A Solution to Fine's Paradox of a Variable
In [2] Kit Fine formulated the paradox of a variable. Roughly, it isolates a conflict between two intuitions concerning the semantic role of a variable. According to the first intuition a semantic role of a variable is exhausted by the range of its values. Consequently, any two variables turn out to be synonymous. On the other hand, in the context of an expression (a formula and/or polynomial), it happens that distinct variables play a different semantic role. Hence, a contradiction. In the first part of the talk we zoom in the structure of Fine’s reasoning and abstract away from the Fine’s example. As the outcome, we extract tacit assumptions, which lead to the paradox. These assumptions are principle of compositionality and a model-theoretic account of the meaning of a first-order formula in a structure. As the next step towards our solution to the paradox we discuss in detail the notion of the semantic role of a variable. We find it ambiguous. We distinguish between the syntactic role of a variable and the semantic role of a variable. The former amounts to cross-identification of argument places in the context of expressions such as a first-order formula and a polynomial. With this distinction at hand we
relate Fine’s paradox to principle of alphabetic innocence. Principle of alphabetic innocence requires that the meaning of a first-order formula must be invariant under each injective renaming of free occurrences of variables. Therefore, it provides a constraint on any possible synonymy relation. This principle is attributed in [9] to Kit Fine. We follow [9] and claim that a solution to Fine’s paradox of a variable calls for an alphabetically innocent semantics for firstorder languages. Admittedly, principle of alphabetic innocence is not a well-known semantic principle. Notwithstanding, we provide a direct vidence from a seminal model-theoretic source [8], which indicates a tacit endorsement of this principle. Preliminary speaking, our strategy of solving Fine’s paradox is to reject the principle of ompositionality and retain the modeltheoretic account of the meaning of a first-order formula. Furthermore, our rejection of compositionality is going to be carefully motivated.
In the second part of the talk we focus on the model-theoretic semantics. We frame our discussion in the historical context and refer to [10]. Therein the full-blooded theory of the meaning of a first-order formula can be found. It identifies the meaning of a formula with the relation it defines. Interestingly, we isolate a serious ambiguity, which is implicit in [10]. Namely, the notion of a relation is understood in a two-fold way. The first reading is purely settheoretic and is assumed in the contemporary model-theory. On the other hand, the second reading incorporates variables in the criteria of identity of relations. Apparently, the consequence of the former is that semantics is non-compositional and the relation of synonymy in a structure does not coincide with the relation of equivalence in a structure. With this at hand, we analyze the standard way of salvaging compositionality [6]. We conclude that it falls short of satisfying the pre-formal reading of compositionality. This curious situation may be perceived as troubling. Apparently, a formula can receive two distinct and well-motivated semantic values. In order put worries to rest, we recall that Dynamic Predicate Logic [5] yields distinct relations of synonymy (though in a different context). Accordingly, we argue that our findings should not be interpreted as a methodological and/or semantic anomaly. Finally, we highlight that [10] contains enormously useful semantic resources. These resources are operations on relations of possibly different lengths, which entered the canon of relational database theory [1].
With these findings at hand, we switch to the contemporary perspective and observe that a semantic value is not assigned to a formula but to a formula in the finite variable context. The latter plays the role of a semantic representation of a formula. We argue that the notion of representation of a formula does not appear to be syntactically and semantically wellunderstood [3]. Consequently, we aim to develop a theory of representation of a formula. We observe that there exists just one formation rule for building representations, which requires that a representation contains all free variables of a formula. Furthermore, this rule does notdecide in what way the relevant set of variables is ordered. We single out two possible orders: (a) canonical order and (b) order of occurrence. Next, we issue few arguments, which support the choice of order of occurrence. Subsequently, we develop the proper grammar, which inductively generates a set of representations of formulas in order of occurrence. Furthermore,
we enquiry into substitution operations for first-order languages. This is due to the fact that the principle of alphabetic innocence is formulated by means a substitution operation. We define an injective replacement of variables in a formula, which is obviously capture avoiding. We prove the substitution lemma for this operation. As a side issue we show how the variant of the relation of alpha-equivalence, proposed in [4], can be generated by means of our variable transformation operation. Next, we proceed towards construal of our alphabetically innocence semantics. It is parasitic on the notion of representation. Admittedly, we follow Tarski’s suit [10]. Namely, we exploit operations found therein but ensure sensitivity to the order of occurrence. We point out subtle differences between our approach and Tarski. Namely, we index negation operations by natural numbers and treat the empty set as a polymorphic relation of any arity. This enables to recover, unlike in the Tarskian case, the law of double
complementation and interdefinability of sentential operators. By resorting to our substitution lemma, we prove that our semantics is alphabetically innocent. Therefore, we provide a solution to Fine’s paradox of a variable. Furthermore, we show that our semantic allows to define the converse relation. Hence, we refute the claim found in [9]. Finally, we critically discuss our semantics. We notice that our semantics inherits features of semantics discussed previously. These features are not a generic consequence of alphabetic innocence. Indeed, they follow from the choice of semantic values of formulas. A more serious upshot is the so-called syntactic contamination of our semantics [11]. Namely, it contains the record of the underlying syntax. Notwithstanding, we argue that in one way or another syntactic contamination besets each model-theoretic variant of the meaning of a formula and set-theoretic interpretations of polynomials. This uneasy feature of our semantics is a consequence of our desire to keep an alphabetically innocent semantics close to the standard Tarskian semantics.
References
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