Classical logic requires each singular term to denote an object in thedomain of quantification—which is usually understood as the setof “existing” objects. Free logic does not. Free logic istherefore useful for analyzing discourse containing singular termsthat either are or might be empty. A term is empty if it either has noreferent or refers to an object outside the domain.

Most free logics have been first-order, their quantifiers ranging overindividuals. Recently, however, some work on higher-order free logicshas appeared. Corine Besson (2009) argues that internalist theories ofnatural kinds require second-order free logics whose quantifiers rangeover kinds, and she finds precedent for this idea ranging as far backas Cocchiarella (1986). Andrew Bacon, John Hawthorne, and GabrielUzquiano (2016) explore the possibility of using a higher-order freelogic to resolve certain intensional paradoxes, but they find thatthis idea faces daunting difficulties. Timothy Williamson (2016)reluctantly concurs. This article, however, focuses mainly onfirst-order logics.

Section 1 lays out the basics of free logic, explaining how it differsfrom classical predicate logic and how it is related to inclusivelogic, which permits empty domains or “worlds.” Section 2shows how free logic may be represented by each of three formalmethods: axiom systems, natural deduction rules and tree rules.Varying conventions for calculating the truth values of atomicformulas containing empty singular terms yield three distinct forms offree logic: negative, positive and neutral. These are surveyed inSection 3, along with supervaluations, which were developed to augmentneutral logics. Section 4 is critical, examining three anomalies thatinfect most free logics. Section 5 samples applications of free logicto theories of description, logics of partial or non-strict functions,logics with Kripke semantics, logics of fiction and logics that are ina certain sense Meinongian. Section 6 takes a glance at freelogic’s history.

• 1. The Basics
  • 1.1 Definition of Free Logic
  • 1.2 How Free Logic Differs from Classical Predicate Logic
  • 1.3 Relation of Free Logic to Inclusive Logic
• 2. Formal Systems
  • 2.1 Axiom Systems
  • 2.2 Natural Deduction Rules
  • 2.3 Tree Rules
• 3. Semantics
  • 3.1 Negative Semantics
  • 3.2 Positive Semantics
  • 3.3 Neutral Semantics
  • 3.4 Supervaluations
• 4. Generic Anomalies
  • 4.1 Problems with Primitive Predicates
  • 4.2 Substitutivity Failures
  • 4.3 Inexpressibility of Existence Conditions
• 5. Some Applications
  • 5.1 Theories of Definite Descriptions
  • 5.2 Logics with Partial or Non-Strict Functions
  • 5.3 Logics with Kripke Semantics
  • 5.4 Logics of Fiction
  • 5.5 Meinongian Logics
• 6. History
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. The Basics

1.1 Definition of Free Logic

Free logic is formal logic whose quantifiers are interpreted in theusual 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 ‘\(2 + 2\)’. Sinceclassical (i.e., Fregean) predicate logic requires that singular terms denote members of \(\bD\), free logic isa “nonclassical” logic. Where \(\bD\) is, as usual, takento be the class of existing things, free logic may be characterized aslogic 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 anabbreviation for ‘logic free of existence assumptions withrespect to its terms, singular and general’. General terms arepredicates. Lambert was suggesting that just as classical predicatelogic 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 generalizesclassical predicate logic by admitting singular terms that denote noexisting thing (‘Aphrodite’, ‘the greatestinteger’, ‘the present king of France’).

Because classical logic’s singular terms must denote existingthings (when, as usual, ‘\(\exists\)’ is read as“there exists”), classical logic is unreliable inapplication to statements containing singular terms whose referentseither do not exist or are not known to. Consider, for example, thetrue statement:

(S)

We detect no motion of the earth relative to the ether,

using ‘the ether’ as a singular term for the light-bearingmedium posited by nineteenth century physicists. The reason why (S) istrue is that, as we now know, the ether does not exist. According toclassical logic, however, (S) is false, because it implies theexistence of the ether. Free logic allows such statements to be truedespite the non-referring singular term. Indeed, it allows evenstatements of the form \({\sim}\exists x\) \(x=t\) (e.g., “theether does not exist”) to be true, though in classical logic,which presumes that \(t\) refers to an object in the quantificationaldomain, they are self-contradictory.

Free logic accommodatesempty singular terms (those thatdenote no member of the quantificational domain \(\bD)\) by rejectinginferences whose validity depends on the classical presumption thatthey must denote members of \(\bD\). Consider, for example, the ruleof universal instantiation (specification): from the premise“Every \(x\) (in \(\bD)\) satisfies \(A\)” we may infer“\(t\) satisfies \(A\).” This rule, whose formalexpression is:

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 resultof 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 principlethat from a formula \(A\) containing a singular term \(t\) we mayinfer that there is something in \(\bD\) that satisfies \(A(x//t)\),the result of replacingone or more occurrences of \(t\) in\(A\) by \(x\):

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

To distinguish terms that denote members of \(\bD\) from those that donot, free logic often employs the one-place “existence”predicate, ‘\(\rE!\)’ (sometimes written simply as‘\(\rE\)’). For any singular term \(t\), \(\rE!t\) is trueif \(t\) denotes a member of \(\bD\), false otherwise.‘\(\rE!\)’ may be either taken as primitive or (inbivalent free logic with identity) defined as follows:

Using ‘E!’ we can express classical logic’s blanketpresumption that singular terms denote members of \(\bD\) as anexplicit premise, E\(!t\), for selected terms \(t\). Thus we canformulate the following weaker analogs of universal instantiation:

and existential generalization:

which are valid in free logic.

1.3 Relation of Free Logic to Inclusive Logic

Classical predicate logic presumes not only that all singular termsrefer to members of the quantificational domain \(\bD\), but also that\(\bD\) is nonempty. Free logic rejects the first of thesepresumptions.Inclusive logic (sometimes also calledempty oruniversally free logic) rejects them both.Thus while inclusive logic for a language containing singular termsmust be free, free logics need not be inclusive.

Many existential assertions—e.g., \(\exists x(x=x),\) \(\existsx(Px \rightarrow Px),\) \(\exists x(Px \rightarrow \forallyPy)\)—are true in all nonempty domains and hence are valid inboth classical logic and non-inclusive free logic. But since allexistentially quantified formulas are false in the empty domain, noneare valid in inclusive logic. Correlatively, since all universallyquantified formulas are true in the empty domain, none areself-contradictory in inclusive logic. Even vacuously universallyquantified formulas (formulas of the form \(\forall xA\), where \(x\)is not free in \(A)\) are true in the empty domain. Hence theschema:

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

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

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

since with \(\bD\) empty and \(A(t/x)\) false, the first but not thesecond is true. Where there is need for such regularities, anon-inclusive free logic may be preferable to an inclusive one. Yetbecause 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, naturaldeduction systems and trees (or, equivalently, tableaux) are among themost common. This section presents all three for the bivalentinclusive form of free logic known as Positive Free Logic (PFL) andmentions some variants. (For the meaning of the term“positive” in this context seeSection 3.2). PFL is formulated in a first-order language \(\bL\) without sentenceletters or function symbols, whose primitive logical operators arenegation (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 thesake of definiteness that the formulas of \(\bL\) are closed (containno unquantified variables) and that they may be vacuously quantified(have the form \(\forall xA\) or \(\exists xA\), where \(x\) does notoccur free in \(A)\). An occurrence of a variable isquantified if it lies within the scope of an operator such as‘\(\forall\)’ or ‘\(\exists\)’ that binds thatvariable; otherwise it isfree.

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 classicalpropositional logic:

Once again, \(A(x/t)\) is the result of replacing all occurrences ofindividual constant \(t\) in \(A\) by the variable \(x\). If there areno such occurrences, then \(A(x/t)\) is just \(A\). In (A1) thevariable \(x\) is not free in \(A\) (since otherwise \(A\) would be anopen formula and formulas of \(\bL\) are closed). However, \(x\) maybe 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 areclassical. (A4) modifies the classical principle:

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

For languages containing the identity predicate, we also need:

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

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

if not (see Sections3.1 and3.2 below). If ‘E!’ is defined in terms of the identitypredicate as indicated inSection 1.2, then (A4) takes the form:

‘\(\rE!\)’ cannot be defined without the identitypredicate (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:

(A\(4')\), like (A4), restricts specification to objects within\(\bD\), but it uses a quantifier instead of ‘\(\rE!\)’ todo so. The quantifier permutation axiom (A\(5')\) is redundant in thepresence of the identity axioms but, as Fine proved in (1983), isindependent 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 thatcontain free variables. These logics follow one of two conventions forvariable assignments. Those that assign to each free variable a memberof \(\bD\) are calledE\(^+\)-logics; those that do not arecalledE-logics. The following specification rule is valid inE\(^+\)-logics but not in E-logics:

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

But since this article employs closed formulas, the distinctionbetween E- and E\(^+\)-logics may here be ignored. (See Williamson(1999) for an illuminating discussion of problems engendered bypermitting 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 ofpropositional logic and identity are as usual. The quantifierintroduction and elimination rules are restricted by use of thepredicate ‘\(\rE!\)’, as follows:

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 (inwhich case it requires no additional rules) or defined in terms of theidentity predicate as inSection 1.2. For non-inclusive logic, we may add a rule that introduces \(\existsx\rE!x\).

2.3 Tree Rules

Jeffrey-style tree rules (Jeffrey 1991) for PFL can be obtained byreplacing the classical rules for existentially and universallyquantified formulas with the following:

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

3. Semantics

Semantics for free logics differ in how they assign truth-values toatomic formulas that areempty-termed—i.e., contain atleast one empty singular term. There are three general approaches:

1. Negative semantics require all empty-termed atomicformulas to be false,
2. Positive semantics allow some empty-termed atomicformulas not of the formE!t to be true, and
3. Neutral (ornonvalent) semantics require allempty-termed atomic formulas not of the formE!t to betruth-valueless.

3.1 Negative Semantics

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

Let the language \(\bL\) be defined as inSection 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 toindividual 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-tupleof 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:

(The metalinguistic symbol ‘\(\Leftrightarrow\)’ means“if and only if.”) A logic adequate to this semantics maybe axiomatized by making three changes to the axioms of PFL. The firstis to add the axiom:

This expresses the convention that an atomic formula cannot be trueunless its terms refer. Second, because all empty-termed identitystatements are false on a negative semantics, (A7) is invalid and mustbe replaced by (A7-). Third, since (A2), (A3), (A-) and (A7-) togetherimply (A5), (A5) may be omitted. The resulting logic is known as NFL(Negative Free Logic). For languages with function symbols, negativefree logic requires in addition thisaxiom of strictness:

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

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

Negative free logic is also peculiar in that it validates theprinciple ofindiscernibility of nonexistents:

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

3.2 Positive Semantics

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

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

Like negative semantics, some positive semantics require each singularterm to denote either a member of \(\bD\) or nothing at all. But thenwhen a term fails to denote, the truth value of an atomic formulacontaining it cannot as usual be a function of its denotation, and theformula must be evaluated in some nonstandard way. To avoid suchirregularity and yet permit empty-termed formulas to be true, otherpositive semantics allow singular terms to denote, and predicates tobe satisfied by, nonmembers of \(\bD\). These nonmembers are collectedinto a second orouter domain \(\bD_o\), in contrast to which\(\bD\) is described as theinner domain. The result is adual-domain semantics.

Positive semantics with dual domains are generally the simplest. Themembers of the outer domain \(\bD_o\) typically represent“non-existing” things. Depending on the application, thesemay be theoretical or ideal entities, error objects (in computerscience), 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 twodisjoint. In a bivalent dual-domain semantics each singular termdenotes an object in \(\bD_o\) though possibly not in \(\bD\). Thus\(\bD\), though not \(\bD_o\), may be empty. Predicates are assignedextensions from \(\bD_o\), and the truth-values of atomic formulas(whether empty-termed or not) are computed in the usual Tarskianfashion: an atomic formula is true if and only if the \(n\)-tuple ofobjects denoted by its singular terms, taken in order, is a member ofthe 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 betrue.

More formally, a dual-domain model for a language \(\bL\) of the sortdefined inSection 2 is a triple \(\langle \bD,\bD_o,\bI\rangle\), where \(\bD\) is apossibly empty inner domain, \(\bD_o\) is a nonempty outer domain suchthat \(\bD \subseteq \bD_o\), and \(\bI\) is an interpretationfunction 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 truthvalues to atomic and quantified formulas as follows:

The clauses for ‘\({\sim}\)’ and ‘ \(\rightarrow\)’ are the same as in negative free logic. PFL with classicalidentity — that is, the logic axiomatized by (A1)–(A7)— is sound and complete with respect to this semantics (Leblancand Thomason 1968).

Dual-domain semantics have been criticized as ontologicallyextravagant. In response, some authors have advocated single-domainpositive semantics, which assign no denotation to empty singularterms. In such semantics empty-termed atomic formulas requireunconventional treatment. Typically such semantics determine thetruth-values of atomic formulas in two different ways: a Tarksi-stylecalculation for formulas whose terms all refer, and a separatetruth-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 calledproto-semantics, but more recently (2007, p. 72) he has characterizedall semantics for positive free logic as “somewhatartificial” and has questioned the logical character of freequantification in general.

3.3 Neutral Semantics

Neutral semantics make all empty-termed atomic formulas not of theformE!t truth-valueless. Truth-valueless formulas are oftensaid to have “truth-value gaps.” Neutral semantics are oftwo types: ordinary neutral semantics, which provide conventions forcalculating the truth values of complex formulas directly from theircomponents, even when there are empty terms, and supervaluationalsemantics, which calculate the truth values of complex formulas byconsidering all the values that their components could have if theirempty terms had referents. Ordinary neutral semantics will beconsidered in this section, supervaluations inSection 3.4.

The uniform policy of making all empty-termed atomic formulastruth-valueless has the advantages of plausibility and simplicity atthe atomic level, but it complicates the evaluation of complexformulas. How are the logical operators to function when some of thevalues on which they usually operate are absent? Some cases are fairlyclear. The negation of a truth-valueless formula, for example, isgenerally 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 \(A \rightarrowB\) true or truth-valueless?
• Let \(A = (B \amp C)\), where \(x\) is free in \(B, B\) be true ofsome but not all members of \(\bD\), and \(C\) be closed andtruth-valueless. Clearly this open formula is either truth-valuelessof every object in \(\bD\) or truth-valueless of some and false ofothers. In either case, is \(\exists xA\) truth-valueless orfalse?

At one extreme, we might want the operators to generate as manyplausible truth values as possible in order to validate as manyclassically valid formulas as we can. At the other, one might arrangethings so thatall empty-termed formulas are truth-valueless,which would produce a very weak logic (Lehman 2001). But however wechoose, many formulas that are valid in both classical predicate logicand the usual forms of free logic—indeed, even in propositionallogic—will become invalid. The law of noncontradiction, forexample:

is truth-valueless whenever \(A\) is (unless we make negations oftruth-valueless statements true) and hence becomes invalid. Of coursethis law and many other standard logical principles remainweaklyvalid—i.e., not false on any model—and it is possibleto construct a logic based on weak validity rather than ordinaryvalidity. But because any such logic will still be weaker thanclassical logic and because its theorems need not even be true, mostlogicians reject this strategy. For more on neutral free logic, seeLehman 1994, 2001, and 2002, pp. 233–237.

3.4 Supervaluations

Neutral semantics can be made to validate all the theorems of standardfree logics by augmenting them with supervaluations. Supervaluationswere first formalized by van Fraassen (1966). The version presentedhere is a variant of Bencivenga’s approach (1981 and 1986).

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

for example, is truth-valueless when \(t\) is nondenoting (assuming anordinary neutral semantics that makes the negation of atruth-valueless formula truth-valueless). Hence in such a semanticsthe 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 itwere, then \(Pt\) would be true. If it were not, then \(Pt\) would befalse. In either case \({\sim}(Pt \amp{\sim}Pt)\) would be true. Thus,since all possible ways of assigning referents to \(t\) agree inmaking \({\sim}(Pt \amp{\sim}Pt)\) true, we should count \({\sim}(Pt\amp{\sim}Pt)\) itself as true. In this way the law ofnoncontradiction can be preserved.

More explicitly, a supervaluation begins with a neutral model \(\bM\)with a single, possibly empty domain. We then construct the set ofcompletions of \(\bM\). These may be regarded as bivalentdual-domain positive models whose inner domain is the domain of\(\bM\), but which also have an outer domain \(\bD_o\) to providereferents for the empty terms. In each completion, singular terms thatarenonempty in \(\bM\) retain their referents, and thosethat 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. Asupervaluation of \(\bM\) is a partial assignment oftruth-values to formulas that makes a formula true if all completionsof \(\bM\) make it true, false if they all make it false, andtruth-valueless if they disagree. A formula is valid on asupervaluational semantics if and only if it is true on allsupervaluations. This semantics validates all and only the theorems ofPFL (Bencivenga 1981, Morscher & Simons 2001, pp.14–18).

Supervaluations employ what Bencivenga (1986) calls a“counterfactual theory” of truth: an empty-termedstatementis true if itwould be true on anyassignment of referents to its empty terms. This has struck manycritics as simply false. Moreover, the logic itself leaves much to bedesired. 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) \(Pt \vdash\rE!t\) isimproperly semantically valid. Therefore, althoughPFL is sound on supervaluational semantics and every semanticallyvalid formula is a theorem of PFL, not all semantically valid sequentsare provable in PFL. In fact, supervaluational consequence is notaxiomatizable by any extension of free logic. This follows from aresult of Woodruff (1984), who has shown that supervaluationalsemantics has many of the undesirable properties of second-ordersemantics. Jerry A. Fodor and Ernest Lapore (1996) argue, furthermore,that the completions needed to construct supervaluations are notmeaning-preserving. Hence, they conclude, two alleged advantages ofsupervaluations—that they explain the meaningfulness ofsentences with truth value gaps and that they allow us to preserveclassical logic—are illusory. Finally, since supervaluations arebuilt from completions that are in effect positive dual-domain models,we may wonder whether the detour through supervaluations is worth thetrouble, since positive dual-domain models alone are simpler and moreadequate to PFL.

4. Generic Anomalies

While problems noted above are specific to particular forms of freelogic, there are anomalies that infect all, or nearly all, forms. Thissection considers three: (1) a cluster of problems related to theapplication of primitive predicates to empty terms, (2) the failure ofsubstitutivitysalva veritate of co-referential expressions,and (3) the inability of free logic to express sufficient conditionsfor existence.

4.1 Problems with Primitive Predicates

In classical logic and in positive free logic any substitutioninstance of a valid formula (or form of inference) is itself a validformula (or form of inference). But in negative or neutral free logicthis is not the case. Asubstitution instance is the resultof replacing primitive non-logical symbols by possibly more complexones of the same semantic type—\(n\)-place predicates with openformulas in \(n\) variables, and individual constants with singularterms—each occurrence of the same primitive symbol beingreplaced by the same possibly complex symbol. The replacement of anoccurrence 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 theindividual constants or variables immediately following \(P\) in thatoccurrence, 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 \(A\), foreach \(i\), \(1\le i\le n\).

Let \(P\), for example, be a primitive one-place predicate. Then ifthe semantics is negative, \(Pt \rightarrow \rE!t\) is valid. But nowconsider the substitution instance \({\sim}Pt \rightarrow \rE!t\), inwhich the open formula \({\sim}Px\) is substituted for \(P\). Thissubstitution instance is false when \(t\) is empty. Hence validformulas may have invalid substitution instances. The same holds forordinary neutral semantics that make conditionals true whenever theirconsequents are true.

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

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

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

Assuming a bivalent positive semantics, which ones should we make trueand 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 ismerely to mimic with empty terms an inference pattern that holds fordenoting terms. To what extent is such mimicry justified? Suppose wedo decide to make ‘\(1/0 \le 1/0\)’ true; should wetherefore make ‘\(1/0 \gt 1/0\)’ false? There are nonon-arbitrary criteria for answering such questions. To a largeextent, of course, the answers don’t matter. There are no factshere; any consistent convention will do. But that’s just theproblem. Some convention is needed, and establishing one can be a lotof bother for nothing.

4.2 Substitutivity Failures

Classical predicate logic has the desirable feature that co-extensiveopen formulas may be substituted for one another in any formulasalva veritate—i.e., without changing thatformula’s truth value. (Open formulas \(A\) and \(B\) in \(n\)free variables \(x_1 ,\ldots ,x_n\) arecoextensive if andonly if \(\forall x_1 \ldots \forall x_n (A \leftrightarrow B)\) istrue.) But, as Lambert noted in 1974, this principle fails for nearlyall free logics with identity. Consider, for example, the formula\(t=t\), where \(t\) is empty, which is an instance of the openformula \(x=x\). Now \(x=x\) is coextensive with both \((x=x \amp\rE!x)\) and \((\rE!x \rightarrow x=x)\), since all three formulas aresatisfied by all members of \(\bD\). Hence if co-extensive openformulas could be exchangedsalva veritate, \((t=t \amp\rE!t)\) and \((\rE!t \rightarrow t=t)\) would have the same truthvalue as \(t=t\). But on nearly all free logics this is not the case.Positive free logic and the supervaluations described inSection 3.4 make \(t=t\) true and \((t=t \amp \rE!t)\) false; negative free logicmakes \(t=t\) false and \((\rE!t \rightarrow t=t)\) true; and anyordinary neutral free logic whose conditionals are true whenever theirantecedents are false makes \(t=t\) truth-valueless and \((\rE!t\rightarrow t=t)\) true. Many find this troubling because, sinceFrege, it has been widely held that (1) extensions of complexlinguistic expressions should be functions of the extensions of theircomponents (so that co-extensive components should be exchangeablewithout affecting the extension of the whole) and (2) the extension ofa formula (or statement) is a truth value.

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

Those who wish to retain (2) may be consoled by the followingobservation: though substitutivitysalve veritate ofco-extensive open formulas fails for nearly all free logics, a relatedbut weaker principle, the substitutivitysalve veritate ofco-comprehensive open formulas, is valid for positive freelogics. Open formulas \(A\) and \(B\) in \(n\) free variables \(x_1,\ldots ,x_n\) areco-comprehensive if every assignment ofdenotations in the outer domain \(\bD_o\) to \(x_1 ,\ldots ,x_n\)satisfies \(A\) if and only if it satisfies \(B\). Among the openformulas mentioned in the previous paragraph, for example, \(x=x\) and\((\rE!x \rightarrow x=x)\) are co-comprehensive in a dual-domainpositive free logic, being satisfied by all members of \(\bD_o\), but\((x=x \amp \rE!x)\) is not co-comprehensive with them, since it issatisfied only by the members of \(\bD\). Unlike co-extensiveness,however, co-comprehensiveness is not expressible in the language ofPFL. But it becomes expressible with the introduction of quantifiersover the outer domain—a strategy considered inSection 5.5.

4.3 Inexpressibility of Existence Conditions

‘Whatever thinks exists,’ ‘Any necessary beingexists’, ‘That which is immediately known exists’:such statements of sufficient conditions for existence are prominentin metaphysical debates. But, somewhat surprisingly, they are notexpressible in free logic. Their apparent form is \(\forall x(A\rightarrow \rE!x)\). But because the universal quantifier ranges justover \(\bD\), which is also the extension of E!, this form is valid infree logic—as it is in classical logic with \(\rE!x\) expressedas \(\exists y\,y=x\). No statement of this form—not even‘all impossible things exist’—can be false. Hence onfree logic all such statements are equally devoid of content. Argumentevaluation suffers as a result. Consider, for example, the obviouslyvalid inference:

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

Unsatisfactory entailments involving “existencehedges”—predications that entail the existence of theirobjects—have recently come up in discussions of neutral freelogic in particular. Daniel Yeakel (2016, p. 379) argues that“on any neutral free logic either some existence hedges willentail some undesired existence claims, or they will not entail somedesired existence claims.” But the example of the previousparagraph works for negative and positive free logics as well. Aremedy is not to be found in free logic alone, but againquantification over the outer domain of a dual-domain semantics mayhelp (seeSection 5.5).

5. Some Applications

This section considers applications of free logic in theories ofdefinite descriptions, languages that allow partial or non-strictfunctions, logics with Kripke semantics, logics of fiction and logicsthat are in a certain sense “Meinongian.” Free logic hasalso found application elsewhere—most prominently in theories ofpredication, programming languages, set theory, logics ofpresupposition (with neutral semantics), and definedness logics. Formore 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 beento the theory of definite descriptions. A definite description is aphrase 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, thedefinite description operator ‘\(\iota\)’, as \(\iotaxA\).Contra Russell, free logic treats definite descriptionsnot as merely apparent singular terms in formulas whose logical formis obtainable only by elaborate contextual definitions, but as genuinesingular terms. Thus, like an individual constant, \(\iota xA\) may beattached to predicates and (under appropriate conditions) substitutedfor variables. For any object \(d\) in the domain \(\bD, \iota xA\)denotes \(d\) if and only if among all objects in \(\bD\), \(d\) andonly \(d\) satisfies \(A\). If in \(\bD\) there is more than oneobject satisfying \(A\), or none, \(\iota xA\) is empty. Thedescription operator therefore satisfies:

(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 Lambert1962; van Fraassen 1991, pp. 8–10).) Adding (LL) tothe free logic defined by (A1)–(A6) and (A7-) gives the minimalfree definite description theory MFD. MFD is the core of virtually allfree description theories, which therefore differ only in theadditional principles they endorse.

There is plenty of room for variation, for MFD fails to specify truthconditions for atomic formulas (including identities) when theycontain empty descriptions, and there are many ways to do it. Makingall atomic formulas containing empty descriptions false yields anegative free description theory axiomatizable by adding (LL) to NFL(Burge 1974, Lambert 2001h). The result is essentially BertrandRussell’s theory of definite descriptions, but with thedescription operator taken as primitive rather than contextuallydefined.

The simplestpositive free description theory makes allidentities between empty terms true. Known as FD2, it may beaxiomatized by adding (LL) and:

to PFL. FD2 is akin to Gottlob Frege’s theory of definitedescriptions; but whereas Frege chose a single arbitrary existingobject to serve as the conventional referent for empty singular terms,FD2 makes this object non-existent. FD2 is readily modeled in adual-domain positive semantics with just one object in the outerdomain.

On FD2 all empty descriptions are intersubstitutablesalveveritate. But this result is subject to counterexamples inordinary 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 ‘isa possible object’ to an empty description. Thus we may prefer amore flexible positive free description theory on which identitiesbetween empty terms may be false. The literature presents a surprisingdiversity of these (Lambert 2001a, 2003c, 2003d, 2003h; Bencivenga2002, pp. 188–193; Lehman 2002, pp. 237–250).

5.2 Logics with Partial or Non-Strict Functions

Some logics employ primitiven-place functionsymbols—symbols that combine with \(n\) singular terms toform a complex singular term. Thus, for example, the plus sign‘+’ is a two-place function symbol that, when placedbetween, say, ‘2’ and ‘3’, forms a complexsingular 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 singularterm that denotes that number’s square. Semantically, theextension of a function symbol is a function whose arguments aremembers of the quantificational domain \(\bD\), and the resultingcomplex term denotes the result of applying that function to thereferents of the \(n\) component singular terms, taken in the orderlisted. Since classical logic requires every singular term (includingthose formed by function symbols) to refer to an object in \(\bD\),for each such function symbol \(f\), it requires that:

Hence classical logic prohibits primitive function symbols whoseextensions are partial functions—functions whose value is forsome arguments undefined. Such, for example, is the binary divisionsign ‘/’, since when placed between two numerals thesecond of which is ‘0’, it forms an empty singular term.Similarly, the limit function symbol ‘lim’ yields an emptysingular term when applied to the name of a non-coverging sequence.Classical logic can accommodate function symbols for partial functionsvia elaborate contextual definitions. But then (as with Russelliandefinite descriptions) the form in which these function symbols areusually written is not their logical form. Free logic provides a moreelegant solution. Because it allows empty singular terms, symbols forpartial functions may simply be taken as primitive.

In applications of free logic involving partial functions, theexistence predicate ‘\(\rE!\)’ is often replaced by thepostfix definedness predicate ‘\(\downarrow\)’. For anysingular term \(t, t\downarrow\) is true if and only if \(t\) has somedefinite 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 canalso readily handlenon-strict functions. A non-strictfunction is a function that may yield a value even if not all of itsarguments 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 astrue. Logics for non-strict functions must be positive because in anegative or neutral logic empty-termed atomic formulas, such as \(f(1,1/0) = 1\), cannot be true. Free logics involving non-strict functionsfind application in some programming languages (Gumb 2001, Gumb andLambert 1991). Such logics may employ a dual-domain semantics in whichthe referents of empty functional expressions such as‘1/0’ are regarded aserror objects—objectsthat correspond in the running of a program to error messages. Thus,for example, an instruction to calculate \(f(1, 1/0)\) might returnthe value 1, but an instruction to calculate \(f(1/0, 1)\) wouldreturn an error message.

5.3 Logics with Kripke Semantics

Kripke semantics for quantifiedmodal logics,tense logics,deontic logics,intuitionistic logics, and so on, are often free. This is because they indextruth to certain objects that we shall call “worlds,” andusually some things that we have names for do not exist in some ofthese worlds. Worlds may be conceived in various ways: they may, forexample, be understood as possible universes in alethic modal logic,times or moments in tense logic, permissible conditions in deonticlogic, or epistemically possible states of knowledge in intuitionisticlogic. Associated with each world \(w\) is a domain \(\bD_w\), ofobjects (intuitively, the set of objects that exist at \(w)\). Anobject may exist in (or “at”) more than one world but neednot exist in all. Thus, for example, Kripke semantics for tense logicrepresents the fact that Bertrand Russell existed at one time butexists no longer by Russell’s being a member of the domains ofcertain “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 madehere: that the same object may exist in more than one world (this isthe assumption oftransworld identity), and that somesingular terms—proper names, in particular—refer not onlyto an object at a given world, but to that same object at every world.Such terms are calledrigiddesignators. Any logicthat combines rigid designators with quantifiers over the domains ofworlds in which their referents do not exist must be free.

Kripke semantics gives predicates different extensions in differentworlds. Thus, for example, the extension of the predicate ‘is aphilosopher’ was empty in all worlds (times) before the dawn ofcivilization and more recently has varied. For rigidly designatingterms, this raises the question of how to evaluate atomic formulas atworlds in which their referents do not exist. Is the predicate‘is a philosopher’ satisfied, for example, by Russell inworlds (times) in which he does not exist—times such as thepresent? The general answers given to such questions determine whethera 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 atomicformula can be true at \(w\) only if all its singular terms havereferents in \(\bD_w\); if not, it is false (in negative semantics) ortruth-valueless (in neutral semantics). In a positive semantics,atomic formulas that are empty-termed at \(w\) may nevertheless betrue at \(w\). Predicates are usually interpreted over the union\(\bU\) of domains of all the worlds, which functions as a kind ofouter domain for each world, so that the extension of an \(n\)-placepredicate \(P\) at a world \(w\) is a subset of \(\bU^n\). Someapplications, however, require predicates to be true of—andsingular terms to be capable of denoting—objects that exist inno world. If so, we may collect these objects into an outer domainthat 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 onlyrelative to a world. Thus \(\exists xA\), for example, is true at aworld \(w\) if and only if some object in \(\bD_w\) satisfies \(A\).Except in intuitionistic logic, where it has a specializedinterpretation, 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 isnon-inclusive—but we shall not do so.

Any of various free modal or tense logics can be formalized by addingto a language \(\bL\) of the sort defined inSection 2 the sentential operator ‘\(\Box\)’. If \(A\) is aformula, so is \(\Box A\). In alethic modal logic, this operator isread “it is necessarily the case that.” More generally, itmeans “it is true in all accessible worlds that,” whereaccessibililty from a given world is a different relation fordifferent modalities: possibility for alethic logics, permissibilityfor deontic logics, various temporal relations for tense logics, andso on. A typical bivalent Kripke model \(\bM\) for such a languageconsists 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 (whichtypically is either \(\bU\) or a superset thereof); and a two-placeinterpretation function \(\bI\) that assigns denotations at worlds toindividual constants and extensions at worlds to predicates. For eachindividual constant \(t\) and world \(w, \bI(t,w)\in \bD_o\). In sucha model, a singular term is a rigid designator if and only if for allworlds \(w_1\) 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 thesemantics is negative or neutral; if it is positive, \(\bI(P,w)\subseteq \bD_o^n\). Truth values at the worlds of a model \(\bM\) areassigned by a two-place valuation function \(\bV\) (where \(\bV(A,w)\)is read “the truth value \(\bV\) assigns to formula \(A\) atworld \(w\)”) as follows:

Under the stipulations that admissible models make all individualconstants rigid designators and that \(\bI(P,w) \subseteq \bD_o^n\),the standard free logic PFL, together with the modal axioms and rulesappropriate to whatever structure we assign to \(\bR\), is sound andcomplete on this semantics.

Modal semantics thus defined call for free logic whenever worlds areallowed to have differing domains—that is whenever we may haveworlds \(u\) and \(w\) such that \(\bD_u \ne \bD_w\). For in that casethere must be an object \(d\) that exists in one of these domains (letit 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 classicallogic) must be true at world \(u\). Such a semantics also requiresfree logic when \(\bD_o\) contains objects not in \(\bU\), for in thatcase rigid designators of these objects are empty in all worlds.Finally, this semantics calls forinclusive logic if anyworld has an empty domain. Thus, given this semantics, the only way tomake the resulting logic unfree is to require that domains befixed—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 Kripkein his ground-breaking (1963) paper on modal logic as one of twostrategies for retaining classical quantification. (The other, moredraconian, strategy was to allow differing domains but ban individualconstants and treat open formulas as if they were universallyquantified.) But such fixed-domain semantics validate the implausibleformula:

which asserts that everything exists necessarily and the equallyimplausible Barcan formula:

(named for Ruth Barcan, later Ruth Barcan Marcus, who discussed it asearly as the late 1940s). To see its implausibility, consider thisinstance: ‘If everything is necessarily a product of the bigbang, then necessarily everything is a product of the big bang’.It may well be true that everything (in the actual world) isnecessarily a product of the big bang—i.e., that nothing in thisworld would have existed without it. But it does not seem necessarythat everything is a product of the big bang, for other universes arepossible in which things that do not exist in the actual world haveother ultimate origins. Because of the restrictiveness andimplausibility of fixed-domain semantics, many modal logicians loosenKripke’s strictures and adopt free logics.

We may also drop the assumption that singular terms are rigiddesignators and thus allownonrigid designators. On thesemantics considered here, these are singular terms \(t\) such thatfor some worlds \(w_1\) and \(w_2, \bI(t,w_1) \ne \bI(t,w_2)\).Definite descriptions, understood attributively, are the bestexamples. Thus the description “the oldest person”designates different people at different times (worlds)—and noone at times before people existed (“worlds” \(w\) atwhich \(\bI(t,w)\) is undefined).

Nonrigid designators, if empty at some worlds, require free logicseven with fixed domains. (Thus classical logic with nonrigiddesignators 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 quantifierrule must differ from that given above, and other adjustments must bemade. For details, see Garson 1991, Cocchiarella 1991, Schweitzer 2001and Simons 2001.

Intuitionistic logic, too, has a Kripke semantics, though special valuation clauses areneeded for ‘\({\sim}\)’, ‘\(\rightarrow\)’ and‘\(\forall\)’ in order to accommodate the special meaningsthese operators have for intuitionists, and ‘\(\Box\)’ isgenerally not used. The usual first-order intuitionistic logic, theHeyting predicate calculus (HPC)—also called the intuitionisticpredicate calculus—has the theorem \(\exists x(x=t)\) and henceis not free. But intuitionists admit the existence only of objectsthat can in some sense be constructed, while classical mathematiciansposit a wider range of objects. Therefore users of HPC cannotlegitimately name all the objects that classical mathematicians can.Worse, they cannot legitimately name objects whose constructibilityhas yet to be determined. Yet some Kripke-style semantics for HPC doallow use of names for such objects (semantically, names of objectsthat “exist” at worlds accessible from the actual worldbut not at the actual world itself). Some such semantics, thoughintended for HPC, have turned out, unexpectedly, not to be adequatefor HPC. An obvious fix, advocated by Posy (1982), is to adopt a freeintuitionistic logic. For more on this issue, see Nolt 2007.

5.4 Logics of Fiction

Because fictions use names that do not refer to literally existingthings, free logic has sometimes been employed in their analysis. Solong as we engage in the pretense of a story, however, there is nospecial need for it. It is true, for example, in Tolkien’sThe Lord of the Rings that Gollum hates the sun, from whichwe can legitimately infer thatin the story there existssomething that hates the sun. Thus quantifiers may behave classicallyso long as we consider only what occurs and what exists “in thestory.” (The general logic of fiction, however, is oftenregarded asnonclassical, for two reasons: (1) a story may beinconsistent and hence require aparaconsistent 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 thestory from what is literally true. For this purpose logics of fictionoften deploy a sentence operator that may be read “in thestory.” Here we shall use ‘\(\mathbf{S}_x\)’ to mean“in the story \(x\),” where ‘\(x\)’ is to bereplaced by the name of a specific story. Anything within the scope ofthis operator is asserted to be true in the named story; what isoutside its scope is to be understood literally. (For a summary oftheories of what it means to be true in a story, see Woods 2006.)

With this operator the statement ‘In the story,The Lord ofthe Rings, Gollum hates the sun’ may be formalized asfollows:

The statement that inThe Lord of the Rings something hatesthe sun is:

This second statement follows from the first, even though Gollum doesnot literally exist. But it does not follow that there existssomething such that it, inThe Lord of the Rings, hates thesun:

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

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 andliteral discourse.

Terms for fictional entities also occur in statements that areentirely literal, making no mention of what is true “in thestory.” Consider, for example, the statement:

Mark Sainsbury (2005, ch. 6) holds that reference failure invariablymakes such statements false and hence that they are best representedin a negative free logic. Others, however—including Orlando 2008and 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 itdeals with an empty singular term, and it must be positive, becauseonly on a positive semantics can empty-termed atomic statements betrue. One must still decide, however, whether the name‘Gollum’ is to be understood as having no referent or ashaving a referent that does not exist.

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

If, on the other hand, terms such as ‘Gollum’ refer tonon-existent objects, then those objects could inhabit the outerdomain of a dual-domain positive free logic. Dumitru (2015), forexample, lays out such a dual-domain semantics for fictional discourseusing free descriptions and compares it with a supervaluationalapproach that also uses free descriptions. In a such a dual-domainsemantics, atomic formulas have their standard truth conditions: (G)is true just in case \(\langle\)Gollum, Gödel\(\rangle\) is amember of the extension of ‘is more famous than’.Moreover, if we allow quantifiers over that outer domain, then‘Something is more famous than Gödel’ (where thequantifier ranges over the outer domain) does follow from‘Gollum is more famous than Gödel’, though‘Thereliterally exists something more famous thanGödel’ (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 donot exist nevertheless have being. His name has been associated withvarious developments in logic. Some free logicians use it to describeany dual-domain semantics. For others, Meinongian logic is somethingmuch more elaborate: a rich theory of all the sorts of objects we canthink about—possible or impossible, abstract or concrete,literal or fictional, complete or incomplete. In this section the termis used to describe logics stronger than the first type but possiblyweaker than the second: positive free logics with an extra set ofquantifiers that range over the outer domain of a dual-domainsemantics.

Whether such logics can legitimately be considered free iscontroversial. On older conceptions, free logic forbids anyquantification over non-existing things (see Paśniczek 2001 andLambert’s reply in Morscher and Hieke 2001, pp. 246–8).But by anybody’s definition, Meinongian logics in the senseintended here at leastcontain free logics when the innerdomain is interpreted as the set of existing things. Moreover, on thestrictly 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 adefense of this definition, see Nolt 2006, pp. 1054–1057.

Historically, quantification over domains containing objects that donot exist has been widely dismissed as ontologically irresponsible.Quine (1948) famously maintained that existence is just what anexistential quantifier expresses. Yet nothing forces us to interpret“existential” quantification over every domain asexpressing existence—or being of any sort. Semantically, anexistential quantifier on a variable \(x\) is just a logical operatorthat takes open formulas on \(x\) into truth values; the value is T ifand only if the open formula is satisfied by at least one object inthe quantifier’s domain. That the objects in the domain have orlack any particular ontological status is a philosophicalinterpretation of the formal semantics. Alex Orenstein (1990) arguesthat “existential” is a misnomer and that we should ingeneral call such quantifiers “particular.” Thatsuggestion is followed in the remainder of this section.

Quantifiers ranging over the outer domain of a dual-domain semanticsare calledouter quantifiers, and those ranging over theinner domaininner quantifiers. If the inner particularquantifier is interpreted to mean “there exists” and themembers of the outer domain are possibilia, then the outer particularquantifier may mean something like “there is possible a thingsuch that” or “for at least one possible thing.” Weshall use the generalized product symbol ‘\(\Pi\)’ for theouter universal quantifier and the generalized sum symbol‘\(\Sigma\)’ for its particular dual. This notationenables us to formalize, for example, the notoriously puzzling butobviously true statement ‘Some things don’t exist’(Routley 1966) as:

Since in a dual-domain semantics all singular terms denote members ofthe outer domain, the logic of outer quantifiers is not free butclassical. With ‘E!’ as primitive, the free innerquantifiers can be defined in terms of the classical outer ones asfollows:

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

Logics with both inner and outer quantifiers have variousapplications. They enable us, for example, to formalize substantivesufficient conditions for existence and hence adequately express theargument ofSection 4.3, as follows:

This form is valid. The co-comprehensiveness of open formulas \(A\)and \(B\) in \(n\) free variables \(x_1 ,\ldots ,x_n\) (seeSection 4.2), can likewise be formalized as:

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

6. History

Inclusive logic was conceived and formalized before free logicperse was. Thus, since inclusive logic with singular terms isdefacto free, the inventors of inclusive logics were, perhapsunwittingly, the inventors of free logic. Bertrand Russell suggestedthe idea of an inclusive logic in (1919, p. 201, n.). AndrezejMostowski (1951) seems to have been among the first to formalize sucha logic. But Morscher and Simons (2001, p. 27, note 3) documentearlier discussions of the idea, and Bencivenga (2014) holds thatJaśkowski (1934) contains, at least implicitly, the firstinclusive logic. Theodore Hailperin (1953), Czeslaw Lejewski (1954)and W. V. O. Quine (1954) made important early contributions. It wasQuine who dubbed such logics “inclusive.”

Henry S. Leonard (1956) was the first to develop a free logicperse, though he used a defective definition of ‘E!’.Karel Lambert began his prolific series of contributions to the fieldin (1958), critiquing Leonard’s definition, and then coining theterm “free logic” in (1960). The early systems of freelogic were positive. Negative free logic was developed by Rolf Schockin a series of papers during the 1960s, culminating in (1968). TimothySmiley suggested the idea of a neutral free logic in (1960), but thefirst thoroughgoing treatment appeared in Lehman 1994. Supervaluationswere described in Mehlberg 1958, pp. 256–260, as a device forhandling, not neutral free logic, but vagueness. But theirformalization and application to free logic began with van Fraassen1966, in which the term “supervaluation” was introduced.Dual-domain semantics were discussed in lectures by Lambert, NuelBelnap and others as early as the late 1950s, but it appears thatChurch 1965 and Cocchiarella 1966 were the first publishedaccounts.

Bibliography

• Antonelli, Gian Aldo, 2000, “Proto-Semantics for PositiveFree Logic,”Journal of Philosophical Logic, 29 (3):277–294.
• –––, 2007, “Free Quantification andLogical Invariance,”Rivista de Estetica (New Series),33 (1): 61–73.
• Bacon, Andrew, Hawthorne, John, and Uzquiano, Gabriel, 2016,”Higher-order free logic and the Prior-Kaplan paradox,“Canadian Journal of Philosophy 46(4–5):493–541.
• Bencivenga, Ermanno, 1981, “Free Semantics” inBoston Studies in the Philosophy of Science, 47: 38–41;revised version reprinted in Lambert 1991, pp. 98–110.
• –––, 1986, “Free Logics,” in D.Gabbay and F. Guenthner (eds.),Handbook of PhilosophicalLogic, vol. III:Alternatives to Classical Logic,Dordrecht: D. Reidel, pp. 373–426
• –––, 2002, “Free Logics,” in D.Gabbay and F. Guenthner (eds.),Handbook of PhilosophicalLogic, 2^nd edition, vol. 5, Dordrecht: Kluwer, pp.147–196. (This is a republication of Bencivenga 1986.)
• –––, 2014, “Jaśkowski’sUniversally Free Logic,”Studia Logica 102:1095–1102.
• Besson, Corine, 2009, “Externalism, Internalism, and LogicalTruth,”Review of Symbolic Logic, 2 (1):1–29.
• Burge, Tyler, 1974, “Truth and Singular Terms,”Noûs, 8: 309–25; reprinted in Lambert 1991, pp.189–204.
• Church, Alonzo, 1965, review of Lambert 1963 inJournal ofSymbolic Logic, 30: 103–104.
• Cocchiarella, Nino B., 1966, “A Logic of Actual and PossibleObjects” (abstract),Journal of Symbolic Logic, 31:688–689.
• –––, 1986,Logical Investigations ofPredication Theory and the Problem of Universals, Napoli, Italy:Bibliopolis.
• –––, 1991, “Quantification, Time andNecessity,” in Lambert 1991, pp. 242–256.
• Dumitru, Mircea, 2015, “A Free Logic for Fictionalism“in Iulian D. Toader, Gabriel Sandu & Ilie Pȃrvu, eds.,Romanian Studies in Philosophy of Science, Springer Verlag,pp. 149–163.
• Dumitru, Mircea and Frederick Kroon, 2008, “What to Say WhenThere Is Nothing to Talk about (Qué decir cuando no hay nada dequehablar),”Crítica: Revista Hispanoamericana deFilosofía, 40 (120): 97–109.
• Feferman, Solomon, 1995, “Definedness,”Erkenntnis, 43 (3): 295–320.
• Fine, Kit, 1983, “The Permutation Principle inQuantificational Logic,”Journal of PhilosophicalLogic, 12: 33–7.
• Fodor, Jerry A., and Ernest Lepore, 1996, “What Cannot beEvaluated Cannot be Evaluated and it Cannot be SupervaluedEither,”Journal of Philosophy, 93 (10):516–535.
• Garson, James W., 1991, “Applications of Free Logic toQuantified Intensional Logic,” in Lambert 1991, pp.111–142.
• Grandy, Richard E., 1972, “A Definition of Truth forTheories with Intensional Definite Description Operators,”Journal of Philosophical Logic, 1: 137–55; reprinted inLambert 1991, pp. 171–188.
• Gumb, Raymond D., 2001, “Free Logic in Program Specificationand Verification,” in Morscher and Hieke 2001, pp.157–93.
• Gumb, Raymond D., and Karel Lambert, 1991, “Definitions inNonstrict Positive Free Logic,”Modern Logic, 7:25–55 and 435–440 (errata).
• Hailperin, Theodore, 1953, “Quantification Theory and EmptyIndividual Domains,”Journal of Symbolic Logic, 18:197–200.
• Hintikka, Jaakko, 1959, “Towards a Theory of DefiniteDescriptions,”Analysis, 19: 79–85.
• Jacquette, Dale, 1996,Meinongian Logic: The Semantics ofExistence and Nonexistence, Berlin: Walter de Gruyter.
• –––, (ed.), 2006,Philosophy of Logic(Series: Volume 5 of theHandbook of the Philosophy ofScience), Amsterdam: Elsevier.
• Jaśkowski, Stanisław, 1934, “On the Rules ofSupposition in Formal Logic,”Studia Logica 1:5–32. Reprinted inPolish Logic 1920–1939, StorrsMcCall (ed.), Oxford, Clarendon Press, 1967, pp. 232–258.
• Jeffrey, Richard, 1991,Formal Logic: Its Scope andLimits, 3^rd edition, New York: McGraw-Hill.
• Kripke, Saul 1963, “Semantical Considerations on ModalLogic,”Acta Philosophical Fennica, 16:83–94.
• Lambert, Karel, 1958, “Notes on E!,”PhilosophicalStudies, 9: 60–63.
• –––, 1960, “The Definition of E! in FreeLogic,” inAbstracts: The International Congress for Logic,Methodology and Philosophy of Science, Stanford: StanfordUniversity Press.
• –––, 1962, “Notes on E! III: A Theory ofDescriptions,”Philosophical Studies, 13(4): 51–59.
• –––, 1963, “Existential ImportRevisited,”Notre Dame Journal of Formal Logic, 4:288–292.
• –––, 1974, “Predication andExtensionality,”Journal of Philosophical Logic, 3:255–264.
• –––, (ed.), 1991,Philosophical Applicationsof Free Logic, New York: Oxford University Press.
• –––, 2001a, “Free Logic and DefiniteDescriptions,” in Morscher and Hieke 2001, pp. 37–47.
• –––, 2001b, “Free Logics,” in LouGoble (ed.),The Blackwell Guide to Philosophical Logic,Oxford: Blackwell Publishing, pp. 258–279.
• –––, 2003a,Free Logic: SelectedEssays, Cambridge: Cambridge University Press.
• –––, 2003b, “Existential Import, E! and‘The’” in Lambert 2003a, pp. 16–32.
• –––, 2003c, “Foundations of the Hierarchyof Positive Free Definite Description Theories” in Lambert2003a, pp. 69–91.
• –––, 2003d, “The Hilbert-Bernays Theory ofDefinite Descriptions” in Lambert 2003a, pp. 44–68.
• –––, 2003e, “Nonextensionality” inLambert 2003a, pp. 107–121.
• –––, 2003f, “The Philosophical Foundationsof Free Logic” in Lambert 2003a, pp. 122–175.
• –––, 2003g, “Predication andExtensionality” in Lambert 2003a, pp. 92–106.
• –––, 2003h, “Russell’s Version ofthe Theory of Definite Descriptions” in Lambert 2003a, pp.1–15.
• Leblanc, Hughes, 1971, “Truth Value Semantics for a Logic ofExistence,”Notre Dame Journal of Formal Logic, 12:153–68.
• Leblanc, Hughes and Richmond H. Thomason, 1968,“Completeness Theorems for Some Presupposition-FreeLogics,”Fundamenta Mathematicae, 62: 125–64;reprinted in Leblanc’sExistence, Truth andProvability, Albany: State University of New York Press, 1982,pp. 22–57.
• Leeb, Hans-Peter, 2006, “State-of-Affairs Semantics forPositive Free Logic,”Journal of Philosophical Logic,35 (2): 183–208.
• Lehman, Scott, 1994, “Strict Fregean Free Logic,”Journal of Philosophical Logic, 23 (3): 307–336.
• –––, 2001, “No Input, No OutputLogic,” in Morscher and Hieke 2001, pp. 147–54.
• –––, 2002, “More Free Logic,” in D.Gabbay and F. Guenthner (eds.),Handbook of PhilosophicalLogic, 2^nd edition, vol. 5, Dordrecht: Kluwer, pp.197–259.
• Lejewski, Czeslaw, 1954, “Logic and Existence,”British Journal for the Philosophy of Science, 5 (18):104–19; reprinted in Dale Jacquette (ed.),Philosophy ofLogic: An Anthology, Oxford: Blackwell, 2002, pp.147–55.
• Leonard, H. S., 1956, “The Logic of Existence,”Philosophical Studies, 7: 49–64.
• Mehlberg, Henryk, 1958,The Reach of Science, Toronto:University of Toronto Press.
• Meyer, Robert K., Ermanno Bencivenga and Karel Lambert, 1982,“The Ineliminability of E! in Free Quantification Theory withoutIdentity,”Journal of Philosophical Logic, 11:229–231.
• Meyer, Robert K. and Karel Lambert, 1968, “Universally FreeLogic and Standard Quantification Theory,”Journal ofSymbolic Logic, 33: 8–26.
• Morscher, Edgar and Alexander Hieke (eds.), 2001,New Essaysin Free Logic: In Honour of Karel Lambert (Applied Logic Series,vol., 23), Dordrecht: Kluwer.
• Morscher, Edgar and Peter Simons, 2001, “Free Logic: AFifty-Year Past and an Open Future,” in Morscher and Hieke 2001,pp. 1–34.
• Mostowski, Andrezej, 1951, “On the Rules of Proof in thePure Functional Calculus of the First Order,”Journal ofSymbolic Logic, 16: 107–111.
• Nolt, John, 2006, “Free Logics,” in Jacquette 2006,pp. 1023–1060.
• –––, 2007, “Reference and Perspective inIntuitionistic Logic,”Journal of Logic, Language andInformation, 16 (1): 91–115.
• Orenstein, Alex, 1990, “Is Existence What ExistentialQuantification Expresses?” in Robert B. Barrett and Roger F.Gibson (eds.),Perspectives on Quine, Cambridge: Blackwell,1990, pp. 245–270.
• Orlando, Eleonora, 2008, “Names without Fictional Objects(Ficción sin metafísica),”Crítica:Revista Hispanoamericana de Filosofía, 40 (120):111–127.
• Parsons, Terence, 1980,Nonexistent Objects, New Haven:Yale University Press.
• Paśniczek, Jacek, 1998,The Logic of IntentionalObjects:A Meinongian Version of Classical Logic,Dordrecht: Kluwer.
• –––, 2001, “Can Meinongian Logic BeFree?” in Morscher and Hieke 2001, pp. 227–36.
• Posy, Carl J., 1982, “A Free IPC is a Natural Logic: StrongCompleteness for Some Intuitionistic Free Logics,”Topoi, 1: 30–43; reprinted in Lambert 1991, pp.49–81.
• Priest, Graham, 2005,Towards Non-Being, Oxford: OxfordUniversity Press.
• –––, 2008,An Introduction to Non-ClassicalLogic: From If to Is, 2^nd edition, Cambridge:Cambridge University Press.
• Quine, W. V. O., 1948, “On What There Is,”Reviewof Metaphysics, 48: 21–38; reprinted as Chapter 1 of Quine1963).
• –––, 1954, “Quantification and the EmptyDomain,”Journal of Symbolic Logic, 19:177–179.
• –––, 1963,From a Logical Point ofView, 2^nd edition, New York: Harper & Row.
• Routley, Richard, 1966, “Some Things Do Not Exist,”Notre Dame Journal of Formal Logic, 7: 251–276.
• –––, 1980,Exploring Meinong’s Jungleand Beyond, Canberra: Australian National Unversity.
• Russell, Bertrand, 1919,Introduction to MathematicalPhilosophy, New York: Simon & Schuster.
• Sainsbury, R. M., 2005,Reference without Referents,Oxford: Clarendon Press.
• Schock, Rolf, 1968,Logics without Existence Assumptions,Stockholm: Almqvist & Wiskell.
• Schweitzer, Paul, 2001, “Free Logic and Quantification inSyntactic Modal Contexts,” in Morscher and Hieke 2001, pp.69–85.
• Shapiro, Stewart, and Alan Weir, 2000,“‘Neo-Logicist’ Logic is Not EpistemicallyInnocent,”Philosophia Mathematica, 3 (8):160–189.
• Simons, Peter, 2001, “Calculi of Names: Free andModal,” in Morscher, Edgar and Alexander Hieke 2001, pp.49–65.
• Smiley, Timothy, 1960, “Sense without Denotation,”Analysis, 20: 125–135.
• van Fraassen, Bas C., 1966, “Singular Terms, Truth ValueGaps and Free Logic,”Journal of Philosophy, 63:481–95; reprinted in Lambert 1991, pp. 82–97.
• –––, 1991, “On (theX)(X =Lambert),”, in W. Spohn, et al. (eds.),Existence andExplanation:Essays Presented in Honor of Karel Lambert,Dordrecht: Kluwer, pp. 1–18.
• Williamson, Timothy, 1999, “A Note on Truth, Satisfactionand the Empty Domain,”Analysis, 59(1):3–8.
• –––, 2016, “Reply to Bacon, Hawthorne, andUzquiano,”Canadian Journal of Philosophy46(4–5): 542–547.
• Woods, John, 2006, “Fictions and their Logic,” inJacquette 2006, pp. 1061–1126.
• Woodruff, Peter W., 1984, “On Supervaluations in FreeLogic,”Journal of Symbolic Logic, 49:943–950.
• Yeakel, Daniel, 2016, “Existence Hedges and Neutral FreeLogic,”Proceedings of the Aristotelian Society, cxvi(3): 379–386.

Academic Tools

	How to cite this entry.
	Preview the PDF version of this entry at theFriends of the SEP Society.
	Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO).
	Enhanced bibliography for this entryatPhilPapers, with links to its database.

Other Internet Resources

[Please contact the author with suggestions.]

Related Entries

descriptions |fiction: fictional entities |logic: classical |logic: intuitionistic |logic: modal |logic: temporal |nonexistent objects |possible objects

Acknowledgments

The author thanks Ian Orr and Eddy Falls for help in researching thisarticle, and Karel Lambert and İskender Taşdelen for usefulcomments on, or corrections to, earlier versions.