
world \(w\)") as follows:

 

\[\begin{aligned} \bV(Pt\_1 \ldots t\_n,w) &= \left\{\begin{array}{l} \rT \Leftrightarrow \langle\bI(t\_1,w),\ldots,\bI(t\_n,w)\rangle \in \bI(P,w); \\ \rF \text{ otherwise.} \end{array}\right. \\ \bV(s=t,w) &= \left\{\begin{array}{l} \rT \Leftrightarrow \bI(s,w)=\bI(t,w); \\ \rF \text{ otherwise.} \end{array}\right. \\ \bV(\rE!t,w) &= \left\{\begin{array}{l} \rT \Leftrightarrow \bI(t,w) \in \bD\_w; \\ \rF \text{ otherwise.} \end{array}\right. \\ \bV({\sim}A,w) &= \left\{\begin{array}{l} \rT \Leftrightarrow \bV(A,w) = \rF\\ \rF \text{ otherwise.} \end{array}\right. \\ \bV(A\rightarrow B,w) &= \left\{\begin{array}{l} \rT \Leftrightarrow \bV(A,w) = F \text{ or } \bV(B,w) = T; \\ \rF \text{ otherwise.} \end{array}\right. \\ \bV(\Box A,w) &= \left\{\begin{array}{l} \rT \Leftrightarrow \text{ for all } u \text{ such that } w\bR u, \bV(A,u) = \rT\\ \rF \text{ otherwise.} \end{array}\right. \\ \bV(\forall xA,w) &= \left\{\begin{array}{l} \rT \Leftrightarrow \text{ for all } d\in \bD\_w, \bV\_{(t,d)}(A(t/x),w) = \rT \\ \quad(\text{where } t \text{ is not in } A \text{ and } \bV\_{(t,d)} \text{ is the}\\ \quad\text{valuation function on the model just like } \bM \\ \quad\text{except that its interpretation function } \bI^\* \text{ is} \\ \quad\text{such that for each world } w, \bI^\*(t,w)= d); \\ \rF \text{ otherwise.} \end{array}\right. \end{aligned}\]


Under the stipulations that admissible models make all individual
constants rigid designators and that \(\bI(P,w) \subseteq \bD\_o^n\),
the standard free logic PFL, together with the modal axioms and rules
appropriate to whatever structure we assign to \(\bR\), is sound and
complete on this semantics.



Modal semantics thus defined call for free logic whenever worlds are
allowed to have differing domains--that is whenever we may have
worlds \(u\) and \(w\) such that \(\bD\_u \ne \bD\_w\). For in that case
there must be an object \(d\) that exists in one of these domains (let
it be \(\bD\_w)\), but not the other, so that any singular term \(t\)
that rigidly designates \(d\) must be empty at world \(u\). Hence
\({\sim}\exists x(x=t)\) (which is self-contradictory in classical
logic) must be true at world \(u\). Such a semantics also requires
free logic when \(\bD\_o\) contains objects not in \(\bU\), for in that
case rigid designators of these objects are empty in all worlds.
Finally, this semantics calls for *inclusive* logic if any
world has an empty domain. Thus, given this semantics, the only way to
make the resulting logic unfree is to require that domains be
*fixed*--i.e., that all worlds have the same domain
\(\bD\), that \(\bD\) be non-empty, and that \(\bD\_o = \bD\).



Just this trio of requirements was in effect proposed by Saul Kripke
in his ground-breaking (1963) paper on modal logic as one of two
strategies for retaining classical quantification. (The other, more
draconian, strategy was to allow differing domains but ban individual
constants and treat open formulas as if they were universally
quantified.) But such fixed-domain semantics validate the implausible
formula:

 

\[ \forall x\Box \exists y(y = x), \]


which asserts that everything exists necessarily and the equally
implausible Barcan formula:

 

\[ \forall x\Box A \rightarrow \Box \forall xA \]


(named for Ruth Barcan, later Ruth Barcan Marcus, who discussed it as
early as the late 1940s). To see its implausibility, consider this
instance: 'If everything is necessarily a product of the big
bang, then necessarily everything is a product of the big bang'.
It may well be true that everything (in the actual world) is
necessarily a product of the big bang--i.e., that nothing in this
world would have existed without it. But it does not seem necessary
that everything is a product of the big bang, for other universes are
possible in which things that do not exist in the actual world have
other ultimate origins. Because of the restrictiveness and
implausibility of fixed-domain semantics, many modal logicians loosen
Kripke's strictures and adopt free logics.



We may also drop the assumption that singular terms are rigid
designators and thus allow *nonrigid designators*. On the
semantics considered here, these are singular terms \(t\) such that
for some worlds \(w\_1\) and \(w\_2, \bI(t,w\_1) \ne \bI(t,w\_2)\).
Definite descriptions, understood attributively, are the best
examples. Thus the description "the oldest person"
designates different people at different times (worlds)--and no
one at times before people existed ("worlds" \(w\) at
which \(\bI(t,w)\) is undefined).



Nonrigid designators, if empty at some worlds, require free logics
even with fixed domains. (Thus classical logic with nonrigid
designators is possible only if we require for each singular term
\(t\) that at each world \(w\), \(t\) denotes some object in
\(\bD\_w\).) On some semantics for nonrigid designators, the quantifier
rule must differ from that given above, and other adjustments must be
made. For details, see Garson 1991, Cocchiarella 1991, Schweitzer 2001
and Simons 2001.



Intuitionistic logic,
 too, has a Kripke semantics, though special valuation clauses are
needed for '\({\sim}\)', '\(\rightarrow\)' and
'\(\forall\)' in order to accommodate the special meanings
these operators have for intuitionists, and '\(\Box\)' is
generally not used. The usual first-order intuitionistic logic, the
Heyting predicate calculus (HPC)--also called the intuitionistic
predicate calculus--has the theorem \(\exists x(x=t)\) and hence
is not free. But intuitionists admit the existence only of objects
that can in some sense be constructed, while classical mathematicians
posit a wider range of objects. Therefore users of HPC cannot
legitimately name all the objects that classical mathematicians can.
Worse, they cannot legitimately name objects whose constructibility
has yet to be determined. Yet some Kripke-style semantics for HPC do
allow use of names for such objects (semantically, names of objects
that "exist" at worlds accessible from the actual world
but not at the actual world itself). Some such semantics, though
intended for HPC, have turned out, unexpectedly, not to be adequate
for HPC. An obvious fix, advocated by Posy (1982), is to adopt a free
intuitionistic logic. For more on this issue, see Nolt 2007.


### 5.4 Logics of Fiction



Because fictions use names that do not refer to literally existing
things, free logic has sometimes been employed in their analysis. So
long as we engage in the pretense of a story, however, there is no
special need for it. It is true, for example, in Tolkien's
*The Lord of the Rings* that Gollum hates the sun, from which
we can legitimately infer that *in the story* there exists
something that hates the sun. Thus quantifiers may behave classically
so long as we consider only what occurs and what exists "in the
story." (The general logic of fiction, however, is often
regarded as *non*classical, for two reasons: (1) a story may be
inconsistent and hence require a
 paraconsistent logic,
 and (2) the objects a story describes are typically (maybe always)
incomplete; that is, the story does not determine for each such object
\(o\) and every property \(P\) whether or not \(o\) has \(P\).)



The picture changes, however, when we distinguish what is true in the
story from what is literally true. For this purpose logics of fiction
often deploy a sentence operator that may be read "in the
story." Here we shall use '\(\mathbf{S}\_x\)' to mean
"in the story \(x\)," where '\(x\)' is to be
replaced by the name of a specific story. Anything within the scope of
this operator is asserted to be true in the named story; what is
outside its scope is to be understood literally. (For a summary of
theories of what it means to be true in a story, see Woods 2006.)



With this operator the statement 'In the story, *The Lord of
the Rings*, Gollum hates the sun' may be formalized as
follows:

 

\[\tag{GHS} \mathbf{S}\_{The\: Lord\: of\: the\: Rings}(\text{Gollum
hates the sun}). \]


The statement that in *The Lord of the Rings* something hates
the sun is:

 

\[ \mathbf{S}\_{The\: Lord\: of\: the\: Rings}\exists x(x \text{ hates
the sun}). \]


This second statement follows from the first, even though Gollum does
not literally exist. But it does not follow that there exists
something such that it, in *The Lord of the Rings*, hates the
sun:

 

\[ \exists x \mathbf{S}\_{The\: Lord\: of\: the\: Rings}(x \text{ hates
the sun}). \]


and indeed that statement is not true, for, literally, Gollum does not
exist. Since the sun, however, exists both literally and in the story,
the statement:

 

\[ \exists x\mathbf{S}\_{The\: Lord\: of\: the\: Rings}(\text{Gollum
hates } x) \]


is true and follows by free existential generalization from (GHS)
together with the true premise '\(\rE!(\text{the sun})\)'.
Thus free logic may play a role in reasoning that mixes fictional and
literal discourse.



Terms for fictional entities also occur in statements that are
entirely literal, making no mention of what is true "in the
story." Consider, for example, the statement:

 

\[\tag{G} \text{Gollum is more famous than Godel.} \]


Mark Sainsbury (2005, ch. 6) holds that reference failure invariably
makes such statements false and hence that they are best represented
in a negative free logic. Others, however--including Orlando 2008
and Dumitru and Kroon 2008--question Sainsbury's treatment,
maintaining that statements like (G) are both atomic and true. If so,
they require a positive free logic. The logic must be free because it
deals with an empty singular term, and it must be positive, because
only on a positive semantics can empty-termed atomic statements be
true. One must still decide, however, whether the name
'Gollum' is to be understood as having no referent or as
having a referent that does not exist.



If 'Gollum' has no referent, then (G) might be handled by
a *single-domain* positive semantics. But that semantics would
have to treat atomic formulas non-standardly; it could not, as usual,
stipulate that (G) is true just in case the pair \(\langle\)Gollum,
Godel\(\rangle\) is a member of the extension of the predicate
'is more famous than'; for if there is no Gollum, there is
no such pair. On such a semantics 'Gollum is more famous than
Godel' would not imply that something is more famous than
Godel.



If, on the other hand, terms such as 'Gollum' refer to
non-existent objects, then those objects could inhabit the outer
domain of a dual-domain positive free logic. Dumitru (2015), for
example, lays out such a dual-domain semantics for fictional discourse
using free descriptions and compares it with a supervaluational
approach that also uses free descriptions. In a such a dual-domain
semantics, atomic formulas have their standard truth conditions: (G)
is true just in case \(\langle\)Gollum, Godel\(\rangle\) is a
member of the extension of 'is more famous than'.
Moreover, if we allow quantifiers over that outer domain, then
'Something is more famous than Godel' (where the
quantifier ranges over the outer domain) does follow from
'Gollum is more famous than Godel', though
'There *literally exists* something more famous than
Godel' (where the quantifier ranges over the inner domain)
does not. Meinongian logics of fiction employ this strategy.


### 5.5 Meinongian Logics



Alexius Meinong is best known for his view that some objects that do
not exist nevertheless have being. His name has been associated with
various developments in logic. Some free logicians use it to describe
any dual-domain semantics. For others, Meinongian logic is something
much more elaborate: a rich theory of all the sorts of objects we can
think about--possible or impossible, abstract or concrete,
literal or fictional, complete or incomplete. In this section the term
is used to describe logics stronger than the first type but possibly
weaker than the second: positive free logics with an extra set of
quantifiers that range over the outer domain of a dual-domain
semantics.



Whether such logics can legitimately be considered free is
controversial. On older conceptions, free logic forbids any
quantification over non-existing things (see Pasniczek 2001 and
Lambert's reply in Morscher and Hieke 2001, pp. 246-8).
But by anybody's definition, Meinongian logics in the sense
intended here at least *contain* free logics when the inner
domain is interpreted as the set of existing things. Moreover, on the
strictly semantic definition used in this article
 (Section 1.1),
 which is also that of Lehman 2002, whether the members of \(\bD\)
exist is irrelevant to the question of whether a logic is free. For a
defense of this definition, see Nolt 2006, pp. 1054-1057.



Historically, quantification over domains containing objects that do
not exist has been widely dismissed as ontologically irresponsible.
Quine (1948) famously maintained that existence is just what an
existential quantifier expresses. Yet nothing forces us to interpret
"existential" quantification over every domain as
expressing existence--or being of any sort. Semantically, an
existential quantifier on a variable \(x\) is just a logical operator
that takes open formulas on \(x\) into truth values; the value is T if
and only if the open formula is satisfied by at least one object in
the quantifier's domain. That the objects in the domain have or
lack any particular ontological status is a philosophical
interpretation of the formal semantics. Alex Orenstein (1990) argues
that "existential" is a misnomer and that we should in
general call such quantifiers "particular." That
suggestion is followed in the remainder of this section.



Quantifiers ranging over the outer domain of a dual-domain semantics
are called *outer quantifiers*, and those ranging over the
inner domain *inner quantifiers*. If the inner particular
quantifier is interpreted to mean "there exists" and the
members of the outer domain are possibilia, then the outer particular
quantifier may mean something like "there is possible a thing
such that" or "for at least one possible thing." We
shall use the generalized product symbol '\(\Pi\)' for the
outer universal quantifier and the generalized sum symbol
'\(\Sigma\)' for its particular dual. This notation
enables us to formalize, for example, the notoriously puzzling but
obviously true statement 'Some things don't exist'
(Routley 1966) as:

 

\[ \Sigma x{\sim}\rE!x. \]


Since in a dual-domain semantics all singular terms denote members of
the outer domain, the logic of outer quantifiers is not free but
classical. With 'E!' as primitive, the free inner
quantifiers can be defined in terms of the classical outer ones as
follows:

 

\[\begin{aligned} \forall xA &\eqdf \Pi x(\rE!x \rightarrow A) \\
\exists xA &\eqdf \Sigma x(\rE!x \amp A). \end{aligned}\]


The outer quantifiers, however, cannot be defined in terms of the
inner.



Logics with both inner and outer quantifiers have various
applications. They enable us, for example, to formalize substantive
sufficient conditions for existence and hence adequately express the
argument of
 Section 4.3,
 as follows:

 

\[ Ti, \Pi x(Tx \rightarrow \rE!x) \vdash \rE!i. \]


This form is valid. The co-comprehensiveness of open formulas \(A\)
and \(B\) in \(n\) free variables \(x\_1 ,\ldots ,x\_n\) (see
 Section 4.2),
 can likewise be formalized as:

 

\[ \Pi x\_1 \ldots \Pi x\_n (A \leftrightarrow B). \]


Richard Grandy's (1972) theory of definite descriptions holds
that \(\iota xA=\iota xB\) is true if and only if \(A\) and \(B\) are
co-comprehensive and thus is readily expressible in a Meinongian
logic. Free logics with outer quantifiers have also been employed in
logics that are Meinongian in the richer sense of providing a theory
of objects (including, in some cases, fictional objects) that is
inspired by Meinong's work (Routley 1966 and 1980, Parsons 1980,
Jacquette 1996, Pasniczek 2001, Priest 2005 and 2008, pp.
295-7).


## 6. History



Inclusive logic was conceived and formalized before free logic *per
se* was. Thus, since inclusive logic with singular terms is *de
facto* free, the inventors of inclusive logics were, perhaps
unwittingly, the inventors of free logic. Bertrand Russell suggested
the idea of an inclusive logic in (1919, p. 201, n.). Andrezej
Mostowski (1951) seems to have been among the first to formalize such
a logic. But Morscher and Simons (2001, p. 27, note 3) document
earlier discussions of the idea, and Bencivenga (2014) holds that
Jaskowski (1934) contains, at least implicitly, the first
inclusive logic. Theodore Hailperin (1953), Czeslaw Lejewski (1954)
and W. V. O. Quine (1954) made important early contributions. It was
Quine who dubbed such logics "inclusive."



Henry S. Leonard (1956) was the first to develop a free logic *per
se*, though he used a defective definition of 'E!'.
Karel Lambert began his prolific series of contributions to the field
in (1958), critiquing Leonard's definition, and then coining the
term "free logic" in (1960). The early systems of free
logic were positive. Negative free logic was developed by Rolf Schock
in a series of papers during the 1960s, culminating in (1968). Timothy
Smiley suggested the idea of a neutral free logic in (1960), but the
first thoroughgoing treatment appeared in Lehman 1994. Supervaluations
were described in Mehlberg 1958, pp. 256-260, as a device for
handling, not neutral free logic, but vagueness. But their
formalization and application to free logic began with van Fraassen
1966, in which the term "supervaluation" was introduced.
Dual-domain semantics were discussed in lectures by Lambert, Nuel
Belnap and others as early as the late 1950s, but it appears that
Church 1965 and Cocchiarella 1966 were the first published
accounts.