In most general terms, free logic is concerned with names that do notdenote. Classical logic requires each singular term to denote anobject in the domain of quantification — which is usually understoodas the set of “existing” objects. Free logic does not.Free logic is therefore useful for analyzing discourse containingsingular terms that either are or might be empty. Varying conventionsfor calculating the truth values of atomic formulas containing emptysingular terms yield three distinct forms of free logic: negative,positive and neutral, which is why we commonly refer to them as freelogics (in the plural) instead.

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 a brief history and motivation of free logic,introduces basic definitions and explains how it differs from relatedlogics. The semantics for the variants of free logics are surveyed inSection 2. Section 3 presents the proof systems for the threevariants, briefly laying out the axiomatization and then focusing onthe currently favored proof-theoretic approaches. Section 4 iscritical, examining three anomalies that infect most free logics.Section 5 samples applications of free logic to theories ofdescription, logics of partial or non-strict functions, logics withKripke semantics, logics of fiction and logics that are in a certainsense Meinongian.

• 1. Brief History and Motivation
  • 1.1 Definition and Variants
  • 1.2 Related Systems
  • 1.3 Language of Free Logic
• 2. Formal Semantics for Free Logics
  • 2.1 Meinongian Semantics
  • 2.2 Supervaluational Semantics
  • 2.3 Negative Semantics
  • 2.4 Generalized Semantics
  • 2.5 Neutral Semantics
• 3. Formal Proof Systems for Free Logics
  • 3.1 Hilbert-style Systems of Free Logic
  • 3.2 Quantified Sequent Calculi
  • 3.3 Positive Free Logic
  • 3.4 Negative Free Logic
  • 3.5 Neutral Free Logic
• 4. Generic Anomalies
  • 4.1 Problems with Primitive Predicates
  • 4.2 Substitutivity Failures
  • 4.3 Inexpressibility of Existence Conditions
• 5. Some Applications
  • 5.1 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
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Brief History and Motivation

A consideration of one question regarding non-denoting names can betraced back to Aristotle (translation Barnes 1984):

It might, indeed, very well seem that the same sort of thing doesoccur in the case of contraries said with combination, ‘Socratesis well’ being contrary to ‘Socrates is sick’. Yetnot even with these is it necessary always for one to be true and theother false. For if Socrates exists one will be true and one false,but if he does not both will be false; neither ‘Socrates issick’ nor ‘Socrates is well’ will be true ifSocrates himself does not exist at all. (13b11–20)

While this passage possibly anticipates the issues free logics areconcerned with, it was in the 20th century that the modern debatebegins. In the first part of the 20th century, some inquiries weremade into a logic with possibly empty domains, orinclusivelogic (called so because it includes this possibility, Quine1954), and thusinquiry is sometimes considered a precursor to the investigations intofree logic. The section 1.2 below discusses the difference betweenfree and inclusive logics, however.

Proper modern free logics, as a family of first-order logics, cameabout as a result of examining the existence assumptions of classicallogics in the middle of the 20th century (Hintikka1959; Lambert 1967, 1997, 2001), with the name due to KarelLambert, which is short for “logic free of existence assumptionswith respect to its terms, singular and general” (we’lltighten up the previously loose talk of ‘names’ and focuson terms instead). But since in modern logic there are hardly anypresuppositions with regard to general terms (although there arenaturally some exceptions, likeBesson 2009), the first point willbe the focus of this exposition.

Consider then the following two arguments.

\[\frac{\text{Bucephalus is a horse.}}{\because \text{ There exists something that is a horse.}}\]

This sounds like a perfectly fine piece of elementary reasoning. Butthis is structurally at leastvery similar to the nextargument:

\[\frac{\text{Sleipnir does not exist.}}{ \because\text{ There exists something that does not exist.}}\]

A useful, albeit artificial, assumption that pervades the standardapproaches to quantified first order logic is that every singular termdenotes an object. Obviously, the second inference is blocked bylimiting singular terms to the denoting ones.

In opposition to this dismissive response “ intuitively,Socratescould have not existed ” (although,see possibilism-actualism debate), free logic chooses to alter thelogical rules for Existential Generalization (and related UniversalSpecification) by explicitly limiting them to the existence predicate\({E!}\).

1.1 Definition and Variants

The definitional hallmarks of a free logic are: (1) it is free ofexistential presuppositions with respect to its singular terms, (2) itis free of existential presuppositions with respect to its generalterms and finally (3) its quantifiers have existential import, or aremore broadly limited to the predicate which delineates the denotingterms, \({E!}\) (most commonly read asexistence, Leonard1956;Baaz and Iemhoff 2006; Maffezioli and Orlandelli 2019, anddefinedness, Feferman1995; Antonelli 2000).

In any case, it is crucial for the understanding of free logic that,in any of its versions, quantification is limited to the\(E!\)-predicate; whereas the particular interpretation of thispredicate may vary and, as we shall see later, this predicate does notneed to even have the time honored status of alogicalpredicate.

As stated previously, point (2) does not have any significant bearingon the discussion, while point (3) represents the limitation ofquantification to \({E!}\). Focusing now on point (1), the questionarises as to the status of atoms containing non-denoting terms, andwhat truth status is assigned to them determines the major variants offree logic.

Positive FL allows for some atoms with terms outside of\({E!}\) to remain true (at the very least self-identities). On theother handnegative FL treats all such atoms as uniformlyfalse (as was the suggestion in Aristotle’s quote above).Finallyneutral FL, assigns a third value to those atoms, andforms a subfamily of its own, depending on the interpretation of thatthird value and/or the meaning and inferential behavior of logicalsymbols.

1.2 Related Systems

Inclusive Logic

Another existence assumption that is frequently employed relates notto terms but to models — namely that the domain is non-empty. Logicswhich allow for the rejection of this assumption (i.e., allow us toinclude the empty domain) are calledinclusive. While theybear a close resemblance to free logics, these two issues areorthogonal to one another and can be considered in separation. To seethis, it is easy to considerempty logic of Hailperin(1953),which contains no singular terms and no elements of the domain. Whileobviously inclusive, this logic is not free, since there are nonon-denoting terms. The converse case of free non-inclusive logics iseasily obtained by additionally specifying this via the addition ofthe following rule to any of the free logics below:

\[\frac{{E!} t, \Gamma \Rightarrow \Delta}{\Gamma \Rightarrow \Delta} \text{NI, t fresh}\]

This is standardly taken to be the case with free logics, so this rulecan be assumed to be added to the systems discussed below.

Classical/Intuitionistic Logic

While most of the presentation here centers on the free logics arisingform the extension of classical propositional base, considerable workhas also gone into investigation of those with intuitionistic bases.Considerations discussed below apply,mutatis mutandis alsofor those bases. In either case the simplest way of obtaining anon-free logic from a free one is simply the addition of the rule

\[\frac{{E!} t, \Gamma \Rightarrow \Delta}{\Gamma \Rightarrow \Delta}\text{NI}^+\]

Notice that in contrast to the previous version of the rule whichrequired anarbitrary term be used, any term can be usedhere. Therefore, in effect this rule eliminates non-denoting terms (itstates, read bottom up, that in any proof search we may add theassumption a term denotes). Important to note is that theintuitionistic version of the rule contains at most one formulas in\(\Delta\), but is otherwise of the same form.

Quasi-free Logic

While the name is recent and due to Indrzejczak(2021a), the logics ofthese type have been considered for a while, with the most notableexample the Logic of Partial Terms (Feferman1995). This logic liessomewhere between classical (mutatis mutandis,intuitionistic) and free logics by, in effect, adding the ruleNI\(^+\) for singular terms, but remaining free for functionalexpressions. Feferman specifically discusses a negative freeextension, but a useful overview of multiple versions (and theiruniform cut elimination) is available in (Indrzejczak2021a).

1.3 Language of Free logic

For the formal presentation of free logics we utilize the language\(\mathcal{L}\), a standard first-order language (without functions),adapted from (Gratzl2010), with the vocabulary defined as

Aformula of \(\mathcal{L}\) is then defined as

2. Formal Semantics for Free Logics

In this section we lay out common semantical approaches to freelogics. We commence with a formal presentation of the three(well-studied) semantical approaches for positive and negative freelogic.

2.1 Meinongian Semantics

Theinner-outer domain, orMeinongian semantics, hasbeen devised to study meta-theoretical properties of positive freelogic. The main ingredients for it are a structure and a valuation,laid out in the following two definitions:

We lay out several examples of (logically) true statements, as well asone of non-entailment.

2.2 Supervaluational Semantics

The supervaluational approach to semantics offers means for reducingthree-valued semantics to a positive one. It is more involved, but hassome advantages over Meinongian semantics given that it does not makeuse of an outer-domain, which can be viewed as ontologically dubious.The price for this is at least more formal complexity and, furthermorethat meta-theoretically it is not compact in the sense of thecompactness theorem; more on this in Example 2 and remark. Theapproach of fully retaining three-valued semantics is explored in thesection on neutral free logic later in this section.

One of the goals of supervaluational semantics is to save the laws ofclassical logic, such as \(A\vee \neg A\), \(\neg(A \wedge \neg A)\)while preserving the spirit of free logic, that is allowing for emptysingular terms and quantification which is tied to the\(E!\)-predicate. This is achieved by setting up a multi-stageprocess, that commences with the following definition:

A stage-2 structure over a stage-1 structure is defined asfollows:

Based on an \(\mathcal{S}_{s2}\)-structure which is in turn acompletion (Morscherand Simons 2001a) of an\(\mathcal{S}_{s2}\)-structure, we define a valuation \({\mathcalV}^{s2}\) as follows:

The remaining definitional clauses for the Boolean cases andexistentially quantified formulas follow as usual. Now with this athand, we can define asupervaluation \(\sigma\) which isbased on a stage-1 structure and its stage-2 superstructures; in orderto indicate the dependence of \(\sigma\) with respect to \({\mathcalV}^{s1}\), we write \(\sigma_{{\mathcal V}^{s1}}\), and define:

Since this semantics is more complicated we define core semanticalnotions in detail and provide some examples.

Next we offer several examples as a way of elaborating on thesedefinitions.

Some remarks about the above examples are in order.

(1) Items 2 and 3 show that strong completeness is not available forlogical superconsequence, and with it that compactness fails.

(2) As the previous example highlights the logical consequencerelation used in supervaluational semantics isglobal asopposed to the more usuallocal. Local consequence statesthat \(\Gamma\) entails \(A\) just in case every model that makes allof \(\Gamma\) true, likewise makes \(A\) true — the quantifier takesthe wide scope. By contrast, global consequence states that if everymodel makes all of \(\Gamma\) true, then every model makes \(A\) true-the implication takes the wide scope instead.

2.3 Negative Semantics

Let us now turn to a negative structure, which is defined asfollows:

2.4 Generalized Semantics

The approach of utilizinggeneralized semantics wasintroduced by Pavlovićand Gratzl (2021) to facilitate meta-theory. In a nutshell, theidea of this approach is that for the proofs of soundness andcompleteness, adescription of a model, rather than afull-blown model, will suffice. This allows for a simplifiedpresentation of semantics (eschewing partial functions) and, given thatit has a greater level of generality, can be applied to differentparticular semantic approaches. Great use of this feature ofgeneralized semantics was made to offer a uniform picture of a varietyof positive and negative free logics.

A positive structure is defined as

The clause \(\varphi({E!})\subseteq \cD\) can be omitted,since \({E!}\) is defined like any other unary predicate by the lastclause of the definition, and not specified as a logical predicate. Asanother consequence of this definition, if \(s=t\) then \({E!} s\) iff\({E!} t\).

To create a model we attach to these structures a valuation function,propositionally classical while incorporating free quantification.

By way of comparison, from these two structures we can likewise defineaclassical structure, namely a structure validating, underthe valuation above, precisely all of classical first order logic:

2.5 Neutral Semantics

A similar investigation is offered for neutral free logics based onstrong (Kleene1938) and weak Kleene conditional. For simplicity thepresentation here is limited to negation and implication,identity-free segment.

Even though there are multiple possible neutral free logics, there isgood reason to see it as a single kind of logic, since the treatmentof quantifiers does not vary between these interpretations, and thesystems under consideration here vary only the type of implicationused.

Notice that the neutral structure here is presented as a negativestructure in order to simplify the presentation of the valuationsbelow.

When it comes to supervaluations of this generalized semantics, acompletion of a neutral structure, \(\varphi '\), is identical to\(\varphi\) it completes, except that \(\varphi'(P^n) \subseteq\varphi'(D)\) (therefore making it a positive structure) such that\(\varphi(P^n) \subseteq \varphi'(P^n)\), and the valuations for atomsdo not contain the case for \(+\) (thus making it moot for logicalsymbols).

We again offer several examples, to illustrate both the functioning ofthe two systems, and also their differences.

3. Formal Proof Systems for Free Logics

Traditional presentation of proof-systems for free logic has moreoften been in Hilbert, or axiom, style and due to this historical factwe first introduce some formalizations of variants of free logic byusing this style. However, further types of proof systems have beendeveloped which facilitated fruitful research in this area; so, mostof this section is devoted to these more useful proof systems. Beforestating systems for Hilbert-style calculi of free logic, we recallsuch a system for classical logic.  It and the rest of thesystems presented here are formulated in a first-order language\(\bL\) without sentence letters or function symbols, whose primitivelogical operators are negation (not) ‘\({\neg}\)’, theconditional (if-then) ‘\(\rightarrow\)’, the universalquantifier (for all) ‘\(\forall\)’, identity‘=’ and ‘E!’, the others being defined asusual. We assume for the sake of definiteness that the formulas of\(\bL\) are closed (contain no unquantified variables) and that theymay be vacuously quantified (have the form \(\forall xA\) or \(\existsxA\), where \(x\) does not occur free in \(A)\). An occurrence of avariable isquantified if it lies within the scope of anoperator such as ‘\(\forall\)’ or‘\(\exists\)’ that binds that variable; otherwise it isfree. Each axiom system hasmodus ponens as the solerule of inference.

3.1 Hilbert-style Systems of Free Logic

In contrast, the Hilbert-style systems for free logics are below,starting with one for the positive variant, PFL. Note in particularAxiom (P4).

Next is a Hilbert-style calculus for negative free logic, NFL.Compared to the one for PFL, note Axiom (N6) and the added Axiom(N9).

For definitions ofderivability, provability, and furthersyntactic concepts and theorems, e.g. deduction theorem, see the entryonclassical logic.

Some theorems. We exhibit some interesting theoremsof free logic, beginning with with positive free logic andHintikka's Law: \(E!s \leftrightarrow \exists x(x = s)\) Theproof from left to right is straightforward:

On the other hand, the proof from right to left is slightly moreconvoluted:

Now moving to negative free logic, we have the stronger claim of

as a theorem, since: \(E!t \to t = t\) is an axiom of type N6, and\(t= t \to E!t\) an axiom of type N9.

In contrast to positive free logic, NFL also proves theindiscernibility of non-existents:

Moreover, indiscernibility of non-existents generalizes to \(\neg E!s\wedge \neg E!t \to (A[s] \to A[t])\) by induction on the complexity of\(A\).

Clearly this formula is not a theorem of PFL as an application of thesoundness theorem shows: let's construct a Meinongian structure with\(D_i = \varnothing,\) and \(D_o = \{0, 1\}\) with \(\varphi(s) = 0,\)and \(\varphi(t) = 1,\) and \(\varphi(P) = \{0\}.\) Then \({\mathcalV}(\neg E!s \wedge \neg E!t \to (P[s] \to P[t])) = \bot,\) and so PFL\(\not \vdash \neg E!s \wedge \neg E!t \to (P[s] \to P[t]).\)

Finally, that truth implies the existential claim likewise holds innegative free logic:

Collapse into classical logic. It is quiteinstructive to ask what happens if positive and negative free logicare combined. Semantically, givenDefinition 3 andDefinition 11 and their respective valuations, the answer might beelusive. However, just like with the generalized semantics,syntactically the answer is straightforward. So, the axioms and rulesof inference are as for PFL and NFL taken jointly.

\(E!t \to (\forall x A \to A)\) follows from axiom 3 and logic, in thelight of NFL-theorem \(E!t \leftrightarrow t = t\), we obtain: \(t=t\to (\forall x A \to A)\) so by PFL axiom P6, the desired result, i.e.\(\forall x A \to A\) follows.

(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.

3.2 Quantified Sequent Calculi

Most of the recent representations of a proof system for free logicshas been in the form of sequent calculi. These approaches are favoredfor their transparency arising from locality — all the relevantinformation for evaluating an inference step is contained locally atwithin the point of application, as well as modularity — extensions donot alter the properties of the base system. For an overview of prooftheory in general see the entry onproof theory, and for its history and the modern developments we utilize here see the entry onthe development of proof theory.

The basic unit of a sequent calculus is asequent \(\Gamma\Rightarrow \Delta\), where \(\Gamma\) and \(\Delta\) are finitemultisets of (closed) formulas, intuitively understood as ``ifeverything left of the arrow holds, then something on the right of thearrow holds''. All the formulas except \(\Gamma\) and \(\Delta\) inthe schematic presentation below are calledactive formulasof the rule if they occur only in the upper sequent(s) andprincipal if they occur in the lower sequent of the rule.

In this section we will present the sequent calculi for the bivalentpositive and negative free logics with and without identity. The baseis the classical propositional calculus G3cp:

Initial sequent: \(P,\Gamma \Rightarrow \Delta, P\),where \(P\) is prime

To obtain the first order quantified calculus G3c, one would add therules for quantifiers:

To add identity, we extend two more rules, for reflexivity of identity(Ref) and the replacement principle (Repl):

The replacement rule is defined only for primes, but it suffices toprove all instances of replacement, so no generality is lost here. Atthe same time, the rule as presented has a geometric form, which meansthat its addition in the present form will be modular, not upsettingthe properties of the base. All of these rules taken togetherconstitute what is most commonly thought of as G3c.

3.3 Positive Free Logic

To obtain a sequent calculus G3pf for positive free logic, we replacethe classical quantifier rules by free quantifier rules, which limitquantification to \({E!}\):

This substitution will suffice to demonstrate two axioms whichcharacterize free logic, of Restricted generalization (RG) andRestricted specification (RS):

\[\forall x {E!} x \tag{RG}\]

\[\forall x A \to ({E!} t \to A[t/x])\tag{RS}\]

The first of these principles corresponds to the rule R\(\forall\),and the second to the rule L\(\forall\). RG states that (free)quantifiers apply to all, and RS that they apply only to, theinstances of \({E!}\). By contrast, the principle of Unrestrictedspecification holds for classical FOL, but not free logics:

\[\forall x A\to A[t/x] \tag{US}\]

One can easily see that this will not be derivable using the freerules due to the presence of the extra condition \(E! t\) in theconclusion of the rule.

These observations on the functioning of the quantifier rules willhold for both positive free logic and negative free logic, but thelatter needs to be distinguished by further modifications we outlinein the next section.

3.4 Negative Free Logic

To obtain a sequent calculus G3nf for negative free logic, we extendthe propositional base of G3cp with the free quantifier rules, muchlike in the positive case (hence the observations of the previoussection still hold), but instead of the standard identity rules we addthe following relational rules:

Relational rules:

\[\frac{{E!} t, P[t], \Gamma\Rightarrow\Delta}{P[t], \Gamma\Rightarrow\Delta}{E!} \qquad \frac{t=t, P[t], \Gamma\Rightarrow\Delta}{P[t], \Gamma\Rightarrow\Delta} =_{Ref}\]\[\frac{s=t, P[s], P[t], \Gamma \Rightarrow \Delta}{s=t, P[t], \Gamma \Rightarrow \Delta}=_{Repl}\]

Notice that the rule for reflexivity of identity has been weakened,now to apply only to terms occurring within atoms. Since the rule for\(E!\) is similarly restricted, we get that \(E! t\) iff \(t = t\).Therefore, instead of the classical reflexivity of identity, we getthe restricted version (remember that free quantification isrestricted):

\[\forall x (x=x) \tag{RRef}\]

All the atomic formulas (even self-identities) are restricted to\(E!\). Therefore, the following principle, that truth implies\({E!}\) (TIE), is characteristic of negative free logic:

\[P[t] \to {E!} t \tag{TIE}\]

Notice that this will not hold for positive free logic when \(P[t]\)is \(t=t\).

Since all the relational rules once again follow the geometric ruleschema, modularity is once again in effect and their additions do notalter the structural properties of the base system.

Positive free logic is the most widely used of the free logics, atleast in part because it represents the least intervention into thefamiliar quantified first-order logic. Compared to it, negative freelogic adds a rule while weakening another, meaning that neither is aproper part of the other. As one consequence, not too surprising ifone observes the form of their respective rules, \({E!} t\) isequivalent to self-identity, \(t=t\), making it also straightforwardlyexpressible in an existence-free language.

3.5 Neutral Free Logic

The proof-theoretic presentation of the system for neutral free logichere, due to Pavlovićand Gratzl (2023) starts off from the sequent calculus for weakKleene logics from (Fjellstad2020). The system there is a five-sided calculus, with thefifth side introduced to account for crispness of formulas (formulasarecrisp when they are either true or false), specificallyin order to deal with the falsity conditions of the universalquantifier (in weak Kleene logics it requires all instances to becrisp).

However, in neutral free logic, quantification is limited to theextension of the predicate \({E!}\), which is itself always crisp(subsequently, \(P(t_1...t_n)\) is crisp iff \({E!} t_i\) for \(1\leqi\leq n\)). As a result, if for some \(t_i\) such that \({E!} t_i\)the instantiated formula \(A[t_i/x]\) is not crisp, then for any such\(t_i\) it is not crisp. Therefore, it follows from \(A[t_i/x]\) beingfalse that it is crisp for every \(t_i\) s.t. \({E!} t_i\), andtherefore the additional crispness condition is not required.

The basic building block of the system is a (dual-)sequent of theform

\[\Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta'\]

where \(\Gamma\), \(\Gamma'\), \(\Delta\) and \(\Delta'\) are againmultisets, which is notational variation of the generalizedpropositional sequent calculi for weak and strong Kleene in (Indrzejczak2021b) (seealso Bochman 1998;Degauquier 2016), extended to quantification.

The vertical bar ‘\(|\)’ is a structural comma (i.e. astructural conjunction between antecedents and a structuraldisjunction between succedents).

In an implication format, the reading of this sequent is “ifeverything in \(\Gamma\) is true and everything in \(\Gamma'\)non-false, then either something in \(\Delta\) is non-false orsomething in \(\Delta'\) is true”, while in thenegative-conjunction reading it states “it is not the case thateverything in \(\Gamma\) is true, everything in \(\Delta\) is false,everything in \(\Gamma'\) is non-false and everything in \(\Delta'\)is non-true”.

Quantifier rules:

\[\begin{aligned} \frac{{E!} t, \forall x A, A[t/x],\Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta'}{{E!} t, \forall x A, \Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta'} L\forall &\\&\\ \frac{{E!} t, \Gamma\ |\ \Gamma' \Rightarrow A[t/x], \Delta\ |\ \Delta'}{\Gamma\ |\ \Gamma' \Rightarrow \forall x A, \Delta\ |\ \Delta'}R\forall^* &\\&\\ \frac{{E!} t,\Gamma \ |\ \forall x A, A[t/x], \Gamma' \Rightarrow \Delta\ |\ \Delta'}{{E!} t,\Gamma \ |\ \forall x A, \Gamma' \Rightarrow \Delta\ |\ \Delta'}L'\forall&\\&\\ \frac{{E!} t, \Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta',A[t/x]}{ \Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta',\forall x A}R'\forall^*&\end{aligned}\]

\({E!}\) rules:

\[\begin{aligned}&\frac{{E!} t,P[t],\Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta'}{P[t],\Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta'}L{E!} && \frac{{E!} t,\Gamma\ |\ \Gamma' \Rightarrow P[t],\Delta\ |\ \Delta'}{\Gamma\ |\ \Gamma' \Rightarrow P[t],\Delta\ |\ \Delta'}R{E!} \\& &&\end{aligned}\]\[\begin{aligned}& \frac{\{{E!} t_{i}\}_{1\leq i\leq n},P(t_1 ... t_n),\Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta'}{\{{E!} t_{i}\}_{1\leq i\leq n},\Gamma\ |\ \Gamma', P(t_1 ... t_n)\Rightarrow\Delta\ |\ \Delta'}L'{E!} &&\\&&&\\&\frac{\{{E!} t_{i}\}_{1\leq i\leq n},\Gamma\ |\ \Gamma' \Rightarrow P(t_1 ... t_n),\Delta\ |\ \Delta'}{\{{E!} t_{i}\}_{1\leq i\leq n}, \Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta', P(t_1 ... t_n)}R'{E!} &&\\&&&\end{aligned}\]\[\begin{aligned}& \frac{{E!} t, \Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta' }{\Gamma\ |\ \Gamma', {E!} t\Rightarrow \Delta\ |\ \Delta'}L\text{Tr}_{{E!}} && \frac{\Gamma\ |\ \Gamma' \Rightarrow {E!} t, \Delta\ |\ \Delta'}{\Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta', {E!} t}R\text{Tr}_{{E!}}\end{aligned}\]

Where \(P\) is atomic; \(P[t]\) is an atom other than \({E!} t\)containing \(t\); \(P(t_1 ... t_n)\) is an atom other than \({E!} t\)containing precisely terms \(t_1 ... t_n\) and in that order; \(t\) isfresh in rules marked with \(*\).

Rules L\({E!}\) and R\({E!}\) tell us, in parallel to the rule\({E!}\) of negative free logic, that in true atoms all terms are\({E!}\), but also that allfalse atoms likewise contain onlyterms that are \({E!}\). So, all crisp atoms contain only denotingterms. Conversely, rules L’\({E!}\) and R’\({E!}\) tell usthat atoms containing only denoting terms are crisp (true or false,respectively). Finally, transfer rules L\(Tr_{E!}\) and R\(Tr_{E!}\)tell us that \({E!}\) atoms themselves are always crisp.

Next, we proceed with identifying the propositional base to attachthese rules to. Two obvious candidates identified in the literature(Priest 2008) areweak Kleene (Bochvar andBergmann 1981) and strong Kleene (Kleene1938) logics, of which the formertends to be more prevalent.

Weak Kleene base

In discussing approaches to neutral free logic(s), Lehmann(1980, 1994, 2002) argues as follows:

The underlying semantic rationale for neutral free semantics isFrege’s functional view of reference: predicates and‘=’ name functions from individuals to truth-values. Iffunctions are operations, as Frege seems to have thought, then thesemantic rules governing subject-predicate and identity constructionsare [such that] where there is no input to an operation, there is nooutput either. The truth-functional connectives name truth-functions,so the same line of thought dictates the weak tables for them. (Lehmann2002, 234)

Following this reasoning the first sequent calculus we observe, G3wnf,will be obtained by adding to the free quantifier and relational rulesweak Kleene base. To make things more manageable, here we restrictourselves to the negation and implication fragment of it:

Initial sequents (is):

\[P, \Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta', P \tag{\(is_1\)}\]\[ \Gamma\ |\ \Gamma', P\Rightarrow P,\Delta\ |\ \Delta'\tag{\(is_2\)} \]\[ P, \Gamma\ |\ \Gamma' \Rightarrow P, \Delta\ |\ \Delta'\tag{\(is_3\)}\]

Propositional rules:

\[\begin{aligned}\frac{\Gamma\ |\ \Gamma'\Rightarrow A,\Delta\ |\ \Delta'}{\neg A, \Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta'}L\neg && \frac{A, \Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta'}{\Gamma\ |\ \Gamma' \Rightarrow \neg A, \Delta\ |\ \Delta'}R\neg\\ &&\\ \frac{\Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta', A}{\Gamma\ |\ \Gamma', \neg A\Rightarrow \Delta\ |\ \Delta'}L'\neg && \frac{\Gamma\ |\ \Gamma', A \Rightarrow \Delta\ |\ \Delta'}{\Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta',\neg A}R'\neg\\ &&\end{aligned}\]

\[\begin{aligned}&\small \frac{A,B, \Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta' \qquad \Gamma\ |\ \Gamma' \Rightarrow A, B, \Delta\ |\ \Delta'\qquad B, \Gamma\ |\ \Gamma' \Rightarrow A, \Delta\ |\ \Delta'}{A\to B, \Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta'}\normalsize L\to\\&\end{aligned}\]\[\begin{aligned}\normalsize\frac{A,\Gamma\ |\ \Gamma' \Rightarrow B, \Delta\ |\ \Delta'}{\Gamma\ |\ \Gamma' \Rightarrow A\to B,\Delta\ |\ \Delta'}R\to&\\&\\\frac{\Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta', A\qquad \Gamma \ |\ B, \Gamma'\Rightarrow \Delta\ |\ \Delta'}{\Gamma \ |\ A\to B, \Gamma'\Rightarrow \Delta\ |\ \Delta'}L'\to&\\&\end{aligned}\]\[\begin{aligned}&\small\frac{\Gamma\ |\ \Gamma', A\Rightarrow B,\Delta\ |\ \Delta'\qquad \Gamma\ |\ \Gamma', A\Rightarrow \Delta\ |\ \Delta', A\qquad \Gamma\ |\ \Gamma', B\Rightarrow \Delta\ |\ \Delta', B}{\Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta', A\to B}\normalsize R'\to\end{aligned}\]

Notice that adding the fourth potential initial sequent,

\[\Gamma\ |\ \Gamma',P(t_1 ... t_n)\Rightarrow \Delta\ |\ \Delta',P(t_1 ... t_n)\tag{\(is_4^*\)}\]

would collapse the system into a bivalent one, since it would statethat it is not the case that an atom is neither true nor false (or, ifan atom is non-false then it is true on the implicational reading),effectively blocking the third value. However, a restricted version ofthat possibility,

\[\{{E!} t_{i}\}_{1\leq i\leq n},\Gamma\ |\ \Gamma',P(t_1 ... t_n)\Rightarrow \Delta\ |\ \Delta',P(t_1 ... t_n)\tag{\(is_4\)}\] is derivable in the present system. The restrictedversion tells us that it is not the case that an atom is neither truenor falseif all the terms therein denote. The same reasoningwill likewise hold for the next system.

Strong Kleene base

On the other hand, \(G3_{snf}\) is obtained by selecting as thepropositional base strong Kleene logic K\(_3\). Note that the systemspresented actually differ only in two rules, with \(G3_{snf}\) presentone containing no three-premise rules:

Initial sequents (is): \[\tag{\(is_1\)} P, \Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta', P\]\[\tag{\(is_2\)} \Gamma\ |\ \Gamma', P\Rightarrow P,\Delta\ |\ \Delta'\]\[ \tag{\(is_3\)} P, \Gamma\ |\ \Gamma' \Rightarrow P, \Delta\ |\ \Delta'\]

Propositional rules:

\[\begin{aligned}\frac{\Gamma\ |\ \Gamma'\Rightarrow A,\Delta\ |\ \Delta'}{\neg A, \Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta'}L\neg && \frac{A, \Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta'}{\Gamma\ |\ \Gamma' \Rightarrow \neg A, \Delta\ |\ \Delta'}R\neg\\ &&\\ \frac{\Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta', A}{\Gamma\ |\ \Gamma', \neg A\Rightarrow \Delta\ |\ \Delta'}L'\neg && \frac{\Gamma\ |\ \Gamma', A \Rightarrow \Delta\ |\ \Delta'}{\Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta',\neg A}R'\neg\end{aligned}\]\[\begin{aligned}\frac{\Gamma\ |\ \Gamma' \Rightarrow A, \Delta\ |\ \Delta'\qquad B, \Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta'}{A\to B, \Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta'} L\to &&\\&&\\\frac{A,\Gamma\ |\ \Gamma' \Rightarrow B, \Delta\ |\ \Delta'}{\Gamma\ |\ \Gamma' \Rightarrow A\to B,\Delta\ |\ \Delta'}R\to&&\\&&\\\frac{\Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta', A \qquad \Gamma \ |\ B, \Gamma'\Rightarrow \Delta\ |\ \Delta'}{\Gamma \ |\ A\to B, \Gamma'\Rightarrow \Delta\ |\ \Delta'}L'\to&&\\&&\\ \frac{\Gamma\ |\ \Gamma', A\Rightarrow \Delta\ |\ \Delta', B}{\Gamma\ |\ \Gamma' \Rightarrow \Delta\ |\ \Delta', A\to B}R'\to &&\end{aligned} \]

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 predicateswith open formulas in \(n\) variables, and individual constants withsingular terms—each occurrence of the same primitivesymbol being replaced by the same possibly complex symbol. Thereplacement of an occurrence of a primitive \(n\)-place predicate\(P\) in some formula \(B\) by an open formula \(A\) with freevariables \(x_1 ,\ldots ,x_n\) is performed as follows: where \(t_1,\ldots ,t_n\) are the individual constants or variables immediatelyfollowing \(P\) 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 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 supervaluations 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 logicwhose conditionals are true whenever their antecedents are false makes\(t=t\) truth-valueless and \((\rE!t \rightarrow t=t)\) true. Manyfind this troubling because, since Frege, it has been widely held that(1) extensions of complex linguistic expressions should be functionsof the extensions of their components (so that co-extensive componentsshould be exchangeable without affecting the extension of the whole)and (2) the extension of a formula (or statement) is a truthvalue.

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 \(\rE!x\)expressed as \(\exists y\,y=x\). No statement of thisform—not even ‘all impossible thingsexist’—can be false. Hence on free logic all suchstatements are equally devoid of content. Argument evaluation suffersas a result. Consider, for example, the obviously valid 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 oftheir objects—have recently come up in discussions ofneutral free logic in particular. Daniel Yeakel (2016, p. 379) arguesthat “on any neutral free logic either some existence hedgeswill entail some undesired existence claims, or they will not entailsome desired 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 intheories of predication, 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 Definite Descriptions

Russellian approach

The most influential theory of definite descriptions of the first halfof the 20th century was developed by Bertrand Russell in his “OnDenoting” and was later incorporated (with more technicalapparatus) into thePrincipia Mathematica (Whiteheadand Russell1910). Russell thought of definite descriptions as incompletesymbols having only meaning in context and none in isolation; more onthis below. This also deprived the definite descriptions of the statusof singular terms.

Whether or not this is the case, there are a couple of rather basicthoughts and lessons that can be borne out: a definite description isof the form: the \(A\), or the \(x\), such that \(A[x]\); where‘the \(x\)’ is the description operator, and‘\(A[x]\)’ is the basis of the description. A definitedescription is (uniquely) satisfied just in case the basis of thedescription is uniquely satisfied. Such a definite description is alsocalledproper. That is, the definite description ‘thecurrent pope’ is uniquely satisfied since there is exactly oneobject in the domain that satisfies ‘being a pope’, i.e.the basis of the definite description. Note that the phrase ‘thebase of a definite description being uniquely satisfied’ meansthat there is exactly one object that satisfies it, or to be even morepainstakingly precise: there is at least one object that satisfies thebasis of the description and there is at most one.

In order to incorporate definite descriptions into (some) formallanguage some used an inverted iota, or an inverted capital I, we justuse \(\imath\), where \(\imath\) is a variable-binding term-formingoperator, such that: \(\imath x A\) is a singular term, if \(A\) is aformula. This admittedly is not in the spirit of Russell, but verymuch in the spirit of free logic that has its core the scientific goalof unification. In this particular case this means that the plethoraof theories on definite descriptions can conveniently be bothdeveloped and compared within the framework of free logic. However,what is in the spirit of Russell is that definite descriptionsessentially occur in two contexts: (1) the \(A\) exists, and (2) the\(A\) is \(F\). Russell’s guiding principles, or as Russell putsit, contextual definitions, are then stated as:

\[E!\imath xA[x] \leftrightarrow \exists x(A[x] \wedge \forall y(A[y] \to x = y))\tag{R1}\]

\[F(\imath xA[x]) \leftrightarrow \exists x(A[x] \wedge \forall y(A[y] \to x = y) \wedge F[x])\tag{R2}\]

In accordance with what Russell tells us about definite descriptionswith a uniquely satisfied basis, the following formulas aretheorems:

\[E!\imath xA[x] \to A[\imath x A[x]]\tag{T1}\]

\[E!\imath xA[x] \to \imath xA[x] = \imath xA[x]\tag{T2}\]

However, Russell’s approach becomes less elegant if it comes tothe negative side, that is, if the basis of a definite description isnot uniquely satisfied. We just mention two here. First, following(R2) negation of the left-hand side is ambiguous given the right-handside of the equivalence, it can either mean:

\[\neg(\exists x(A[x] \wedge \forall y(A[y] \to x = y) \wedge F[x]))\]

or

\[\exists x(A[x] \wedge \forall y(A[y] \to x = y) \wedge \neg F[x])\]

In order to disambiguate Russell inventedscope operators todistinguish the two different readings, in which he did not succeedconsistently; more work on the formal aspects of Russell’stheory are found e.g. (Scales1969; Czermak 1974;Donald Kalish 1992; Lambert 1997; Gratzl 2015). The secondinelegance is that, given that Russell denies definite descriptionsthe status of singular terms, he needs a philosophical theory aboutthe logical structure of sentences and terms, which is interesting initself, but also cumbersome and furthermore leads to the fact thatRussell’s approach is not closed under substitution, which isnot advantageous given that uniformity is a goal.

Hilbert and Bernays Approach

Another way that would remain in classical logic but treats definitedescriptions radically different is found in Hilbert & Bernays intheirGrundlagen der Mathematik, Vol. 1, 1934. In theirapproach expressions of the form \(\imath xA\) are accepted assingular terms just in case it is provable in the formal system, orsimply logic, that the basis of \(\imath xA\), i.e. \(A\), is uniquelysatisfied. That is, both formulas are provable:

\[\exists xA[x], \ and \ \forall x \forall y(A[x] \wedge A[y] \to x = y)\]

If these formulas are provable, then \(A[\imath xA[x]]\) is among theprovable statements of the formal system in question. Now, thisapproach has been critiqued on occasion for e.g. mingling theformation rules of formulas and terms with provability of the formalsystem, for not being natural — not even for mathematics for which ithas been originally developed for. The former point of critique can belifted, see e.g. (Lambert1999); others were not soimpressed by the syntactic quagmire left behind by Hilbert and Bernaysand developed quite elaborated results, e.g. (Schütte1971).

On the positive side, there are two points: (1) it does not prove\((***)\) (see below) and it proves \(t = \imath x(x = t)\). The lastformula is also known asFrege’s Law, (FL), which Fregeintroduced in hisGrundgesetze; that is to say, in adifferent context but (FL) is what we have nowadays in classicalfirst-order logic. Clearly, classical logic or even a free logic and(FL) alone will not result in an interesting account of definitedescriptions.

Free Description Theories

Taking stock, there are options within classical logic to developquite divergent theories on definite descriptions, all of which havetheir own merits and drawbacks. The three theories outlined havebackground theories, or philosophical attitudes, of their own and so,as the argument goes, are not directly comparable within one commonframework. Fortunately, here free logic offers a way out.

Lambert’s Law is the minimal principle (at least up to now) alldescription theories based on free logic seem to agree upon:

\[\forall x(x = \imath yA[y] \leftrightarrow \forall y(A[y] \leftrightarrow x = y))\tag{LL}\]

Following Lambert(1997), it is also instructive to determine which effect (LL)has if it is combined with classical logic; where it is assumed thatclassical logic is closed under substitution for all singular terms.From (LL) and axiom 3 (or as is also known: universal instantiation)and tautologies, it follows that (this closely follows Lambert1997):

\[t = \imath xA \to A[t/x]\tag{*}\]

The following is a theorem:

\[\imath x(P(x) \wedge \neg P(x)) = \imath x(P(x) \wedge \neg P(x)\tag{**}\]

But now contradiction follows from \((*), (**)\), and modusponens:

\[P(\imath x(P(x) \wedge \neg P(x))) \wedge \neg P(\imath x(P(x) \wedge \neg P(x)))\tag{***}\]

Now, further inductive support for the usefulness of free logic isthat free logicians drew the morals of adopting a version of freelogic in which contradiction can be avoided; suffice it to say that inthe face of antinomy and it appropriate error analysis, typically someroutes are open for investigation, one that varies universalinstantiation is at least a viable and systematic option.

We already introduced Hintikka’s Law (HL), i.e.

\[E!t \leftrightarrow \exists x(x = t)\tag{HL}\]

earlier. So, adding (HL) to PFL, Russell’s (R1), (R2), (T1), and(T2) are obtainable.

However, NFL is more in tune with Russell’s account (sincedefinite descriptions are, in fact, treated as genuine singularterms), for this end NFL is accompanied with (LL) and a version ofaxiom 8 of NFL including definite descriptions, i.e.:

\[\iota xA[x] = \iota xA[x] \to E!\iota xA[x]\]

NFL enjoys the theorem ofindiscernability of non-existents,which has the following instance (given that NFL enriched withdefinite descriptions is closed under substitution):

\[\neg E!\imath xA \wedge \neg E!\imath xB \to (B[\imath xA] \to B[\imath xB])\]

Free description theories leaning more towards the positive side offree logic are, for instance, theories building on (LL) and

\[\forall x(A[x] \leftrightarrow B[x]) \to \imath xA[x] = \imath xB[x]\tag{Coext}\]

In (Morscher and Simons2001b) the truth of this principle is called self-evident. Thepredicates ‘having kidneys’ and ‘having aheart’ are co-extensional, which is an empirical truth, andaccording to (Coext) then implies that the \(x\) having a lung = the\(x\) having a heart — a truth of the status of being obvious might bechallenged. Whereas, still having positive intuitions the predicates‘being a Labrador full up with food’ and ‘being aMartian’ are co-extensional, both being empty, the then part:the \(x\) being a Labrador full up with food = the \(x\) being aMartian seems to be more in line with PFL. It isnoteworthy for Hilbert and Bernays (Coext) is a theorem given that theuniquenss-conditions are provable for both \(A\), and \(B\).

A principle related to (Coext) and still maintaining PFL is:

\[\neg E!s \wedge \neg E!t \to s = t\tag{Negexid}\]

Both (Coext) and (Negexid) are reflections of thechosenobject-idea which assigns empty singular terms and arbitraryobject. Under this assumption one is back to the classical existenceassumptions, and so classical logic. The chosen-object theory is dueto Frege, the formulation used is due to Carnap (Carnap1964): \[\forall x(x = \imath yA[y] \leftrightarrow\]\[(\exists z(\forall y(A[y] \leftrightarrow y = z) \wedge z = x)\] \[\vee \neg \exists z(\forall y(A[y] \leftrightarrow y = z) \wedge z=*))\tag{Chosob}\]

Technical investigations on the chosen object theory are found in(Donald Kalish 1992)and more recently and more proof-theoretically motivated in (Indrzejczak2019). After a period of lull, where logical research ondefinite descriptions was not so pronounced, the research intodefinite descriptions has lately again picked up e.g. (Kürbis2021, 2022).

5.2 Logics with Partial or Non-Strict Functions

Some logics employ primitiven-place functionsymbols—symbols that combine with \(n\) singularterms to form a complex singular term. Thus, for example, the plussign ‘+’ 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 valueis for some arguments undefined. Such, for example, is the binarydivision sign ‘/’, since when placed between two numeralsthe second of which is ‘0’, it forms an empty singularterm. Similarly, the limit function symbol ‘lim’ yields anempty singular term when applied to the name of a non-covergingsequence. Classical logic can accommodate function symbols for partialfunctions via elaborate contextual definitions. But then (as withRussellian definite descriptions) the form in which these functionsymbols are usually written is not their logical form. Free logicprovides a more elegant solution. Because it allows empty singularterms, symbols for partial functions may simply be taken asprimitive.

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 aserrorobjects—objects that correspond in the running of aprogram to error messages. Thus, for example, an instruction tocalculate \(f(1, 1/0)\) might return the value 1, but an instructionto calculate \(f(1/0, 1)\) would return 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)—butnot others (the present, for example, or all future times). Twonatural assumptions are made here: that the same object may exist inmore than one world (this is the assumption oftransworldidentity), and that some singular terms—propernames, in particular—refer not only to an object at agiven world, but to that same object at every world. Such terms arecalledrigiddesignators. Any logic that combinesrigid designators with quantifiers over the domains of worlds in whichtheir 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 asthe present? The general answers given to such questions determinewhether 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 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 trueof—and singular terms to be capable ofdenoting—objects that exist in no world. If so, we maycollect 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 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 1.3 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,” whereaccessibility 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 wemay have worlds \(u\) and \(w\) such that \(\bD_u \ne \bD_w\). For inthat case there must be an object \(d\) that exists in one of thesedomains (let it be \(\bD_w)\), but not the other, so that any singularterm \(t\) that rigidly designates \(d\) must be empty at world \(u\).Hence \({\sim}\exists x(x=t)\) (which is self-contradictory inclassical logic) must be true at world \(u\). Such a semantics alsorequires free logic when \(\bD_o\) contains objects not in \(\bU\),for in that case rigid designators of these objects are empty in allworlds. Finally, this semantics calls forinclusive logic ifany world has an empty domain. Thus, given this semantics, the onlyway to make the resulting logic unfree is to require that domains befixed—i.e., that all worlds have the samedomain \(\bD\), 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 nothingin this world would have existed without it. But it does not seemnecessary that everything is a product of the big bang, for otheruniverses are possible in which things that do not exist in the actualworld have other 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 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 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 theintuitionistic predicate calculus—has the theorem\(\exists x(x=t)\) and hence is not free. But intuitionists admit theexistence only of objects that can in some sense be constructed, whileclassical mathematicians posit a wider range of objects. Thereforeusers of HPC cannot legitimately name all the objects that classicalmathematicians can. Worse, they cannot legitimately name objects whoseconstructibility has yet to be determined. Yet some Kripke-stylesemantics for HPC do allow use of names for such objects(semantically, names of objects that “exist” at worldsaccessible 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 byPosy (1982), is to adopt a free intuitionistic logic. For more on thisissue, 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—includingOrlando 2008 and Dumitru and Kroon 2008—questionSainsbury’s treatment, maintaining that statements like (G) areboth atomic and true. If so, they require a positive free logic. Thelogic must be free because it deals with an empty singular term, andit must be positive, because only on a positive semantics canempty-termed atomic statements be true. One must still decide,however, whether the name ‘Gollum’ is to be understood ashaving no referent or as having 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 orconcrete, literal or fictional, complete or incomplete. In thissection the term is used to describe logics stronger than the firsttype but possibly weaker than the second: positive free logics with anextra set of quantifiers that range over the outer domain of adual-domain semantics.

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, Meinongianlogics in the sense intended here at leastcontain freelogics when the inner domain is interpreted as the set of existingthings. Moreover, on the strictly semantic definition, which is alsothat of Lehman 2002, whether the members of \(\bD\) exist isirrelevant to the question of whether a logic is free. For a defenseof 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,an existential quantifier on a variable \(x\) is just a logicaloperator that takes open formulas on \(x\) into truth values; thevalue is T if and only if the open formula is satisfied by at leastone object in the quantifier’s domain. That the objects in thedomain have or lack any particular ontological status is aphilosophical interpretation of the formal semantics. Alex Orenstein(1990) argues that “existential” is a misnomer and that weshould in general 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).

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Beginning with the 2025 update, Norbert Gratzl and Edi Pavlovićhave taken responsibility for updating and maintaining this entry.