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Dedicated to Diego Marconi, teacher and friend, on the occasion of his sixtieth birthday.

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1Henry Leonard and Karel Lambert first introduced so-called presupposition- free (or just simply: free) logics in the 1950’s in order to provide a logical framework allowing for non-denoting singular terms (be they descriptions or constants) such as “the largest prime” or “Pegasus” (see Leonard 1956 and Lambert 1960). Of course, ever since Russell’s paradigmatic treatment of definite descriptions (Russell 1905), philosophers have had a way to deal with such terms. A sentence such as “the F is G ” involving the description “the F ” is interpreted as saying that there is one and only one F , and that any such F’s are G’s. On such an account, predications involving non-denoting descriptions (most famously, for instance, “the present king of France is bald”) turn out to be false, precisely because there are no objects satisfying the descriptions, uniquely or otherwise. In a classically Russellian move, this treatment can then be extended to possibly non-denoting individual constants by construing them as standing in for definite descriptions, or clusters thereof.

2Russell’s treatment of descriptions, however, does not take them seriously qua singular terms. Descriptions are not explained, but rather explained away, by showing how to rephrase any sentence containing a description by means of a synonymous sentence that does not contain it. While this move deals nicely with the case of non-denoting descriptions, it does nothing to do justice to any intuitions according to which “Pegasus” is a genuine singular term, as is “the winged horse” or “the largest prime.”

3The point of free logic is precisely to allow us - should we wish to do so - to employ a linguistic framework in which those intuitions are taken seriously, and distinguish denoting from non-denoting terms, and even perhaps to give us some account of their referential failure. We would like, for instance, to be able to tell at least the beginning of a story as to why descriptions such as “the winged horse” or “the golden mountain” fail contingently, whereas “the largest prime” or “ the unique x such that x does not equal x” fail necessarily. (Giving such an account would thus require mixing free and modal logics, as free logicians have often proceeded to do.)

4Many variants of free logics have been introduced since Leonard’s first forays in the mid-1950s. An extensive survey can be found in Bencivenga 2002 and Lehman 2002. But all variants agree on their treatment of bound variables, in that they cohere with Quine’s (1948; 1953) injunction that to be is to be the value of a bound variable. In other words, all variants of free logics assume the existence of some domain D of objects, over which bound variables are taken to range. The domain D comprises the set of “existents” — presumably the medium-sized physical objects we are all acquainted with, together with whatever theoretical, fictional, or otherwise unattainable entities upon which we choose to bestow actual existence.

5It follows that in free logic it is possible to introduce an existence predicate by means of an explicit definition. To claim that a term t denotes an object in D one only has to assert

6x (x = t),

7which is often abbreviated as E!(t). There is of course a long-standing philosophical tradition, going back at least to Kant, according to which existence is not a predicate on a par with “is bald”; again, it was Russell the first to claim that to make an existence claim such as “Unicorns exist” is not to ascribe a property to any object(s), but rather to make a second-order claim according to which the propositional function “x is a unicorn” is exemplified. Free logic takes instead the opposite view, and claims that existence is indeed a first-order predicate, albeit perhaps one that needs to be distinguished from real predicates such as “is bald” (see Bencivenga 1980, Miller 2002, Parsons 1980, Pears 1967, Zalta 1988).

8Then we have the further question of how to deal with non-denoting terms, whose referents (if any) cannot be in D. The many different variants of free logics differ precisely in their answers to this question. Before we turn to these various answers, notice, though, that D is often allowed to be empty, thereby representing the possibility that there might indeed be nothing at all. Unsettling as such a state of affairs might be for the faint metaphysical heart, its consequences in point of logic are even more curious. While of course a lot depends on the precise details of the formalism, a prima facie case can be made that over such a domain all existential claims are false, whereas all universal ones are true. Even more disturbing for the classically trained logician is then the fact that over an empty domain vacuous quantification is no longer idle: for, when x is not free in φ, is false but y is true. Versions of free logics that allow for the possibility of an empty domain are called inclusive.

9Logicians have further distinguished among variants of free logics based on the way they handle atomic predications involving non-denoting terms such as, for instance, “Pegasus is a horse.” One option is to assume that none of these predications have a truth-value. The intuition is that for an atomic predication of the form P (a) to be true the object denoted by a needs to fall within the extension of P, and for the predication to be false the object denoted by a needs to fall outside the extension of P. If neither of these conditions is met (because there is no object denoted by a, then P (a) is neither true nor false, i.e., truth- valueless or exhibiting a truth-value gap. This way of looking at the issues dates back at least to Frege, and found its strongest proponent perhaps in Strawson (1949), according to whom a precondition for a sentences having a truth value is that the singular terms in that sentence denote. When the precondition fails, the sentence is truth-valueless.

10Many free logicians are however committed to the opposing view, bivalence, i.e., the idea that sentences are either true or false, as long as they are grammatical. Accordingly, they are faced with the problem of assigning a truth-value to atomic predications involving non-denoting terms. Among bivalent free logics, a few adopt the solution of stipulating that all such predications are simply false, including “Pegasus is winged” or even “Pegasus = Pegasus”. For instance one such proposal is due to Scott (1967) who assigned one and the same object to all the non-referring terms, thereby validating all identities t = t' involving such terms (including self-identities).

11In what follows I will be concerned with positive free logics of the inclusive variety, i.e., free logics that allow for an empty domain as well as for at least some atomic predications (including identity) involving non-denoting singular terms to be true. Before proceeding with some of the more formal material, it will be useful to review some basic facts concerning definite descriptions and some of the surrounding issues.

12Definite descriptions are expressions of the form “the so-and-so” that purport to refer to one and only one individual. Russell raised the issue of the interpretation of sentences containing definite descriptions that fail to pick out exactly one individual. As mentioned, he proposed to re-interpret sentences involving definite descriptions in term of explicitly given existence and uniqueness conditions, in such a way that the truth conditions of a sentence such as “the φ is ψ” are given by

13x [y (y = x ↔ φ(y)) ˄ ψ(x)].

14The above formula says that there is one and only one φ, and that such a φ is a ψ. In fact, Russell introduced a description operator ι that binds a variable and a formula (just like a quantifier), but returning a singular term (unlike quantifiers, which return formulas). The term ιxφ(x) represents “the φ”, and is to be defined contextually by means of the displayed formula above.

15Free logicians agree with this treatment of ιxφ(x) when the existence and uniqueness conditions are met (although they might insist that ιxφ(x) should be construed as a genuine singular term not to be explained in context). However they disagree with Russell’s treatment of ιxφ(x) when the conditions are not met and ι x φ(x) is thus irreferential.

16Positive free logicians, in particular, are concerned with a number of principles involving potentially preferential terms that could plausibly be considered for adoption. Here is a sample list, ordered by increasing logical strength:

  1. . ιxφ(x) = ιxφ(x) ;

  2. . x [x = ιyφ(y) ↔ z(z = x ↔ φ(z))] (Lambert’s Law);

  3. . t = ιx (x = t) ;

  4. . φ(ιxφ( x) ).

17Being inconsistent, this last principle is clearly the strongest of them all, as it can be easily seen by taking φ( X ) to be ψ( X ) ¬ ψ( X ). Even when the logical hierarchy among these principles is established, intuitions differ as to their status. Intuitions being what they are, there seems to be little one can do to settle the issue here, except perhaps note that the weakest of these is also the one that appears to be most plausible. Lamberts law gives identity conditions for definite descriptions, and appears to be similar to other abstraction principles found in the literature, beginning most (in)famously, with Frege’s “Basic Law V”. The third principle states that t is the unique x that equals t, which is obviously valid when t denotes, but questionable when it does not.

18Just as in the case of classical logic, investigations into (positive) free logic first began proof-theoretically. People were at first concerned with giving a set of axioms for positive free logic, to be assessed (like most axiom systems) based on the plausibility of their consequences and their fit with the relevant intuitions. The first such is due to K. Lambert, and it comprises the following axiom schemata:

  1. All classical tautologies;

  2. . x (φ → ψ) → (xφ) →xψ);

  3. . x (φ(x) → (E!(t) → φ(t));

  4. . xyφ → y;

  5. . t = t;

  6. . t = t' → (φ(t) → φ(t')) ;

  7. If φ is an axiom and x a variable, then is an axiom.

19Interestingly, Lambert has shown that in the context of the other axioms, universal instantiation (3) is equivalent to

20. y (xφ) → φ(y)).

21Intuitively, it is clear why this axiom achieves the desired effect: given that the universal quantifier is constrained to range over the domain D of existents, Lamberts axiom asserts that universal instantiation is admissible only for terms denoting objects in D.

22We now turn to the task of reviewing several proposals (some old, some new) to provide a semantics for positive free logic. The first proposal we consider is van Fraassen’s (1966) supervaluational semantics. Supervaluations were indeed developed in connection with free logic, but were soon applied to the analysis of a number of linguistics phenomena, such as vagueness, in which truth-value gaps would seem to appear.

23Van Fraassen’s starting point is the notion of a partial interpretation , defined to be a triple (D, Π, ρ) where D is a possibly empty domain, Π provides an interpretation for the non-logical constants of the language (e.g., subsets of Dn to n-place predicate symbols, etc.), and ρ is a partial reference function assigning objects in D to some, all, or none of the individual constants of the language.

24The idea of supervaluations rests on the notion that partial interpretations can be ordered under extension: we say that ' extends if and only if ' agrees with on the items to which the latter assigns a truth-value (while more sentences are possibly assigned a truth value on '). We can then say that ' is a completion of if and only if it is a maximal interpretation extending . Completions are obviously classical interpretations in the sense of ordinary first-order logic. (Details are purposely left vague here, since the properties of supervaluations depend on exactly how extensions are defined, but such a semantic approach will not be our focus.)

25Then, given a partial interpretation , we say that φ is supertrue on if and only if it is true on every completion of ; superfalse on if and only if it is false on every completion of ; and supervalueless otherwise. On these definitions it is clear that many classical validities are recovered, which was exactly part of the motivation. For instance, a sentence such as P( a ) ˅ ­ P( a ) turns out supertrue on every , even when a is non-denoting in (and so the sentence is truthvalueless on ).

26However intuitively attractive supervaluations might be, supervalidity turns out to be non-axiomatizable. It was indeed shown by Woodruff (1984) that supervalidity is Π11-complete (intuitively this can be seen by considering that supertruth contains a hidden universal quantification over all possible extensions of a predicate). Essentially because of this reason, we will not return to this semantic proposal.

27The most widely adopted semantic framework for positive free logic is the so-called “outer-domain” semantics, which was first proposed by Leblanc & Thomason (1968) and by Meyer and Lambert (1968) (at about the same time Belnap independently had an unpublished formulation of outer-domain semantics independently as well). In outer-domain semantics, one has two distinct domains of objects, the first one of which (the “inner” domain) comprises the existent objects, and the second one (the “outer” domain) all remaining objects. Accordingly, an interpretation is a structure (D0, D1, Π, ρ) where: each Di is possibly empty, subject to the requirement that D0 D1; as before, ρ assigns objects from D0 D1 to the constants; and Π provides interpretations for the predicate symbols. The crucial feature of outer-domain semantics is that in defining satisfaction conditions for quantified formulas, assignments to the variables are constrained to take values in the inner domain D0. Then, this does justice to Quine’s dictum concerning the values of the variables. Notice also that whenever the inner domain is empty, no assignment s to the variables has any x-variants, so that again universally quantified formulas are true and existentially quantified ones are false.

28While outer-domain semantics is complete for Lambert’s axiomatization of positive free logic, it is also philosophically suspect. The reason is that in the eyes of many, non-existent objects should be — well — non-existent. The semantics is also questionable from a more technical point of view, in that it seems to reduce free logic to an exercise in restricted quantification (as Church famously pointed out). Similar considerations prompted Lambert to express the following desiderata:

It would be nice to have a gapless, bivalent semantic development for positive free logic in which the model structure is of the single domain variety, but the interpretation function applied to singular terms is partial, and doesn’t appeal to senses or things of that kind. There is such a semantics for negative free logic, ... but not one for positive free logic — at least there is not one in which the truth-values of statements with irreferential singular terms approximate the ones we have clear intuitions about. So there is an open problem for you. (Lambert 1997: 80)

29Proto-semantics for positive free-logics was developed mainly in response to these considerations of Lambert’s. Proto-semantics does indeed provide a “gapless, bivalent” and “extensional” approach of the “the single domain variety” (see Antonelli 2000).

30Proto-semantics revolves around the notion of a proto-interpretation for the language. As before, let D be a possibly empty domain, and ρ a partial reference function assigning objects from the domain to some subset of the constants of the language. Then, a proto-interpretation π is a function assigning to each predicate P a signed extension, where a signed extension is a pair of the form (S, +) or (S, -), where S D x... x D as before, and the two signs + and are arbitrary markers. Let us say that proto-interpretations π and π' are equivalent if for any predicate P, we have that π(P) and π'(P) differ at most on the sign of the extension of P .

31Having defined proto-interpretations, now we can proceed to define the actual interpretations for the language: an interpretation Π assigns to each n-tuple t1,... ,tn of terms a proto-interpretation π, subject to the following two requirements:

  1. If π and π' are both in the range of Π, then π is equivalent to π';

  2. If Π(t) = Π(t') then Π(t1... = Π(t1... t';

32We often write Πt1…tn for Π(t1... tn). Finally, we define a model M to be a structure (D, Π, ρ, ≡) satisfying the following principle of indiscernibility: a necessary condition for t1 ≡ t2 to hold in M is that both

  1. If either one of ρ(t1) and ρ(t2) is defined then so is the other one, in which case ρ(t1) = ρ(t2); and

  2. If both ρ(t1) and ρ(t2) are undefined, then Πt1 = Πt2 .

33Notice that the relation is treated more like a non-logical constant - indiscernibility - than true (metaphysical) identity. (Incidentally, the above fixes a mistake in Antonelli 2000 pointed out by the late Gumb 2001.)

34We finally define the truth of a sentence in a model. In fact, we will give a simultaneous recursive definition of truth in a model for arbitrary sentences as well as reference for arbitrary terms (atomic or descriptions). Assume that the model M is saturated, in the sense that every d D is denoted by a constant Cd. We first deal with the case for atomic sentences:

  1. If ρ(t1) is defined for 1 ≤ i ≤ n, and Πt1…tn = (S, ±) then P(t1... tn) holds in M if and only if ( ρ ( t1 ), ... , ρ (tn)) S;

  2. If ρ(t1) is undefined for some 1 ≤ i ≤ n, then P ( t1... tn) holds in M if and only if Πt1…tn = (S, +) (i.e., if the extension assigned by Π t1…tn to P has a positive sign).

35The definition of truth for the atomic case is straightforwardly extended to Boolean combinations of atomic sentences by means of truth-functional connective. The case for the quantifiers is handled substitutionally: the sentence x ψ(x) holds in M if and only if ψ(Cd) holds in M for every d D. And finally we can extend the function ρ to definite description, as follows: if there is a unique d D such that ψ(Cd ) holds in M, then ρ(ιxψ( x )) = d, and ρ(ιxψ( x )) is undefined otherwise.

36This completes the description of proto-semantics. It is worth noting that the semantic framework is well-behaved from a meta-logical point of view: one can prove a completeness theorem with respect to Lambert’s axiomatization given above; in fact standard first-order results (compactness, Craig interpolation, Beth definability) extend rather naturally to proto-semantics, and one can even read off an exponential upper bound on the increase in size of the interpolant over the classical case (see Antonelli 2000 for details). Indeed, proto-semantics has been viewed as “the most philosophically palatable standard semantics to date” (Gumb 2001).

37We will come back to an evaluation of these proposals at the end of the paper, but in order to assess the special kind of quantification implicit in free logic it will be necessary to digress and take a more abstract look at quantifiers in general. The modern view of quantifiers undoubtedly originates with Frege, who in section 21 of his Grundgesetze der Arithmetik asks us to consider how a statement such as x(x2 = 4) is obtained from the general from xφ( x ) by replacing x2 = 4 for φ( x) (where φ( x ) is now viewed as a place-holder for first-level “concepts”). Accordingly the general quantifier form x φ(x) is, in Frege’s terminology, a “second-level concept” or a concept of concepts. In particular, the existential quantifier can be identified with the second-level concept under which all and only the (first-level) concepts fall that are exemplified (a view later picked up by Bertrand Russell).

38In spite of these pioneering remarks, the modern view of the quantifiers did not really emerge until relatively recently, especially with the work of Mostowski (1957) and Montague (1970,1973). Both linguists and logicians have been interested in generalized quantifiers, the former as tools for the representation of natural language, and the latter as expressive pieces of logical machinery.

39In modern terminology, a quantifier Q over a domain D is just a collection of subsets of D, for instance:

  1. . ∀ = {D};

  2. . ∃ ={X ⊆ D: X ≠ ∅};

  3. . ∃!k ={X ⊆ D: card (X) = k};

  4. John ={X ⊆ D: John ∈ X}.

40According to this representation, a formula of the form Q xφ(x) is true (in a model M) if and only if the extension of φ(x) in M belongs to the quantifier Q; in the case where Q is , for instance, xφ(x) is true if and only the extension of φ(X) belongs to the quantifier , i.e., if such an extension is just D : in other words, if and only if every object in D falls within the extension of φ(X), as desired.

41Similar considerations show that the other quantifiers mentioned above deliver intuitively correct truth conditions as well: the existential quantifier can be identified with the collection of all the non-empty subsets of D, so that xφ( x ) is true if and only if the extensions of φ( x ) belongs to , i.e., if such an extension is non-empty. Somewhat more interesting is the last case mentioned above, which shows that singular terms can be regarded as quantifiers as well (these are called “Montagovian individuals”). So the term John can be identified with the collection of all subsets of the domain having John as a member. It follows that “John runs” is true if and only if the extension of “run” belongs to the quantifier John, i.e., if and only if John belongs to the extension of “runs”.

42It is also worth mentioning that an equivalent notion of quantifiers is due to Lindström, according to whom a quantifier can be viewed as a class of structures. Let the language L contain a quantifier Q. Then we have seen that for each L-model M, Q determines a family of subsets A of D. Write Q(A) whenever A belongs to Q. Now let instead L' be the expansion of L by a new predicate symbol P . Then Q can as well be identified with the class of all L'-structures of the form (M, A), where A belongs to Q. Then we have that Q(A) holds (in the old sense) if and only if (M, A) belongs to Q (in the new sense). Moreover, Q x φ(x) holds in M if and only if x (P x ↔ φ( x)) holds in (M, A), for every A in Q.

43All quantifiers we have considered so far are “unary” in that they apply to single subsets of the domain. However, many quantifiers are best viewed as “binary” quantifiers, applied to ordered pairs of subsets of the domain:

  1. “All A’s are B’s”: All ={(A, B): A B};

  2. “Some A’s are B’s”: Some = {( A, B): A ∩ B ≠ ∅};

  3. “Most A’s are B’s”: Most = {( A ,B): card (A ∩ B) > card ( A - B)}.

44Now it is true that some binary quantifiers can be represented by means of a unary quantifier applied to a Boolean combination of their two arguments. For instance, as we teach beginning logic students, All can be represented by means of applied to a Boolean combination of A and B (viz., ( D - A) B). But this is just a coincidence (and an unhappy one at that, because it hides an important point from the students). As a case in point, the quantifier Most cannot be so represented.

45Binary quantifiers of the form Q(A, B) can have several important properties. One of them is conservativity, a property enjoyed by virtually all quantifiers that can be expressed in natural language by means of one word (such as “All”, “Some”, etc.). Formally, conservativity is the following property: Q(A, B ) holds if and only if Q (A, A ∩ B) holds. For instance, “All A’s are B’s” is equivalent to “All A’s are A’s that are Bs” and “Some As are B’s” is equivalent to “Some A’s are A’s that are B’s”. (The one exception to the above claim is the quantifier “Only”, which is not conservative - a fact taken by some to be a sign that, appearance to the contrary notwithstanding, “Only” is not really a natural-language quantifier, or at least not in the same sense as “All” and “Some”.) Among the other properties we should mention also right monotony, if Q( A, B) and B C then also Q(A, C) (the quantifiers “All” and “Most” have this property); and right anti-monotony. If Q(A, B) and C B then also Q( A, C) (the quantifiers “No” and “Few” have this property).

46From our point of view, a particularly important property is invariance under permutation. As before, let Q be a quantifier over some domain D (suppose Q is unary, for definiteness) and let π be a permutation of D, i.e., a function that maps D onto D in a 1-1 fashion. The permutation lifts to subsets of D in a natural way, by putting: π [A] = {π ( d ): d A). A quantifier Q is invariant under permutation if and only if A belongs to Q if and only if π [A] belongs to Q.

47Notions and operations that are invariant under permutation in this sense are considered to be “logical” notions - a tradition that originated in logic with a lecture Tarski delivered in 1956 (see Tarski 1986), but that can be traced back to Klein’s Erlanger Program. The idea is that logic is maximally general, and therefore independent of the subject matter; it would seem to follow that logical notions should not be sensitive to permutations of the domain - that one ought to be able to interchange the objects in a one to one fashion, leaving the logical notions unaffected. And in fact, notice that the ordinary quantifiers , , as well as their binary counterparts “All” and “Some” are invariant in this sense. If a subset A of the domain is non-empty (or universal, or having at least k members, ...) then the same holds of its image π [A]. In contrast, Montagovian individuals, i.e., quantifiers of the form

48{X D : a X},

49for some object a D, are not invariant in this sense (as can be seen by exchanging a with some other object b ≠ a ). And indeed, facts about individuals are not “logical” and therefore perturbed by permutations of the domain. While the adequacy of invariance as a logicality criterion can be (and indeed has been) questioned (for instance, Feferman 1999 points out that it does not allow comparisons across domains), there seems to be consensus among philosophers of logic that invariance provides at least a prima facie necessary condition for the logical character of some notion or operation.

50It is useful at this point to further digress (before we return to our main topic) and consider briefly second-order quantifiers. In the modern abstract approach, second-order quantifiers are identified with collections of (or, in the general case, relations over) first-order quantifiers. Consider the sentence P φ(P): this sentence is true if and only if the set of subsets A D that satisfy φ(P) is nonempty. The second-order existential quantifier can then be identified with the collection of all non-empty subsets of the power-set of the domain; and similarly the second-order universal quantifier can be identified with the set containing the power-set of the domain as its only member.

51It is well known that second-order quantifiers (on this analysis) have expressive power that far exceeds that of ordinary first-order logic - the flip of this fact being that second-order logic lacks the nice meta-theoretic properties that we have come to expect from the first-order predicate calculus (axiomatizability, completeness, compactness, upward and downward Löwenheim-Skolem, etc). Partly to address these shortcomings, logicians (beginning with Henkin 1950) have developed a general interpretation of the second - and higher - order quantifiers. On such an interpretation, for instance, the second-order existential quantifier no longer ranges over all non-empty subsets of the power-set of the domain, but only over some collection thereof. Such an approach regains all the pleasant features of first-order logic (at the cost of expressive power, of course) and is tantamount to switching to a multi-sorted version of first-order logic (with one sort of variables for individuals and another sort for sets thereof, with predication — i.e., membership — no longer viewed as a logical constant). This is a well-motivated and completely natural way to generalize the meaning of a quantifier by weakening its expressive power.

52Having established this, it will be perhaps somewhat unexpected that a similar maneuver can be carried out at the first-order level as well. It is indeed a bit surprising that apparently this move has not appeared before in the literature. Just as logicians have provided a non-standard or general interpretation for the second-order quantifiers, the same can be done for their first-order counterparts!

53Let L be a first order language comprising the usual Boolean connectives and an assortment of predicate symbols (including identity), as well as a quantifier symbol *. We provide an interpretation for the quantifier by allowing it to range over some collection of non-empty subsets of the domain (i.e., not necessarily the collection of all such subsets). This means that each model M will now come equipped with a collection E* of non-empty subsets of D, with the satisfaction clause for the quantifier now stating that * x φ(x) is true in M if and only the extension of φ(x) (i.e., the collection of those d D that satisfy φ(x)) is a member of E*. This is a perfectly adequate truth definition for our language. Having defined *, this automatically determines a dual quantifier *, defined as ¬ * x ¬ φ(x). Indeed it is not difficult to see that just as * ranges over a collection of subsets omitting ∅, so * will range over a collection of subsets containing D.

54It is also important to notice that neither * nor, therefore, its dual * are permutation invariant. This is because there will be collections E* of non-empty subsets of the domain (over which * is taken to range), that will not be fixed under all permutations. For instance, if E* contains (say) {a} but not {b}, then any permutation exchanging a and b will move E*.

55What is the logic of *? The answer to this question is immediate once we realize that general models for * and outer-domain models give rise to the same set of validities. In fact given an outer-domain model M of the form (D0, D1, Π, ρ) we can define a generalized model by putting

56E* = {X D: X ∩ D0 ≠ }.

57Conversely, given a general model equipped with a collection E* , we can obtain the inner domain D0 as a choice-set for E*. In other words, we can obtain D0 by selecting a member from each set in E* (possibly with repetitions); the members so selected can be regarded as witnesses for the existential claims satisfied by E* (not unlike in Henkin’s 1949 famous completeness proof). While this mapping will not in general preserve truth, it will preserve validity, in the sense that if a sentence fails in a general model it will also fail in the corresponding outer-domain model, and vice versa. Now we see that the logic of ∃* is positive free logic.

58So our quest for a natural and well-motivated semantics for positive free logic has led us to * and *. These quantifiers arise from their standard counterparts by a maneuver broadly accepted by logicians and philosophers. After many, somewhat artificial attempts (including proto-semantics) a case can now be made that these general first-order models occupy a privileged position among the various proposals for positive free logic. Unfortunately, * and * are not invariant under permutation, which can lead us to question the logical character of free quantification.

59Of course, we have also known all along that outer-domain semantics is non-invariant in this sense (all that one needs to do is permute objects in such a way that the extension of some formula φ that meets D0 no longer does so after the application of π). But outer-domain semantics was not only artificial, but also philosophically suspect, so that failure of invariance could arguably be considered the least of its problems. Now, instead, we have a proposal that can reasonably be regarded as the right account of free quantification, and that now turns out to be questionable on logical grounds.

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G. Aldo Antonelli, «Free quantification and logical invariance»Rivista di estetica, 34 | 2007, 61-73.

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G. Aldo Antonelli, «Free quantification and logical invariance»Rivista di estetica [Online], 34 | 2007, online dal 30 novembre 2015, consultato il 13 juin 2024. URL:; DOI:

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