Abstract
Tarski’s model set theoretical analysis of logical truth presupposes a reduction principle, according to which, if a universally quantified sentence is true, then all of its instances are logically true. Etchemendy, in a recent critique, rejects the reduction principle on the basis of what he finds to be intuitive counterexamples. He proposes a philosophical diagnosis of his sense of the failure of Tarski’s account due to its commitment to the principle. Etchemendy’s objections to the reduction principle are avoided when Tarski’s quantificational criterion of logical truth is applied to a Meinongian domain of existent and nonexistent objects, rather than a referentially extensional domain of existent entities only. The conclusion is not that Tarski intended a Meinongian object theory domain for his analysis of logical truth, but that Etchemendy’s criticisms inadvertently show this to be its proper semantic application where the objections are forestalled.
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- 1.
See Etchemendy 1990, 95–124.
- 2.
Tarski 1956 [1936], 416–7: ‘One of the concepts which can be defined in terms of the concept of satisfaction is the concept of model. Let us assume that in the language we are considering certain variables correspond to every extra-logical constant, and in such a way that every sentence becomes a sentential function if the constants in it are replaced by the corresponding variables. Let L be any class of sentences. We replace all extra-logical constants which occur in the sentences belonging to L by corresponding variables, like constants being replaced by like variables, and unlike by unlike. In this way we obtain a class L′ of sentential functions. An arbitrary sequence of objects which satisfies every sentential function of the class L′ will be called a model or realization of the class L of sentences…’ [selected emphases added].
- 3.
The difficulties of distinguishing between logical and extralogical terms are considerable, and I do not mean to downplay the problems involved. For purposes of my countercriticism of Etchemendy’s attack on Tarski, however, I have no need to offer an exact criterion. Where there is a potential for serious controversy about the classification of a term, as with the identity sign, I have, for the sake of argument, granted Etchemendy’s claim that the term is logical, as affording his counterexamples the maximum benefit of doubt. Etchemendy 1990, 111–24. See also his 1988, 69: ‘However, as long as the quantifiers are treated as logical constants, Tarski’s analysis always leaves the domain of quantification fixed. Because of this, sentences like (15) [(∃x)(∃y)(x ≠ y)] will come out logically true on Tarski’s account … This [is] simply because on the present selection of logical constants, there are no nonlogical constants in the sentence to replace with variables. Thus, such sentences are logically true just in case they happen to be true; true, of course, on the intended interpretation.’
- 4.
Tarski 1956, 418–20: ‘Underlying our whole construction is the division of all terms of the language discussed into logical and extra-logical. This division is certainly not quite arbitrary. If, for example, we were to include among the extra-logical signs the implication sign, or the universal quantifier, then our definition of the concept of consequence would lead to results which obviously contradict ordinary usage … Perhaps it will be possible to find important objective arguments which will enable us to justify the traditional boundary between logical and extra-logical expressions. But I also consider it to be quite possible that investigations will bring no positive results in this direction, so that we shall be compelled to regard such concepts as “logical consequence”, “analytical statement”, and “tautology” as relative concepts which must, on each occasion, be related to a definite, although in greater or less degree arbitrary, division of terms into logical and extra-logical.’
- 5.
Sher 1991, 45–52, critically examines Tarski’s definition of logical consequence independently of Etchemendy’s reduction principle attribution. She comes close to recognizing the principle, when she writes, 45: ‘DEFINITION MC The sentence X is a material consequence of the sentences of the class K iff at least one sentence of K is false or X is true. Tarski’s statement first seemed to me clear and obvious. However, on second thought I found it somewhat puzzling. How could all material consequences of a hypothetical first-order logic λ become logical consequences?’ What Etchemendy regards as a false presupposition of Tarski’s theory, Sher, in modified form, considers as a potentially false conclusion. Sher blocks the objection that in some model for λ (4) (There are exactly two things) follows as a Tarskian logical consequence from the contingently false (2) (There is exactly one thing). She argues that for Tarski there is a model for λ with a universe of cardinality α for arbitrary α. Sher concludes, 45–6: ‘Thus in particular λ has a model with exactly one individual. It is therefore not true that in every model in which (2) is true, (4) is true too. Hence, according to Tarski’s definition, (4) is not a logical consequence of (2).’ Although this defense is not directed against an objection that explicitly invokes the reduction principle, it clearly addresses the same concern about overgeneration of logical consequence that Etchemendy summarizes by attributing the principle to Tarski. Sher’s solution involving Tarski’s thesis of the unlimited cardinalities of models for formal languages by itself moreover appears inadequate when applied to some of Etchemendy’s counterexamples. Consider the dilemma Etchemendy builds around the claim that for Tarski either ∃x∃y(x ≠ y) or ¬∃x∃y(x ≠ y) must be true in any λ (regardless of the size of its model’s domains), but that Tarski’s account overgenerates the specifically logical consequences of λ if either formula is true. The problem goes to the heart of the present discussion, where the issue in avoiding intuitive objections to Tarski’s theory of logical consequence is not merely the cardinality of a model’s domains vis-à-vis the population of the actual world, but rather its constitution by exclusively existent or existent and nonexistent objects.
- 6.
Wittgenstein 1922, 5.5351, similarly rejects the (substantive) definition of (something, p’s being a) ‘proposition’ in Russell 1903, 15, as ‘p implies p’. We could as well say equivalently, ∀p[p is a proposition ↔ [p → p]], or, for that matter, ‘p ↔ p’. A proposition p is anything that can imply or be implied. It is anything to which a truth function can relate among any choice of such objects, relating to itself in the simplest and logically most foundational case in the implicational tautology, by which Russell tries to define the concept of proposition. The trouble is that we are then assuming that something is a proposition just in case it can be the argument of a truth function, whereas the domain and range of truth functions, such as Russell’s ‘implies’, are defined over what must then be a predetermined set of propositions. We must have propositions in order to define truth functions, so how can we turn exclusively to truth functions in order to define the concept of a proposition? To put the problem another way, for Russell in 1903, in order to define the concept of a proposition, we must invoke the concept of implication; whereas implication is defined as a formal semantic relation between truth-value-bearing propositions. Russell in 1927 takes the concept of proposition as primitive, and the earlier definition of 1903 is not followed. See Tarski 1986, for an attempt to generalize principles for identifying logical terms from the formalization of a geometrical theorem.
- 7.
In his later 1986, Tarski understood identity and the existential quantifier as purely logical ‘notions’, according to the invariance under the domain-reflexive one-one transformation (function) criterion. The argument here, without entering into the necessary morass of details, seems questionable. Tarski refers to an elegant result he achieved in 1936 in collaboration with Lindenbaum, but which considers only the Principia Mathematica calculus of (existent) ‘individuals’. Invariance arguably is not guaranteed for transformations involving non-(existent entity)-designating terms, at least for the standard extensionally interpreted ‘existential’ quantifier, if not also for the identity predicate or functor, and hence not for what is logically the truly widest class of transformations. It appears again that Tarski’s later criterion for the logical-extralogical term distinction, as a prerequisite for trouble-free analysis of logical consequence, works properly only if invariance under all transformations obtains for a Meinongian semantic domain of existent and nonexistent objects. Otherwise, as Etchemendy complains, Tarski’s criterion merely fortuitously gives correct results for contingently or accidentally populated domains, such as the domain of individuals. Tarski does not notice the problem because he does not consider transformations for formal languages with non-(existent entity)-designating singular terms. The difficulty surfaces in another guise, when Tarski writes, 1986, 152: ‘Are set-theoretical notions logical notions or not? Again, since it is known that all usual set-theoretical notions can be defined in terms of one, the notion of belonging, or the membership relation, the final form of our question is whether the membership relation is a logical one in the sense of my suggestion. The answer will seem disappointing. For we can develop set theory, the theory of the membership relation, in such a way that the answer to this question is affirmative, or we can proceed in such a way that the answer is negative. So the answer is: “As you wish”!’ Since identity and the existential quantifier are also definable in terms of set theoretical membership, it should follow by parity of reasoning that the status of these terms or ‘notions’ as logical or extralogical is equally ambivalent, confirming the suspicions of informal inquiry into their standard meanings. See Stoll 1979 [1963], 26 and 196. A less formal definition is found, for example, in Quigley 1970, 2, as an axiom of elementary set theory, ‘A = B iff (for all z, z M A iff z M B) [M = ∈]’.
- 8.
My understanding of the role of logical constants in Tarskian semantics agrees in important respects with Sher’s 1991 rationale, especially Chap. 3, 36–66, ‘To Be a Logical Term’. See also McCarthy 1981. I am uncertain about the material adequacy of Sher’s analysis, in light of the difficulty in Tarski’s 1986 characterization of logical notions observed in preceding note 7. Sher writes, 1991, 63: ‘In the lecture ‘What are Logical Notions?’ Tarski proposed a definition of ‘logical term’ that is coextensional with [my] condition (E).’ Condition (E) is Sher’s general characterization of formality as invariance under isomorphic structures, and an essential component of her criterion for logical constants in a first-order logic. According to Tarski’s definition, Sher maintains, ‘…the truth functional connectives, standard quantifiers, and identity are logical terms…’
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Jacquette, D. (2015). Tarski’s Quantificational Semantics and Meinongian Object Theory Domains. In: Alexius Meinong, The Shepherd of Non-Being. Synthese Library, vol 360. Springer, Cham. https://doi.org/10.1007/978-3-319-18075-5_10
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