free logic

FREE LOGIC: A FIFTY-YEAR PAST AND AN OPEN FUTURE

1. DEFINITION AND TYPES OF FREE LOGIC

The term ‘free logic’, as introduced by Karel Lambert in 19601, is short for‘presupposition-free logic’ or, more explicitly, for ‘logic free of existentialpresuppositions’. Since existential presuppositions are connected withterms, this paraphrase was understood as ‘logic free of existential presup-positions with respect to its singular and general terms’. Modern logic inthe sense of standard first order predicate or quantificational logic withidentity (QL=, for short) is almost fully free with respect to its generalterms or predicates, with one exception, namely universal terms or predi-cates like ‘Px ∨ ¬Px’ or ‘x = x’. In standard systems of QL=, ‘∃x(Px ∨ ¬Px)’and ‘∃x(x = x)’ are theorems. If we read such a formula as ‘somethingexists’ this seems to express a matter of ontology rather than logic; takingit as a law of logic is therefore “a defect in logical purity” as already notedby Russell2. Giving up this presupposition results in a logic called inclus-ive logic by Quine3 or empty logic by some free logicians. The main con-cern of free logic was and is therefore existential presuppositions with re-spect to singular terms. In standard systems of QL= we usually have forevery singular term t and every variable v the theorem ∃v(v = t). And instandard quantification theory without identity (QL) we still have forevery formula A the theorem A(t/v) → ∃vA and its dual ∀vA → A(t/v).

If we allow t to be a singular term not referring to an existent object andwe take A to be the predicate or general term ‘x does not exist’, these the-orems are instantiated by obvious falsehoods. This, however, is the caseonly if we interpret the quantifiers as having existential import and beingread as ‘for every existing thing’ and ‘for at least one existing thing’ (or‘there exists something’). That explains why this understanding of quanti-fication becomes part of the definition of free logic.

Even if the main aim of free logic is to eliminate existential presupposi-tions with respect to singular terms, a definition of free logic has to takeinto account also general terms, not only because otherwise it would be in-complete, but also because otherwise it would be incoherent: existentialpresuppositions with respect to singular terms could return through theback door via general terms containing singular ones like ‘x = t’. An ad-

EDGAR MORSCHER AND PETER SIMONS

1

Alexander Hieke and Edgar Morscher (eds.),New Essays in Free Logic, 1–34.

© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

equate definition of free logic therefore has to include three components:

A logical system L is a free logic iff(1) L is free of existential presuppositions with respect to the singular

terms of L, (2) L is free of existential presuppositions with respect to the general

terms of L, and(3) the quantifiers of L have existential import.

A logical system L is a universally free logic iff(1) L is a free logic, and(2) L is an inclusive (or empty) logic.

According to this definition, free logic is not a particular logical systembut rather a whole family of systems so that we can also use the pluralform and speak of “free logics”. Free logics in this sense are logical sys-tems which allow singular terms to be empty or non-denoting insofar asthey do not refer to existent things, and at the same time the theorems ofsuch a system remain logically true even if the singular terms occurring inthem are empty. Here it is important to notice that in some versions of freelogic a singular term may be empty, in not referring to any existing object,and yet referential, in referring to a non-existing object.

There are three types of free logic to be distinguished in this context de-pending on whether or not elementary sentences containing empty singu-lar terms do or do not have certain truth-values. An elementary or logicallysimple sentence is a sentence containing no logical operator (i.e. no truth-functional connective and no quantifier). Now we can define:

A logical system L is a negative free logic iff L is a free logic and everyelementary sentence of L containing at least one empty singular term isfalse.

A logical system L is a positive free logic iff L is a free logic and thereis at least one true elementary sentence of L containing at least oneempty singular term.

Among the systems which are free but neither negative nor positive onlyone type is attractive enough to get a name of its own, and is usually de-fined as follows:

2 EDGAR MORSCHER AND PETER SIMONS

A logical system L is a neutral free logic iff L is a free logic and everyelementary sentence of L containing at least one empty singular term(with the only exception perhaps being ‘t exists’) has no truth-value at all.

2. SOME MOTIVATIONS BEHIND FREE LOGIC

The interest in problems of free logic increased and the first works in thefield were published at the same time as the first steps were taken towardsthe development of a semantics for modal logics which later became wellknown by the title ‘Possible World Semantics’. This seems to be no merecoincidence since there is an overlapping area of common motivationbehind the two developments. Also in different fields of modal logic (likealethic, epistemic and deontic logic) we very often deal with things, per-sons, actions, situations etc. which do not actually exist, whether they donot yet exist or no longer exist or never exist. In modal logic therefore itis common to use singular terms which do not denote an object existing inevery possible world.

The reasons for developing systems of free logic are, however, largelyindependent of such modal considerations. They also concern the occur-rence of empty singular terms in non-modal contexts and some of thesesentences nevertheless seem to be obviously true. Introducing a singularterm for an allegedly existing planet, chemical substance, particle, numberetc. should not in any way depend on our knowledge that the purported ob-ject allegedly denoted by the singular term in fact exists. This motive de-pends on the common aim of keeping our scientific language, in particularits vocabulary and its formation rules, independent of the facts we want todescribe by it and independent of our knowledge of these facts. If so, weneed a language containing singular terms t for which ∃v(v = t) is not atheorem and which therefore can occur in a sentence A(t/v) without ∃vAbeing deducible from it. But this is just a free logic as defined above. Sim-ilar arguments apply to a language which serves the purpose of expressingfictions, myths and fairy tales in which (empty) names of non-existing per-sons occur without turning all these sentences into truth-valueless expres-sions (as Frege did) or into falsehoods. Another solution for this problemwas offered by Russell who eliminated most singular terms from a lan-guage and replaced them by definite descriptions. This, however, seems tobe a quite unnatural solution.

There are also philosophical motives which led to the development offree logic. An adequate discussion of traditional arguments for the exist-

FREE LOGIC: A FIFTY-YEAR PAST AND AN OPEN FUTURE 3

ence of God or of Descartes’ “Cogito ergo sum” is only possible in a lan-guage allowing empty singular terms. Otherwise these problems are turnedinto trivialities and their alleged solutions into petitiones principii. Thesame holds for evaluative or normative contexts of ethics in which veryoften non-existing persons, actions and situations are taken into account.

3. SYNTACTICAL SYSTEMS OF FREE LOGIC

3.1 THE LANGUAGES FL, FL–, FL= AND FL+ OF FREE LOGIC

We describe the formal language FL in the usual way by listing its vocabu-lary and specifying its formation rules, thereby defining its formulas.

The vocabulary of FL is the same as that of QL=, augmented by thesymbol ‘E!’ for existence. It therefore consists of the following signs:

(1) the descriptive symbols of FL(1a) the n-place (n = 1, 2,…) predicates of FL: P, Q, R,…(1b) the individual constants of FL: a, b, c,…

(2) the logical symbols of FL(2a) the (individual) variables of FL: x, y, z,…(2b) the connectives of FL: ¬, →(2c) the (universal) quantifier of FL: ∀(2d) the logical predicates of FL: =, E!

(3) the auxiliary signs of FL: (, )

By dropping the existence predicate E! we get the vocabulary of a re-stricted language FL=, and by dropping also the identity symbol and theindividual constants (i.e. by omitting all symbols of (1b) and (2d)) we getthe vocabulary of an even more restricted language FL–. On the otherhand, we get the vocabulary of an enlarged language FL+ by adding thefollowing clause (2e):

(2e) the description operator of FL+: ι

A symbol of FL, FL–, FL= or FL+ which is either an individual constant oran individual variable will be called an individual symbol. A single sign ora finite sequence (string) of such signs will be called an expression of oneof these languages. Expressions which are either individual constants ordefinite descriptions (which will be introduced below) are called indi-


vidual names and correspond to what we usually call singular terms whenwe do not refer to a particular language. Expressions which are either indi-vidual variables or individual names will be called individual terms. Asusual, we will call not only a single sign ∀ or ι standing alone a quantifieror description operator, respectively, but also an expression of the form∀v or ιv where ∀ or ι, respectively, is prefixed to a variable v.

In the metalanguages of our languages we will use the connectives, theuniversal quantifier, the logical predicates and the auxiliary signs auto-nymously, whereas we introduce the following kinds of metavariables forsigns and expressions of FL+ (and accordingly for the other languageswhich are less rich than FL+):

(1) for n-place predicates: Pn

(2) for individual names: t, t1, t2,… (3) for (individual) variables: v, v1, v2,… (4) for individual terms (i.e. individual names or variables): s, s1, s2,…(5) for expressions, in particular for formulas: A, B, B1, B2… (6) for classes of formulas: C

We can now define a (well-formed) formula of FL recursively by thefollowing formation rules:

(1) If Pn is an n-place predicate of FLand s1, s2,…, sn are individual terms of FLthen Pns1s2…sn is a formula of FL.

(2) If s1 and s2 are individual terms of FLthen s1 = s2 is a formula of FL.

(3) If s is an individual term of FLthen E!s is a formula of FL.

(4) If A and B are formulas of FLthen ¬A and (A → B) are also formulas of FL.

(5) If A is a formula of FL and v is an (individual) variable of FLthen ∀vA is a formula of FL.

(6) Nothing is a formula of FL except by virtue of (1)–(5).

A formula A of FL is an elementary formula of FL iff A contains no con-nective and no quantifier, i.e. iff A is of one of the forms (1), (2) or (3).And a formula A of FL is an atomic formula of FL iff A is elementary anddoes not contain a logical predicate, i.e. iff A is of the form (1). The def-inition of a formula of FL= results from the definition above by omitting


clause (3) and the definition of a formula of FL– by omitting also clause(2) from the definition above.

For FL+ we augment the definition of a formula by the following clause(0) for definite descriptions thereby turning the definition into a simultan-eous recursive definition of terms and formulas of FL+:

(0) If A is a formula of FL+ and v is a variable of FL+

then ιvA is a definite description (and thereby an individual name and an individual term) of FL+.

In a formula ∀vA of FL the subformula A is called the scope of thequantifier ∀v. An occurrence of a variable v in a formula A of FL is boundiff this occurrence of v is within a subformula ∀vB of A (i.e. iff this occur-rence of v is within the scope of a quantifier ∀v or within the quantifier ∀vitself). And an occurrence of a variable v in a formula A of FL is free iff itis not bound. A variable v is bound in a formula A of FL iff v has at leastone bound occurrence in A, and it is free in A iff v has at least one free oc-currence in A. A formula A of FL is an open formula or a sentence form ofFL iff there is at least one variable free in A, and A is a closed formula ora sentence of FL iff A is not an open formula, i.e. iff there is no variablefree in A.

Logical constants other than ¬, → and ∀ can be used as abbreviationsand introduced into FL via the usual definitions as follows:

(D1) (A ∧ B) :↔ ¬(A → ¬B)(D2) (A ∨ B) :↔ ¬A → B(D3) (A ↔ B) :↔ (A → B) ∧ (B → A)(D4) ∃vA :↔ ¬∀v¬A

In our metalanguage we will use two kinds of substitution operations,which we will abbreviate by A(s2/s1) and A(s2//s1), respectively. With re-spect to A(s2/s1) we have to distinguish three cases depending on which ofs1 and s2 is an individual name and which is a variable. A(t/v) is the resultof replacing every free occurrence of v in A by the individual name t. A(v/t)is the result of replacing every occurrence of an individual name t in A bya variable v which is free for t in A. A(w/v) is the result of replacing everyfree occurrence of v in A by a variable w which is free for v in A. We saythat a variable v – or an individual name t containing the variable v – isfree [to be substituted] for an individual name t' or an individual variable


w, respectively, in a formula A iff no occurrence of t' or no free occurrenceof w, respectively, is within the scope of a quantifier ∀v in A. Finally,A(s2//s1) is the result of replacing zero or more occurrences of the indi-vidual term s1 in A by the individual term s2.

3.2. THE AXIOMATIC SYSTEM NFL OF NEGATIVE FREE LOGIC

Like any free logic, NFL must weaken the principle of Universal Specifi-cation of QL=, i.e. ∀vA → A(t/v). If an existence predicate E! is available,as in NFL, the standard Specification axiom can be replaced by ∀vA →(E!t → A(t/v)). In a negative free logic we must guarantee that an elemen-tary sentence is false as soon as an individual name (i.e. singular term) init is empty. This can be obtained by axiomatically requiring in NFL that forevery elementary formula A(t/v) we have A(t/v) → E!t as a theorem. Vari-ous formulations of negative free logic have been proposed, the first byRolf Schock4 and a later one by Ronald D.Scales5. The system NFL pre-sented below comes very close to a formulation due to Tyler Burge6.

We formulate the axiomatic basis of NFL as a set of axiom schemata.The axioms of NFL are all and only those closed formulas (sentences) ofFL which are described by the following schemata:

(NA0) If A is a formula of FL which is a substitution instance of a the-orem of the propositional calculus, then the universal closure ofA is an axiom of NFL

(NA1) A → ∀vA(NA2) ∀v(A → B) → (∀vA → ∀vB)(NA3) ∀vE!v(NA4) ∀vA → (E!t → A(t/v))(NA5) For every elementary formula A of FL in which the variable v is

free: A(t/v) → E!t

(NA6) ∀v(v = v)(NA7) t1 = t2 → (A → A(t2//t1))(NA8) For every axiom A of NFL, for every individual constant t oc-

curring in such an axiom and for every variable v which is freefor t in A:∀vA(v/t)

The axiomatic basis of NFL will be completed by Modus Ponens as itssingle rule of inference:


(NR) For any closed formulas A and B of FL:from A → B and A to infer B

The metalogical concepts of NFL are defined as usual: a derivation in NFLof a closed formula A of FL from a class C of closed formulas of FL is asequence B = ⟨B1, B2,…, Bn⟩ of closed formulas Bk (1 ≤ k ≤ n) of FL suchthat Bn = A and for each Bk ∈ B: Bk ∈ C or Bk is an axiom of NFL or forsome i and j (i < k, j < k) Bi = Bj → Bk, i.e., Bk follows from precedingformulas in the sequence by (NR). A closed formula A of FL is derivablein NFL from a class C of closed formulas of FL (abbreviated: C D

NFLA)

iff there is a derivation in NFL of A from C. A proof in NFL of a closedformula A of FL is a derivation in NFL of A from C with C = ∅. A closedformula A of FL is provable in NFL or a theorem of NFL (D

NFLA) iff there

is a proof of A in NFL. A class C of closed formulas of FL is consistent inNFL iff there is no formula A of FL such that C D

NFLA and C D

NFL¬A.

3.3. THE AXIOMATIC SYSTEMS PFL, PFL= AND PFL–

OF POSITIVE FREE LOGIC

A positive free logic is a free logic without the characteristic feature of anegative free logic as expressed by axiom (NA5) of NFL. A system ofpositive free logic PFL results therefore from NFL by dropping (NA5). Atthe same time we can now replace axiom (NA6) by the scheme t = t,because in positive free logic this is true for all singular terms t includingempty ones. The axiomatic basis of PFL can thus be represented as follows:

(PA0) If A is a formula of FL which is a substitution instance of a the-orem of the propositional calculus, then the universal closure ofA is an axiom of PFL

(PA1) A → ∀vA(PA2) ∀v(A → B) → (∀vA → ∀vB)(PA3) ∀vE!v(PA4) ∀vA → (E!t → A(t/v))(PA5) –––(PA6) t = t(PA7) t1 = t2 → (A → A(t2//t1))(PA8) For every axiom A of PFL, for every individual constant t oc-

curring in such an axiom and for every variable v which is freefor t in A:∀vA(v/t)


Again, Modus Ponens is the only rule of inference for PFL:

(PR) For any closed formulas A and B of FL:from A → B and A to infer B

The metalogical definitions are as for NFL. A system similar to PFL wasfirst formulated by Karel Lambert7. PFL shares with NFL the main fea-tures of any free logic: the attribution of existential import to the quantifiersas expressed by axiom (PA3) and (NA3), respectively, and a weakenedform of Universal Specification (or alternatively of Existential Generaliza-tion) as expressed by axiom (PA4) and (NA4), respectively. The latter dis-tinguishes systems of free logic from standard systems of QL=. In a lan-guage not containing the existence predicate E! or a substitute for it othermeans must be used for weakening Universal Specification or ExistentialGeneralization, respectively. Such a system of positive free logic for arestricted language FL= which differs from FL in not including the logicalpredicate E! results from PFL by dropping the axioms concerned with E!and allowing Universal Specification only for variables. In contrast toNFL and PFL, we allow that in PFL= open formulas are included amongthe axioms which are described as follows:

(P=A0) If A is a formula of FL= which is a substitution instance of atheorem of the propositional calculus, then the universal clo-sure of A is an axiom of PFL=

(P=A1) For every variable v which is not free in A:A → ∀vA

(P=A2) ∀v(A → B) → (∀vA → ∀vB)(P=A3) –––(P=A4) For every formula A of FL=, for every variable v and for every

variable w which is free for v in A:∀vA → A(w/v)

(P=A5) –––(P=A6) s = s(P=A7) s1 = s2 → (A → A(s2//s1))(P=A8) For every axiom A of PFL= and every variable v:

∀vA

Modus Ponens is the only rule of inference of PFL=, but in contrast to NFLand PFL, in PFL= it must be allowed to apply to open formulas:


(P=R) For any formulas A and B of FL=:from A → B and A to infer B

The metalogical definitions of NFL and PFL must be adapted for PFL= insuch a way that open formulas are no longer excluded as the bearers of thedefined predicates. PFL= is a free but not an inclusive logic. By requiringvariable v in axiom (P=A4) to be free in A and/or allowing it to be replacednot only by an individual variable w but also by an individual constant twe can get other versions of a free and/or inclusive logic as shown byHugues Leblanc and Richmond H.Thomason8.

An even more restricted set of axioms for language FL– results fromPFL= by dropping axioms (P=A6) and (P=A7) and restricting the set ofaxioms as well as the rule of inference (Modus Ponens) again to closedformulas, as in NFL and PFL. The universal closure of (P=A4) therebyserves as the axiom of Universal Specification. In addition an axiom forquantifier permutation is needed9. The axiomatic basis of such a systemPFL– can therefore be reduced to the following schemata:

(P –A0) If A is a formula of FL– which is a substitution instance of atheorem of the propositional calculus, then the universal clo-sure of A is an axiom of PFL–

(P –A1) A → ∀vA(P –A2) ∀v(A → B) → (∀vA → ∀vB)(P –A3) –––(P –A4) For every formula A of FL–, for every variable v and for every

variable w which is free for v in A:∀w(∀vA → A(w/v))

(P –A5) –––(P –A6) –––(P –A7) –––(P –A8) –––(P –A9) ∀v∀wA → ∀w∀vA

The rule of inference (Modus Ponens) and the metalogical definitions ofPFL– are the same as for NFL and PFL. Of course, PFL– can be completedto a quantification theory with identity by adding appropriate axioms foridentity whereby axiom (P –A9) becomes redundant. A system almostidentical with PFL– was published as the first axiomatic formulation offree logic in an article by Karel Lambert in 196310. Of special interest isaxiom (P –A4) of PFL– which appeared independently in the same year in


Kripke’s system of quantified modal logic11. In spite of the two systemsbeing independent from one another historically as well as systematically,this coincidence indicates an overlapping area of shared problems.

PFL= has been proved sound and complete by Hugues Leblanc andRichmond H.Thomason12 and PFL– by Leblanc and Robert K.Meyer13.

Whereas several systems of positive as well as of negative free logichave been developed from the beginning of free logic, the development ofneutral free logic was confined for a long time to mere outlines like thoseby Timothy Smiley14 and Brian Skyrms15. In the meantime, however, amore detailed treatment of neutral free logic has been put forward by ScottLehmann16.

4. SEMANTICAL SYSTEMS OF FREE LOGIC

Several semantic approaches have been developed for free logic. The fol-lowing three are now well established:

(1) semantics with a partial interpretation function and a total valuationfunction

(2) semantics with an inner and an outer domain(3) supervaluation semantics

Each of these approaches specifies its own type of models M consisting ofdomains D and interpretation functions I and associated with valuationfunctions V. In the semantical systems presented below the interpretationfunction I is always defined on the set of descriptive symbols, i.e. non-logical predicates and individual constants, of FL. It is total iff it is definedfor all of them, and partial iff it is not total. In a semantics for free logic Idoes not assign an existing object to each individual constant. I thereforeassigns to some individual constant t of FL either a non-existing object orno object at all; in the second case I(t) remains undefined and I thereforeis a partial function. In the semantical systems described below the valua-tion functions V based on the interpretation functions I are always definedon the set of closed formulas of FL. They can be either total or partial.

The first type of semantics described in what follows will use a partialinterpretation function, whereas its valuation function is total, assigning atruth-value to each formula of FL. The second type of semantics uses atotal interpretation function and a total valuation function. The third typeof semantics uses a partial and a total interpretation function and a total


and two partial valuation functions. We will describe the three approachesbriefly in what follows. For the different types of models we will use dif-ferent superscript numerals (speaking of M1-models, M2-models, etc.); thesame superscripts are used for variables ranging over such models (M1,M2,…) as well as for corresponding interpretation functions (I1, I 2,…) andassociated valuation functions (V1, V 2,…).

4.1. SEMANTICS WITH A PARTIAL INTERPRETATION FUNCTION

AND A TOTAL VALUATION FUNCTION

In a semantics of the first type an M1-model is defined as an ordered pairconsisting of a possibly empty set D and a partial interpretation functionI1, i.e.

M1 = ⟨D, I1⟩

such that:

(1) for every individual constant t of FL: either I1 does not assign any-thing at all to t and I1(t) thereby remains undefined, or I1(t) ∈ D;

(2) for every n-place predicate Pn of FL: I1(Pn) ⊆ Dn;(3) for every object d ∈ D there is an individual constant t of FL such

that I1(t) = d.

Condition (3) states that the interpretation function I1 of an M1-model pro-vides a “full” (or complete) interpretation of the associated domain D.

We then define truth and falsehood in a model M1 for every closedformula A of FL by defining a total valuation function V1 from the set ofclosed formulas of FL into the set {T, F} of truth-values as follows:

(1) V1(Pnt1t2…tn) = T iff for every ti (1 ≤ i ≤ n): I1(ti) is defined and⟨I1(t1), I1(t2),…, I1(tn)⟩ ∈ I1(Pn).

(2) V1(t1 = t2) = T iff I1(t1) is defined and I1(t2) is defined and I1(t1) = I1(t2).

(3) V1(E!t) = T iff I1(t) is defined.(4) V1(¬A) = T iff V1(A) ≠ T.(5) V1(A → B) = T iff V1(A) ≠ T or V1(B) = T or both.(6) V1(∀vA) = T iff for every individual constant t: if I1(t) is defined

then V1(A(t/v)) = T.(7) V1(A) = F iff V1(A) ≠ T.


Clause (6) above shows that the interpretation of quantifiers is substitu-tional. It therefore requires that the interpretation functions of the modelsto be used provide a “full” or complete interpretation. We can drop thiscondition if we use a Kanger-Mates17 version of (6), reading:

(6') V1(∀vA) = T iff there is an individual constant t not occurring in Asuch that for every model M1' = ⟨D', I1'⟩ which differs from M1 atmost with respect to t: if I1'(t) is defined then V1'(A(t/v)) = T.

The semantic concepts are defined in the usual way: a closed formula A ofFL is logically true iff for each M1-model M1, V1(A) = T. A closed formulaA of FL is a logical consequence of a class C of closed formulas of FL ifffor each model M1: if V1(B) = T for each B ∈C then V1(A) = T. And a classC of closed formulas of FL is satisfiable iff there is at least one model M1

such that V1(B) = T for each B ∈ C.If we define M1-models and the corresponding valuation functions V1 in

this way, the system NFL of negative free logic will turn out to be ade-quate, i.e. sound and complete, relative to such a semantics. By a changeof conditions (1) and (2) in the definition of the valuation function V1 wecan adapt M1-models in such a way that they can be used for proving theadequacy of systems of positive free logic, as was done by Hugues Le-blanc and Robert K.Meyer18.

4.2. OUTER DOMAIN SEMANTICS

On this approach, an M2-model is defined as a triple:

M2 = ⟨Do, Di, I 2⟩

where Do and Di are two disjoint and possibly empty sets of objects, Do iscalled the outer domain and Di is called the inner domain, whose union isnon-empty:

(1) Do ∩ Di = ∅(2) Do ∪ Di ≠ ∅

We define D as this union: D := Do ∪ Di. The interpretation function I 2 isa total function for which we require:


(1) for every individual constant t of FL: I 2(t) ∈ D;(2) for every n-place predicate Pn of FL: I 2(Pn) ⊆ Dn;(3) for every object d ∈ Di there is an individual constant t of FL such

that I 2(t) = d.

The valuation function V 2 is also total and assigns a truth-value, i.e. T orF, to each closed formula of FL relative to an interpretation function I 2 ofan M2-model M2 and is defined recursively as follows:

(1) V 2(Pnt1t2…tn) = T iff ⟨I 2(t1), I 2(t2),…, I 2(tn)⟩ ∈ I 2(Pn).(2) V 2(t1 = t2) = T iff I 2(t1) = I 2(t2).(3) V 2(E!t) = T iff I 2(t) ∈ Di.(4) V 2(¬A) = T iff V 2(A) ≠ T.(5) V 2(A → B) = T iff V 2(A) ≠ T or V 2(B) = T or both.(6) V 2(∀vA) = T iff for every individual constant t: if I 2(t) ∈ Di

then V 2(A(t/v)) = T.(7) V 2(A) = F iff V 2(A) ≠ T.

Models of this type have been developed by Nuel Belnap and Karel Lam-bert. M2-models can serve as a semantical basis for different systems offree logic, but they are used in particular for positive free logic. PFL= hasbeen proved to be adequate with respect to M2-models by Hugues Leblancand Richmond H.Thomason19.

4.3. SUPERVALUATION SEMANTICS

In the first type of semantics we used models with partial interpretationfunctions I1 so that I1(t) was not defined for every individual constant t.Each way of assigning truth-values to elementary sentences containingsuch empty individual constants has a certain kind of arbitrariness, be itthat all of them are assigned F (like in the semantics of section 4.1) or allof them are assigned T or that some are assigned T and some F. If, how-ever, we decide to assign no truth-value at all to such a sentence, say A,then ¬A will usually have none either. But then A ∨ ¬A would not turn outtrue, which is a clear violation of a classical law of logic. In the semanticsdescribed in 4.1 an elementary sentence containing an empty individualconstant therefore always gets the truth-value F thereby making a systemof negative free logic like NFL provably adequate relative to such asemantics. The disadvantage of such a system, however, is that it is unnat-ural in the following respect: if we allow empty singular terms at all, we


would like to have at least some elementary sentences containing them tobe true. This results in a system of positive free logic like PFL. A seman-tics relative to which PFL is adequate is the inner/outer domain semanticsas described in section 4.2. But it has the disadvantage of requiring anouter domain of non-existing entities, against all our realistic intuitions.There are other versions of inner/outer domain semantics which avoidsuch Meinongian consequences by means of technical devices; they oftentherefore look less natural. The question is therefore how to develop asemantics appropriate for a positive free logic without falling into thedrawbacks of an inner/outer domain semantics. A solution to this problemis offered by supervaluation semantics. It starts with models of the sametype as in the first approach, but in opposition to this approach it allowselementary sentences containing empty singular terms to be truth-value-less. In order to avoid this resulting in a rejection of classical laws of logic,these models are “completed” in order to fill, at least temporarily, thetruth-value gaps remaining in the first “round” of valuation. Supervalu-ation semantics therefore starts with a third type of models, namely M3-models, defined as follows:

M3 = ⟨D, I 3⟩

where D is again a possibly empty set of objects and I 3 is a partial inter-pretation function like I1 for which we require just as for M1-models:

(1) for every individual constant t of FL: either I 3 does not assign any-thing at all to t and I 3(t) thereby remains undefined, or I 3(t) ∈ D;

(2) for every n-place predicate Pn of FL: I 3(Pn) ⊆ Dn;(3) for every object d ∈ D there is an individual constant t of FL such

that I 3(t) = d.

The valuation function V 3 associated with M3-models, however, is (in con-trast to V1) also a partial function (like I 3) and its domain is restricted toelementary formulas of FL. V 3 is therefore a partial function from closedelementary formulas of FL into the set {T, F} of truth-values; it is definedas follows:

(1a) If for every ti (1 ≤ i ≤ n), I 3(ti) is defined, then V 3(Pnt1t2…tn) = T iff ⟨I 3(t1), I 3(t2),…, I 3(tn)⟩ ∈ I 3(Pn), and V 3(Pnt1t2…tn) = F iff ⟨I 3(t1), I 3(t2),…, I 3(tn)⟩ ∉ I 3(Pn).


(1b) If for at least one ti (1 ≤ i ≤ n), I 3(ti) is undefined, then V 3(Pnt1t2…tn) is undefined.

(2a) If both I 3(t1) and I 3(t2) are defined, then V 3(t1 = t2) = T iff I 3(t1) = I 3(t2), and V 3(t1 = t2) = F iff I 3(t1) ≠ I 3(t2).

(2b) If either I 3(t1) or I 3(t2) is undefined but the other is defined, then V 3(t1 = t2) = F.

(2c) If neither I 3(t1) nor I 3(t2) is defined, then V 3(t1 = t2) is undefined.

(3) V 3(E!t) = T iff I 3(t) is defined, and V 3(E!t) = F iff I 3(t) is undefined.

We define now the concept of a completion (i.e. a complete supermodel)of an M3-model:

M4 = ⟨D', I4⟩ is a completion of M3 = ⟨D, I 3⟩ iff(1) D' ≠ ∅,(2) D ⊆ D',(3) for every n-place predicate Pn: I 3(Pn) ⊆ I4(Pn),(4) for every individual constant t: if I3(t) is defined, then I4(t) = I3(t), (5) for every individual constant t: I4(t) ∈ D'.

Clauses (1)–(4) of this definition express that M4 is a supermodel of M3,and clause (5) says that I4 is a total function and M4 is herewith “com-plete”.

The valuation function V4 relative to an M4-model M4 from the point ofview of an M3-model M3 of which M4 is a completion is a total functionfrom all the closed formulas of FL into the set {T, F} of truth-values. V4

therefore depends on V 3 and should therefore be read as V 3+1. It is definedas follows:

(1) If A is a closed elementary formula of FL and V 3(A) is defined,then V4(A) = V 3(A).

(2) If A is a closed elementary formula of FL and V 3(A) is undefined,then V4(A) is determined independently of V 3 in the usual way forcomplete models as follows:

(2a) If A is a closed elementary formula of the form Pnt1…tn , then V4(Pnt1…tn) = T if ⟨I4(t1),…, I4(tn)⟩ ∈ I4(Pn), and V4(Pnt1…tn) = F if ⟨I4(t1),…, I4(tn)⟩ ∉ I4(Pn).


(2b) If A is a closed elementary formula of the form t1 = t2, then V4(t1 = t2) = T if I4(t1) = I4(t2), and V4(t1 = t2) = F if I4(t1) ≠ I4(t2).

[(2c) If A is a closed elementary formula of the form E!t, then V 3(E!t) isalways defined and therefore taken care of by clause (1), i.e. foreach individual constant t: V4(E!t) = V 3(E!t).]

(3) V4(¬A) = T iff V4(A) = F.(4) V4(A → B) = T iff V4(A) = F or V4(B) = T or both.(5) V4(∀vA) = T iff for each individual constant t: if V4(E!t) = T then

V4(A(t/v)) = T.

On the basis of valuation functions V4 which complete V 3 we define thesupervaluation S(M3) as a partial function from closed formulas of FL intothe set {T, F} of truth-values as follows:

(1) S(M3)(A) = T iff V4(A) = T for every completion M4 of M3.(2) S(M3)(A) = F iff V4(A) = F for every completion M4 of M3.(3) S(M3)(A) is undefined otherwise, i.e. iff

V4(A) = T for at least one completion M4 of M3, and V4'(A) = F for at least one completion M4' of M3.

We can then define logical truth and logical consequence in terms ofsupervaluations in the following way: a closed formula A of FL is logicallysupertrue iff for all M3-models M3: S(M3)(A) = T. A closed formula B ofFL is a logical superconsequence of a class C of closed formulas of FL ifffor all M3-models M3: if S(M3)(B) = T for each B ∈ C then S(M3)(A) = T.And a class C of closed formulas of FL is supersatisfiable iff there is atleast one model M3 such that S(M3)(B) = T for each B ∈ C.

It can be shown that if A is a sentence of FL which is a tautology ofpropositional logic, then A is logically supertrue. Indeed it was, amongother things, to guarantee this result that supervaluation semantics was de-veloped. On the other hand, the classical relation between logical truth andlogical consequence can get disturbed in supervaluation semantics insofaras a closed formula B of FL may be a logical superconsequence of a closedformula A without the corresponding conditional A → B being logicallysupertrue. So, e.g., ∃vP1v is a logical superconsequence of P1t, whereasP1t → ∃vP1v is not logically supertrue. Therefore the weak semantic com-pleteness of a calculus (with respect to logical supertruth) does not needany more to coincide with its strong semantic completeness (with respectto logical superconsequence).


Supervaluation semantics was developed and used by Bas van Fraassen toprove the soundness and completeness of PFL=20. The version of super-valuation presented above was used by Ermanno Bencivenga, Karel Lam-bert and Bas van Fraassen for proving the soundness and completeness of atableaux version of PFL21. Brian Skyrms outlined how to adapt supervalu-ation semantics in such a way that it can be used for neutral free logic22.

4.4. FURTHER SEMANTICAL APPROACHES FOR FREE LOGIC AND COMPARISON

The three semantical approaches which we have described were developedfor and within free logic but have found interesting applications also out-side free logic. They are the three semantic standard approaches for freelogic but not the only semantic approaches developed for free logic. Thereis also, e.g., the so-called story semantics23. Its M5-models M5 = ⟨D, I5, S⟩contain in addition to the domain D and the interpretation function I5 astory S. S is thereby a possibly empty set of atomic formulas of FL each ofwhich contains at least one individual constant t to which I5 does notassign a member of D so that I5(t) is undefined. The story S in such amodel supplies truth-values to at least some of the elementary formulas ofFL containing empty individual constants, thereby filling some of thetruth-value gaps among them.

Of the three standard approaches each one has its merits as well as itsdrawbacks. The first approach with “bare” partial interpretation functionsand without any other auxiliary devices suffers from the arbitrariness ofthe assignments of truth-values to elementary sentences (like, e.g., assign-ing the truth-value F to all of them, as in negative free logic) if we do notwant to violate basic classical logical principles. The outer domain seman-tics suffers either from its dubious ontological presuppositions or from theartificiality of the objects in the outer domain. And supervaluation seman-tics is formally more complex and less transparent than the two other ap-proaches and, moreover, it interferes with the usual relation betweenlogical truth and logical consequence. None of these standard approachesis completely satisfying. The perfect “all-and-everything” approach – ifthere is one at all – has yet to be discovered. From a logical point of view,however, these things may be considered mere matters of taste. The mainpurpose of a semantics for logic consists in providing motives for selectingcertain axioms for formal systems for which proofs of their adequacy, i.e.soundness and completeness, then can be given by means of such asemantics. Each of the standard semantic approaches for free logic servesthis purpose.


5. FREE DESCRIPTION THEORIES

One major motivation for developing free logic has always been to providea basis for theories of definite descriptions. These are phrases of the form‘the so and so’, where the phrase is singular (not plural) and purports torefer to a single thing (in contrast to generic uses of such phrases as in‘The whale is a mammal’ which usually purport to refer to more than onlyone single object). The best known theory of definite descriptions in logicis that of Russell’s “On Denoting”, later incorporated with additional tech-nical apparatus into Principia Mathematica. Russell’s answer to the prob-lems posed by definite descriptions was to deny them the status of singularterms and to regard an expression of the form ‘the so and so’ as needing tobe eliminated in context, where the most important principles governingthis elimination are

(R1) The so and so exists iff exactly one thing is a so and so.(R2) The so and so is such and such if and only if there is exactly one

so and so and it is such and such.

Using the symbols we have already introduced for the language FL+ andits metalanguage, these can be symbolized respectively as

(R1') E!ιvA ↔ ∃v(A ∧ ∀w(A(w/v) → w = v))(R2') B(ιvA) ↔ ∃v(A ∧ ∀w(A(w/v) → w = v) ∧ Bv)

In these formulas we assume that the variable v is free in A. In a definitedescription ιvA of FL+ A is called the scope of the description operator ιvor the basis of the definite description ιvA.

Russell’s principles tell us how to treat descriptions whose scopes areuniquely fulfilled. In particular the following intuitive consequences result:

(R3) E!ιvA → A(ιvA/v)(R4) E!ιvA → ιvA = ιvA

The theory also tells us that there are two ways in which E!ιvA can fail tobe true: one if no object at all fulfils the scope A and the other if more thanone object does. We call descriptions whose scope is uniquely satisfiedproper descriptions and ones whose scope is, for either of the two reasonsgiven above, not uniquely fulfilled improper descriptions.


On the other hand Russell’s theory has one technical and two philosoph-ical drawbacks. The technical drawback is that in operation the theory re-quires elaborate notational conventions about marking scope, which inPrincipia Mathematica itself were not formulated consistently. This can beovercome fairly easily by notational changes. The philosophical draw-backs are more important. The first is that because the identity ιvA = ιvAis false if the scope is not uniquely fulfilled, i.e. if the description is im-proper, the logical principle of self-identity t = t would be violated, pro-vided that we allowed descriptions as substituends for the singular term t.Russell gets round this by regarding definite descriptions as not beinggenuine singular terms at all, but “improper symbols”, which may looklike singular terms but in fact are not, so they cannot congruously be sub-stituted for the singular term t in the principle of self-identity above. Thishowever immediately reveals the second drawback, which is precisely thatRussell fails to treat what look like singular terms and behave like singularterms as singular terms. He is forced to make a distinction between the reallogical form of a sentence and its apparent or grammatical form, and there-fore to propose that the grammatical form of a sentence may mislead us asto its true logical form. While some early analytic philosophers such asWittgenstein and Ramsey regarded this as Russell’s major contribution tophilosophical logic, it is by no means apparent that Russell is right. So ifwe can find a theory of descriptions in which they are treated as genuinesingular terms then we can overcome this drawback of Russell’s theory.Free logic with definite descriptions, as provided for by the language FL+,provides just such a logical framework, and is preferable for that reason.

Russell was not the first modern logician to formulate a theory of de-finite descriptions. He was preceded by Frege in the latter’s Grundgesetzeder Arithmetik (vol.1) of 1893. There Frege provided what he called “asubstitute for the definite article”. Unlike Russell, however, Frege took de-scriptions to be genuine singular terms, subsuming both them and simplenames like ‘France’ under the general title of ‘proper names’. Frege wasnot prepared for scientific purposes to allow proper names or singularterms to be empty. Empty singular terms arise in natural languages in atleast two ways. One is that there are proper names which lack a bearer, aswith ‘Zeus’ or ‘Vulcan’. The other is when we have improper descriptions,as with ‘the leader of the Greek gods’ or ‘the planet closer to the Sun thanMercury’, which are contingently empty, or indeed necessarily empty de-scriptions such as ‘the greatest prime number’ or ‘the least rapidly con-verging series’. Frege’s solution to this difficulty was to provide a referentfor descriptions that would otherwise be improper. In fact he suggested


two ways to do this. One was simply to stipulate that all otherwise im-proper descriptions designate an arbitrarily stipulated object *, such as theempty set ∅ or the number 0 or the truth-value F. So if nothing is A andnothing is B then ιvA = ιvB, and similarly if many things are A and manyare B. The second solution, which relied on Frege’s having set theory orsomething including it (his theory of value-ranges) in his logical theory,was to stipulate that if a predicate A is uniquely fulfilled, then ιvA denotesthe unique object denoted by t such that A(t/v), and if A is not uniquely ful-filled, then ιvA denotes the set {v | A} of things that do fulfil it. This isclearly a different theory to the chosen object theory, because it means thatfor example ιx(x is a dog) ≠ ιx(x is a cat), since the set of dogs and the setof cats are distinct.

For the definite article Frege gave only one axiom, namely

(FrL) t = ιv(v = t)

And he effectively used definite descriptions only once, in the derivationof the naïve comprehension principle of set theory (Theorem I of Grund-gesetze). As this fact indicates, the definite article was implicated by asso-ciation with the assumptions leading to the paradox Russell discovered inFrege’s system, but the fault lies squarely elsewhere, with the theory ofvalue-ranges (extensions of functions, including sets, the extensions ofconcepts). Frege in fact prefaced his definite descriptor not to an open for-mula as we have done but to a name of a value-range.

Now even apart from the inconsistency of his system, Frege was underno illusions about the artificiality of the stipulations by which he closedwhat would otherwise be gaps in reference, hence his use of the term‘substitute for the definite article’ rather than simply ‘definite article’. Buthis reason was that he thought sentences containing empty terms wouldlack reference themselves, and since for him the reference of a sentencewas a truth-value this would mean having truth-value gaps in the midst ofserious science. So in his own terms Frege’s solution is reasonable sincehe was not attempting anything like a linguistic analysis of actual usage,rather a scientifically better substitute.

Nevertheless, there are technical and philosophical drawbacks to Frege’stheory as well. Technically Frege’s axiom (FrL) is rather weak and doesnot suffice to prove the obvious truth of identity of coextensionals

(IdCoex) ∀v(A ↔ B) → ιvA = ιvB


Philosophically the problem is that whether or not the theory is acceptablein its own terms, it cannot be used to analyse or explicate the actual sin-gular definite article, because of its blatant artificiality (in either version).The whole point is that certain descriptions are indeed improper, whereasFrege fudges things so there are none.

Enter free description theory. Because free logic liberates us from theexistential presupposition shared by Frege and Russell, that all “real” sin-gular terms have to refer, we are free to formulate description theorieswhich allow improper descriptions to be genuine singular terms lackingreference, and yet not have the logical system in which they are embeddedfall down around our ears. A further advantage is that by having descrip-tions as genuine terms, albeit sometimes empty ones, we can compareFrege’s ideas directly with Russell’s without having to add the rider thatRussell does not “really” have descriptions as logical units at all. Russell’stheory of descriptions, suitably filtered through a free logic, turns out to becloser to negative free logic, while Frege’s is closer to positive free logic.This also opens the hunting season for interesting and/or acceptable prin-ciples of description theory which may not conform to either historicalprecedent. In fact there is a spectrum of free description theories, just asthere is a spectrum of modal logics or a spectrum of set theories, with dif-ferent principles.

All free description theories have the following principle in common,which we shall call by its most frequently used name of Lambert’s Law:

(LaL) ∀v(v = ιwA ↔ ∀w(A ↔ w = v))

This principle assures us that a description is proper just when we expect,namely when its scope is uniquely fulfilled. Further, on the assumption ofHintikka’s Law of the equivalence of singular existence and the existenceof an individual:

(HiL) E!t ↔ ∃v(v = t)

we can derive all the Russellian formulas (R1)–(R4). To get the negative free logic effect of Russell’s own theory we have to

have a free logic without the principle of self-identity and adopt the axiom

(NDT) ιvA = ιvA → E!ιvA

mindful that it is its contrapositive that makes the Russellian character plain.


To get the effect of Fregean positive free description theory we add(LaL) to a positive free logic with the principle of self-identity. To get thefurther effect of Frege’s first (chosen object) theory we need to add theaxiom

(ChO) (¬E!t1 ∧ ¬E!t2) → t1 = t2

We can also get this effect as a special case for descriptions by adding theprinciple of Identity of Coextensionals (IdCoex) above.

Karel Lambert in particular has devoted much time to formulating anddiscussing the differences among and relative merits of these and otherfree description theories and it is not our intention here to recapitulate thisdiscussion in detail. One novel suggestion worth following might be tohave a theory with more than one description operator, say one Russellianand one Fregean, in play at the same time. Suffice it to say that free logicprovides the ideal proving ground or framework for trying out one’s intu-itions about definite descriptions.

6. HISTORICAL DEVELOPMENT OF FREE LOGIC

Free logic started getting organized as a field of research on its own rightabout fifty years ago. Its roots, however, reach back to the ancients, at leastto Aristotle, to mention only one clear example. The classic passage in theCategories 13b reads as follows:

“It might, indeed, very well seem that the same sort of thing doesoccur in the case of contraries said with combination, ‘Socrates iswell’ being contrary to ‘Socrates is sick’. Yet not even with these is itnecessary always for one to be true and the other false. For if Socratesexists one will be true and one false, but if he does not both will befalse; neither ‘Socrates is sick’ nor ‘Socrates is well’ will be true ifSocrates himself does not exist at all.”24

This passage expresses the basic idea of negative free logic. Indeed itserved as a kind of motto to Ronald Scales’ doctoral dissertation25 inwhich he developed a system of negative free logic.

Aristotle is just one prominent example of a logician anticipating ideasof free logic in earlier time. Such ideas can be found throughout the his-tory of logic from ancient to modern time. Free logic in its strict sense,


however, is a child of modern formal logic. Articles by Henry S.Leo-nard26, Karel Lambert27, Nakhnikian & Salmon28, Jaakko Hintikka29, andby Hugues Leblanc and Theodore Hailperin30, published in the fifties ofthe last century, are commonly rated as the first publications in free logic.

The main reason for the development of free logic was the problemarising from allowing empty singular terms to enter the vocabulary ofstandard systems of QL or QL=. Empty singular terms obviously do notharmonize with certain well-established rules and laws of QL like Existen-tial Generalization (or its dual, Universal Specification). Not every solu-tion of this problem, however, leads to free logic. There are three re-sponses which consist in a mere change of the linguistic framework of QLor QL=, leaving untouched its rules and laws. This can be done by eitherreplacing all singular terms by definite descriptions and by analyzing themaway, e.g., à la Russell’s theory of descriptions; or by eliminating everyempty singular term from the language; or by not allowing empty singularterms to be substituted for variables and thereby excluding them from theapplication of certain logical laws and rules, in particular of ExistentialGeneralization. The first of these three responses does not take singularterms serious enough; the second and the third response suffer from thefact that they must take empty singular terms, even if they can be definedonly via semantics or theoremhood, as a syntactical category of its own.Free logic, on the other hand, does not take the laws and rules of standardQL or QL= as sacrosanct, but is ready to change these laws and rules if,and as much as, necessary, thereby making manifest their existentialpresuppositions. This was done by Henry S.Leonard and Karel Lambert byexplicitly adding the existential sentence E!t as an antecedent to Exist-ential Generalization and Universal Specification for every singular term t(and not only for empty ones). The question remained as to how to definethe existential predicate E!. As interesting as Leonard’s definition wasfrom a metaphysical point of view, it proved unsatisfactory from a logicalpoint of view because he used a modal operator plus quantification overpredicates in his definition of E! 31. It did not take long, however, for thisproblem to be settled definitively by E!t :↔ ∃v(v = t) 32. Hintikka as wellas Leblanc and Hailperin handled the problem without an existence pre-dicate by restricting the application of Existential Generalization to indi-vidual variables (or to bound individual variables, respectively) and byexcluding individual constants (or free individual variables, respectively)from its application33. It was shown that in these systems the weakenedform of Existential Generalization, i.e. ∃v(v = t) → (A(t/v) → ∃vA), is prov-able34 whereas the unweakened version A(t/v) → ∃vA is not35. These first


systems of (positive) free logic were natural deduction systems. The firstaxiomatic system of free logic, due to Karel Lambert, was PFL–, which didnot include an existence predicate nor did it make use of the identity pre-dicate. At the same time the first systems of negative free logic includingan existence predicate were developed by Rolf Schock. The first versionof PFL with E! as a primitive symbol is again due to Karel Lambert36.

The development of a semantics accompanying the formal systems offree logic was prompt. It was Russell’s main antagonist Alexius Meinongwhose theory of objects was the main inspiration for the inner/outerdomain semantics. But such a semantics can be formulated also withouthaving the problematic consequences of Meinong’s philosophy. Such adevelopment of an inner/outer domain semantics was envisaged independ-ently by Nuel Belnap and Karel Lambert already at the very beginning offree logic, but remained unpublished37. A new impetus for such a develop-ment came from Alonzo Church38, and finally such a semantics wasrealized in several ways. In one approach, described in section 4.2, whichis due to Hugues Leblanc and Richmond H.Thomason39, the inner and theouter domain are disjoint sets. In Nino B.Cocchiarella’s approach40 theinner domain is a subset of the outer domain. A third version was de-veloped by Dana Scott41 primarily for definite descriptions and resemblesFrege’s (and Carnap’s) theory of the chosen object.

If we want to do without outer domains some singular terms do not getassigned an object at all, the interpretation function thereby becoming apartial function. In this case the question arises of how to evaluate sen-tences containing such referential “holes”. If we want to avoid truth-valuegaps and save classical logical principles like bivalence, we must settlethese questions by convention. Supervaluation semantics searches for amore natural solution to this problem. It was originally created by Bas C.van Fraassen42 and improved to the form described in section 4.3 by Er-manno Bencivenga43.

7. FUTURE PERSPECTIVES

Free Logic has helped in solving problems in different fields of scienceand philosophy. It has attained interesting results for both its own aims andthe aims of other areas. Many questions and problems of free logic stillremain undecided and wait for a solution.


Free logic is developing today in different directions. We have alreadymentioned in section 5 open problems concerning free theories of definitedescriptions. One of the questions to be answered concerns the exact lo-gical relations among the theories already available. Dealing with suchproblems might also lead to new innovative ideas for the semantics of freelogic. Furthermore, free logic is spreading into different related fields. Onthe one hand it looks for closer connection to modal logic. On the otherhand it seeks cooperation with and application to computer science andother fields like artificial intelligence, cognitive science and linguistics.The papers in this volume serve the purpose of leading into these newareas of cooperation and application of free logic. They show how afterhalf a century of work free logic is still fresh and alive44.


NOTES

1. Lambert (1960).2. Russell (1919), p.203.3. Quine (1954), p.177: “inclusive quantification theory (i.e., inclusive of the empty

domain)”; cf. also Ja∂kowski (1934), Carnap (1937), pp.140 f., Church (1951), Mo-stowski (1951), Hailperin (1953), Schneider (1958), (1961), Leblanc/Meyer (1969).

4. Schock (1964a), (1964b), (1964c), (1968); cf. also (1962), (1965), (1980a), (1980b). 5. Scales (1969).6. Burge (1974), pp.311 ff.7. Lambert (1967), p.139, and in more detail Meyer/Lambert (1968), p.9.8. Leblanc/Thomason (1968).9. I.e., axiom (P–A9) below. Originally, (P–A9) was assumed to be derivable from the

rest of the axioms. This was questioned, however, by Trew (1970), and finally the in-dependence of (P–A9) was established in Fine (1983). This explains why some ver-sions of PFL– do not and others do contain (P–A9). Cf. Leblanc (1982), pp.12, 74(n.7), 117 (n.17), 450 (n.20).

10. Lambert (1963a), pp.290 f.11. Kripke (1963), p.89; cf. the remarks in Bencivenga (1989b), p.128 (pp.18 f. of the

reprint) and in Leblanc (1981), pp.123 f., where Leblanc addresses the passage inKripke (1963) as Kripke’s “flirt” with free logic.

12. Leblanc/Thomason (1966), (1968).13. Leblanc/Meyer (1970a), (1970b).14. Smiley (1960).15. Skyrms (1968).16. Lehmann (1994).17. Kanger (1957), p.3, Mates (1972), p.60; cf. Skyrms (1981).18. Cf., e.g., Leblanc/Meyer (1970a).19. Leblanc/Thomason (1968).20. van Fraassen (1966a).21. Bencivenga/Lambert/van Fraassen (1991), pp.157–167.22. Skyrms (1968).23. Lambert/van Fraassen (1972), pp.179 ff.24. Aristotle (1984), p.21.25. Scales (1969).26. Leonard (1956).27. Lambert (1958 ) ff.28. Nakhnikian/Salmon (1957).29. Hintikka (1959a), (1959b).30. Leblanc/Hailperin (1959).31. Leonard (1956), p.58. The improvements proposed by Rescher (1957), p.67, and

by Nakhnikian/Salmon (1957), p.539, turned out unsatisfactory too. Cf. also Re-scher (1959), pp.163 f.

32. Cf. Hintikka (1959), pp.133, 134, Leblanc/Hailperin (1959), p.239; cf. Kripke(1963), p.90.


33. Cf. Hintikka (1959), pp.129–131, Leblanc/Hailperin (1959), pp.240–242.34. Hintikka (1959), pp.133 f., Leblanc/Hailperin (1959), p.242. In Hintikka’s version

t is thereby taken to be a free individual variable and in Leblanc/Hailperin an indi-vidual constant.

35. Leblanc/Hailperin (1959), p.242.36. Lambert (1965).37. Cf. Bencivenga (1986), p.422 (n.16). The notion of an outer domain is due to

Joseph S.Ullian; cf. Leblanc/Thomason (1968), reprint in Leblanc (1982), p.55 (n.4).38. Church (1965).39. Leblanc/Thomason (1966).40. Cocchiarella (1966).41. Scott (1970).42. van Fraassen (1966a), (1966b), (1968).43. Bencivenga (1980b), (1981); cf. also Meyer/Lambert (1968), Skyrms (1968) and

Woodruff (1984).44. Our survey aims to introduce the reader to the field of free logic and open the way

into the following papers. We have based it in particular on Leblanc (1982), pp.3–16, Bencivenga (1986), Lambert (1991b) and Lambert (1997). Our work wassupported by the Spezialforschungsbereich SFB F012 of the Austrian Fonds zurFörderung der wissenschaftlichen Forschung at the University of Salzburg. Forvaluable suggestions and improvements we are indebted to Alexander Hieke and toHannes Leitgeb.


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