Non-logical Constants Research Articles

Dasgupta's Detonation

Dasgupta's Detonation

Logical Form, Truth Conditions, and Adequate Formalization

Abstract I discuss Andrea Iacona’s idea that logical form mirrors truth conditions, and that logical form, and thus truth conditions, are in turn represented by means of adequate formalization. I criticize this idea, noting that the notion of adequate formalization is highly indefinite, while the pre-theoretic idea of logical form is often much more definite. I also criticize Iacona’s claim that certain distinct sentences, with the same truth conditions and differing only by co-referential names, must be formalized by the same formula (in the same context). I criticize this claim, noting that it imposes implausible demands on adequate formalization. Finally, I offer some brief remarks on the connection between Iacona’s ideas and the distinction between logical and non-logical constants.

LOGICAL CONTEXTUALITY IN FREGE

AbstractLogical universalism, a label that has been pinned on to Frege, involves the conflation of two features commonly ascribed to logic: universality and radicality. Logical universality consists in logic being about absolutely everything. Logical radicality, on the other hand, corresponds to there being the one and the same logic that any reasoning must comply with. The first part of this paper quickly remarks that Frege’s conception of logic makes logical universality prevail and does not preclude the admission of different contexts of discourse. The paper then aims to make it clear how Frege’s universalism can make sense of contextuality. Drawing on a suggestion made by Frege in his discussion of Hilbert, it shows that a properly Fregean notion of model can be devised. Taking up a suggestion from Wilfrid Hodges and William Demopoulos that the non-logical constants of a formal language can be compared to indexicals, this paper shows,paceHodges and Demopoulos, that such an understanding of non-logical constants is not beyond Frege’s horizon. A formal framework, based on the modern tool of fibrations, is set out to explain and justify this point. This framework allows one to compare Frege and Tarski, by formalizing Frege’s suggestion and by presenting Tarski’s semantics in a generalized setting.

First-order modal logic in the necessary framework of objects

AbstractI consider the first-order modal logic which counts as valid those sentences which are true on every interpretation of the non-logical constants. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, Timothy Williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. He notes that only the cardinalities of these sets matters, and that not all pairs of infinite sets determine the same logic. I use so-called two-cardinal theorems from model theory to investigate the space of logics and consequence relations determined by pairs of infinite sets, and show how to eliminate the assumption that worlds are individuals from Williamson's argument.

Summary

What is the relation between modal logic and modal metaphysics? As a technical discipline, modal logic is in effect a branch of applied mathematics that studies extensions of standard formal languages by additional sentence operators, independently of any particular interpretation of those operators. Modal metaphysics studies the nature and extent of possibility and necessity, where those terms are understood in a strong non-epistemic ‘metaphysical’ sense: something is necessary just in case it would have obtained no matter what, and possible just in case it does not necessarily not obtain. Informally, metaphysicians can read the operators □ and ◊ of modal logic as expressing such necessity and possibility, although that reading plays no official role in the technical development. When the modal language also contains quantifiers, metaphysicians will naturally read them as unrestricted. That metaphysical reading of the modal operators and quantifiers will be assumed in what follows. Metaphysicians want to know which formulas of the modal language are metaphysically universal , in the sense of being true on every interpretation of the non-logical constants and free variables in them. The metaphysically universal formulas, whichever they are, constitute the core metaphysical theory of the most general structure of modal reality. We can use modal logic as a technical aid in studying that theory.

EXPLANATORINESS: CAUSE VERSUS CRAIG

In "Toward a Theory of the Process of Explanation", Ilpo Halonen and Jaakko Hintikka (2003) present two interlocking theses, one positive, one negative. Their positive thesis is that philosophical inquiries into the nature of explanation should begin with a theory that explicates the logical form of the process of explaining, [ms. 4] Given that explanations are the product of such processes, a theory of explanation will emerge as a con sequence of the theory of explaining. Their negative thesis is that the nature and importance of what they call the "covering law model" of explanations has been dramatically misunderstood. By analyzing the logical form of the process of explanation, Halonen and Hintikka endeavor to show that covering laws should be viewed not as premises, but rather as 'summaries' of explanatory arguments. The core of Halonen and Hintikka's positive thesis is a theory of the process of explaining which the authors dub the "interrogative nomological" model. Since the interrogative-nomological model is essen tially a special case of Hintikka's interrogative model of inquiry, [ms. 4] the explanatory process overall is viewed as an interrogative "argument", aimed at finding an answer to the question why E?, where E is the ex planandum. Such argument begins with a (possibly informal) background theory T, then proceeds with the explainer posing "questions" to nature (experiments) and nature yielding "answers" (results of experiments). At least a part of this process is aimed at establishing some suitable initial conditions / such that the explanandum can be logically derived from T and /. Accordingly, while explaining does contain a deductive element, it is not a purely deductive enterprise, [ms. 44] Halonen and Hintikka emphasize that the explanandum (and, presum ably, the explanans) can have any logical form whatsoever, [ms. 27] What is critical in explanations is the behavior of non-logical constants, [ms. 27] The constants in the explanandum fall into two mutually exclusive, but not necessarily exhaustive, classes: the critical and the queried constants. The critical ones are, so to speak, the values of the parameters that specify

Modality, invariance, and logical truth

Let us sum up. We began with the question, “What is the interest of a model-theoretic definition of validity?” Model theoretic validity consists in truth under all reinterpretations of non-logical constants. In this paper, we have described for each necessity concept a corresponding modal invariance property. Exemplification of that property by the logical constants of a language leads to an explanation of the necessity, in the corresponding sense, of its valid sentences. I have fixed upon the epistemic modalities in characterizing the logical constants: to be a logical constant in the language of a population is to be invariant over a modality describing complete possible epistemic states of that population (or an idealized analogue thereof). The grounds for this characterization are these: (1) It leads, I believe, to an extensionally reasonable demarcation of the logical constants, including clear cases and excluding clear non-cases. It gives a principled criterion for deciding unclear cases. (2) It provides an analysis of the topic-neutrality of logic. (3) It leads to an explanation of the epistemic necessity of the logical truths in terms of the topic-neutrality of the logical constants. All the same, it is reasonable to ask, even if the suggested demarcation of logic is extensionally correct, whether it can reasonably be expected to be fundamental. The epistemic invariance of an expression is a rather striking property, one which we should want to explain. What is missing, then, is an explanation of the distinguishing epistemic properties of the constants in terms of more fundamental properties involving their understanding and use. It would be these that properly define the nature, not just the extent, of logic.

Popper's theory of deductive inference and the concept of a logical constant

This paper deals with Popper's little-known work on deductive logic, published between 1947 and 1949. According to his theory of deductive inference, the meaning of logical signs is determined by certain rules derived from ‘inferential definitions’ of those signs. Although strong arguments have been presented against Popper's claims (e.g. by Curry, Kleene, Lejewski and McKinsey), his theory can be reconstructed when it is viewed primarily as an attempt to demarcate logical from non-logical constants rather than as a semantic foundation for logic. A criterion of logicality is obtained which is based on conjunction, implication and universal quantification as fundamental logical operations.

Logic in the twenties: the nature of the quantifier

We are often told, correctly, that modern logic originated with Frege. For Frege clearly depicted polyadic predication, negation, the conditional, and the quantifier as the bases of logic; moreover, he introduced the idea of a formal system, and argued that mathematical demonstrations, to be fully precise, must be carried out within a formal language by means of explicitly formulated syntactic rules.Consequently Frege has often been read as providingallthe central notions that constitute our current understanding of quantification. For example, in his recent book on Frege [1973], Michael Dummett speaks of ”the semantics which [Frege] introduced for formulas of the language of predicate logic.” That is, “An interpretation of such a formula … is obtained by assigning entities of suitable kinds to the primitive nonlogical constants occurring in the formula … [T]his procedure is exactly the same as the modern semantic treatment of predicate logic” (pp. 89–90). Indeed, “Frege would therefore have had within his grasp the concepts necessary to frame the notion of the completeness of a formalization of logic as well as its soundness … but he did not do so” (p. 82).This common appraisal of Frege's work is, I think, quite misleading. Even given Frege's tremendous achievements, the road to an understanding of quantification theory was an arduous one. Obtaining such understanding and formulating those notions which are now common coin in the discussion of logical systems were the tasks of much of the work in logic during the nineteen-twenties.

The Logico-Linguistic Mind-Brain Problem and a Proposed Step towards its Solution

This paper argues that if a person's beliefs are idealized as a set of sentences (theoretical, observational, and mixed) then the device of Ramsey sentences provides a treatment, of the mind-brain problem, that has at least four noteworthy characteristics. First, sentences asserting correlations between one's own brain state and one's own “private” experiences are, on such treatment, reconstrued as neither causal, coreferential, nor as meaning postulates, but as clauses in an overall hypothesis (Ramsey sentence) whose only nonlogical constants have “private” meanings. Second, sentences asserting psycho-physical correlations in general, or in other individuals, remain theoretical and susceptible to scientific reduction, though not prejudged to be so. Third, communication between persons having mutually exclusive, “private,” observational vocabularies can be made intelligible. Fourth, it becomes possible in principle (though not necessary) that the world could ultimately be given a total description in a scientific language, Lφ, without mental primitives. However, such a language would be susceptible to interpretation by a given individual using a metalanguage whose primitives (not among those of Lφ) were privately observational to that individual. His procedure in accomplishing such an interpretation would be to construct a Tarskian truth definition in a metalanguage MLφ and then to incorporate it, by Ramsifying its descriptive terms, in his own overall Ramsey sentence. The physical language, Lφ, while not containing a person's primitives, would, of course, be able to define their physical correlates, thus providing a certain sort of mutual mirroring, and a certain sort of relativity.

The decision problem for mathematical structures of quantum theory

The purpose of this letter is to enunciate some results concerning the ~!ecision problem relative to a well-known mathematical structure of quantum theory. Such a structure, originally proposed by BIRKHOFF and yon NEU~ANN (1) and later developed by PI toN (3) and EMCH (3) together with JAVCH (4,5), is essentially the structure of the <( propositions ~ (*) on a given quantum system and turns out to be the same of a complete, atomic, orthocomplemented, weakly modular, satisfying-the-covering-law lattice ((( system of propositions ~, in the nomenclature introduced by PIRON). Let us denote with TQ~ the theory of the above systems of propositions ; we will prove that TQM is undecidable, in a sense that will be specified below. To begin with, it is useful to recall some definitions which are of common use in mathematical logic; they arc essentially due to TARSKI (6). The theories discussed in the following are all formalized within the lower predicate calculus, and they have the same logical constants (i.e. quantifiers, connectives, symbol of equivalence) and rules of inference. Let us call T anyone of these theories; we assume further tha t T is specified by its nonlogical constants (i.e. symbols of operations, and so on) and nonlogical axioms (i.e. relations). We suppose known to the reader the definitions of a formula, a sentence (i.e. a formula without free variables), a provable sentence (in terms of nonlogical constants and nonlogical axioms), together with the definitions of consistency, completeness, and finite axiomatizabili ty of a theory T. Moreover, we suppose also known the concept of a general recursive set.

Provability with finitely many variables

In first-order logic with equality but with finitely many variables, no finite schema suffices to give a sound and complete axiomatization of the universally valid sentences. The proof uses a rather deep result from algebraic logic. The purpose of this note is to give a rather obvious consequence of a recent theorem of J. S. Johnson [5] (whose proof is based on Monk [7]). The consequence has to do with provability in languages which differ from ordinary first-order languages with equality just in having only finitely many individual variables. Such languages have previously been investigated in Henkin [2 ], Henkin-Tarski [3 ], Jaskowski [4], and Pieczkowski [8]. Our result is that it is impossible to write down finitely many schemata which will give a notion of proof that is sound and complete. See below for a more precise formulation; in particular, the notion of schema is made explicit. What makes our result an easy consequence of Johnson's theorem is the fairly general knowledge of certain connections between logically valid sentences in these restricted first-order languages, and equations which hold identically in each representable polyadic equality algebra. The main portion of the note is devoted to an exposition of these connections. Thanks are due to the referee for aid in the formulation of these connections. We employ the usual set-theoretic notation. f*X is the f-image of the set X. co is the set of all natural numbers. We assume throughout that 3 -<a <& (Our languages will have a variables; the case a <2 is considered in Henkin [2].) ?a is the first-order language with equality with the sequence (vi:i<a) of individual variables and with the sequence (Ri:i<4) of nonlogical constants, where Ri is an a-ary relation symbol for each i<co. We treat 7, -, V, and = as primitive logical symbols; V, A/, +*, and 3 are defined in the usual way. An 5a-structure is a structure 21= (A, Ri)i<,, where A 70 and RiCaA for each i<cw. If Received by the editors November 10, 1969 and, in revised form, March 13, 1970. AMS 1969 subject classifications. Primary 0216, 0218; Secondary 0248.

Satisfaction for n-th order languages defined in n-th order languages

In this paper we show that for n-th order languages L′ with p nonlogical constants, 2 ≦ n &lt; ω, 0 ≦ p &lt; ω, a notion of satisfaction can be defined in an n-th order language containing one additional nonlogical constant, say S. By the usual methods we also show that this notion cannot be defined in L′. Hence, in its present formulation, Beth's Theorem in [1] for first order languages has no analogue for L′.Our defining expression is such that, given any values of the other nonlogical constants and any appropriate m-tuple, it allows us to determine whether or not the m-tuple belongs to the value S of S without considering the totality of objects which are of the same type as S. Whether every definition in an n-th order language is equivalent to one thus “predicative”, and hence whether there is a formulation of Beth's Theorem which generalizes to higher orders, we do not know.The falsehood for L′ of the analogue of Beth's Theorem implies the falsehood of an analogue of an interpolation theorem for first order languages. The above definability of satisfaction for L′ implies a result on finite axiomatizability in slightly richer languages. Details are given in § 4.

On weak completeness of intuitionistic predicate logic

Suppose the ri-placed relation symbols Pi, 1 ≦ i ≦ k, are all the non-logical constants occurring in the closed formula , also written as , of Heyting's predicate calculus (HPC). Then HPC is called complete for provided , i.e.Here D ranges over arbitrary species, and over arbitrary (possibly incompletely defined) subspecies of ;

On recursively enumerable and arithmetic models of set theory

In this note we shall prove a certain relative recursiveness lemma concerning countable models of set theory (Lemma 5). From this lemma will follow two results about special types of such models.Kreisel [5] and Mostowski [6] have shown that certain (finitely axiomatized) systems of set theory, formulated by means of the ϵ relation and certain additional non-logical constants, do not possess recursive models. Their purpose in doing this was to construct consistent sentences without recursive models. As a first corollary of Lemma 5, we obtain a very simple proof, not involving any formal constructions within the system of the notions of truth and satisfiability, of an extension of the Kreisel-Mostowski theorems. Namely, set theory with the single non-logical constant ϵ does not possess any recursively enumerable model. Thus we get, as a side product, an easy example of a consistent sentence containing a single binary relation which does not possess any recursively enumerable model; this sentence being the conjunction of the (finitely many) axioms of set theory.

Finite Axiomatizability using additional predicates

By a theory we shall always mean one of first order, having finitely many non-logical constants. Then for theories with identity (as a logical constant, the theory being closed under deduction in first-order logic with identity), and also likewise for theories without identity, one may distinguish the following three notions of axiomatizability. First, a theory may be recursively axiomatizable, or, as we shall say, simply, axiomatizable. Second, a theory may be finitely axiomatizable using additional predicates (f. a.+), in the syntactical sense introduced by Kleene [9]. Finally, the italicized phrase may also be interpreted semantically. The resulting notion will be called s. f. a.+. It is closely related to the modeltheoretic notion PC introduced by Tarski [16], or rather, more strictly speaking, to PC∩ACδ.For arbitrary theories with or without identity, it is easily seen that s. f. a.+ implies f. a.+ and it is known that f. a.+ implies axiomatizability. Thus it is natural to ask under what conditions the converse implications hold, since then the notions concerned coincide and one can pass from one to the other.Kleene [9] has shown: (1) For arbitrary theories without identity, axiomatizability implies f. a.+. It also follows from his work that : (2) For theories with identity which have only infinite models, axiomatizability implies f. a.+.

The classical and the ω-complete arithmetic

We consider two formal systems for the theory of (natural) numbers, both of which are applied second-order functional calculi with equality and the description operator. The two systems have the same primitive symbols, rules of formation, and axioms, differing only in the rules of inference.The primitive logical symbols of the systems are the improper symbols (,), the prepositional connectives ∨, &amp;, ⊃, ≡, ~, the quantifiers ( ), (E), the equality symbol =, the description operator ι,-infinitely many distinct individual (or number) variables, and for each positive integer k infinitely many distinct k-place function variables. Our systems have in addition the following four primitive nonlogical (or arithmetical) constants:0, 1, +, ×.The classes of “number formulas” (nfs) and “propositional formulas” (pfs) are defined inductively as the least classes of formal expressions (i.e. of concatenations of primitive symbols) satisfying the following conditions:(1) 0, 1, and the number variables are nfs.

Comparison of subtheories

Let ZF denote the Zermelo-Fraenkel set theory and ZFI the theory obtained from it by the addition of an axiom which asserts the existence of at least one inaccessible number. If ZF is consistent then ZFI contains number-theoretic theorems which are not theorems of ZF, e.g. the arithmetical sentence which asserts the consistency of ZF-Con(ZF). Mostowski [4] introduced a sentence of the theory of real numbers which can be proved in ZFI but cannot be proved in ZF. Let S1 and S2 be two theories such that the relation of Si to S2 is like that of ZFI to ZF. The present paper will show to what extent the subtheories of S1 contain more theorems than the corresponding subtheories of S2. We shall use the terminology of Tarski-Mostowski-Robinson [8].2 A set of symbols of the first order predicate calculus with the usual formation rules for terms and formulae, as laid out in [8, pp. 6-7], will be called a standard formal language. Each theory with standard formalization is formalized in a standard formal language. If R is a standard formal language and P an unary predicate (P need not be a symbol of R) we construct a new formal language R(P) by the relativization of quantifiers in R to P (see [8, p. 24]). If A is a sentence of R we denote by A (P) the sentence obtained from A by relativizing the quantifiers in it to P. Let Q be a theory with standard formalization. We define the notion of an interpretation of R in Q as in [8, pp. 20-21, 29]. This interpretation is obtained by a new theory with standard formalization Q(R) which is formalized in the language which consists of the symbols of both Q and R(P). The valid sentences of Q(R) are exactly those which are derivable from the set which consists of the valid sentences of Q and possible definitions of the nonlogical constants of R(P) in Q. Given a particular interpretation of R in Q we can define in the language R a theory with standard formalization Q/R (which may be called the theory induced by Q in R) by the following stipulation: A sentence A of R is valid in Q/R

Two theories with axioms built by means of pleonasms

This paper contains examples T1 and T2 of theories which answer the following questions:(1) Does there exist an essentially undecidable theory with a finite number of non-logical constants which contains a decidable, finitely axiomatizable subtheory?(2) Does there exist an undecidable theory categorical in an infinite power which has a recursive set of axioms? (Cf. [2] and [3].)The theory T1 represents a modification of a theory described by Myhill [7]. The common feature of theories T1 and T2 is that in both of them pleonasms are essential in the construction of the axioms.Let T1 be a theory with identity = which contains one binary predicate R(x, y) and is based on the axioms A1, A2, A3, B1, B2, B3, B4, Cnm which follow.A1: x = x. A2: x = y ⊃ y = x. A3: x = y ∧ y = z ⊃ x = z.(Axioms of identity.)B1: R(x, x). B2: R(x, y) ⊃ R(y, x). B3: R (x, y) ∧ R(y, z) ⊃ R(x, z).(Axioms of equivalence.)B4: x = y ⊃ [R(z, x) ≡ R(z,y)].Let φn be the formulawhich express that there is an abstraction class of the relation R which has exactly n elements.Let f(n) and g(n) be two recursive functions which enumerate two recursively inseparable sets [5], and call these sets X1 and X2.We now specify the axioms Cmm.It is obvious that the set composed of the formulas A1−A3, B1−B4, Cnm (n,m = 1,2, …) is recursive.The theory T1 is essentially undecidable; for if there were a complete and decidable extension T′1 (of it, then the recursive sets Z = {n: φn is provable in T′1} and Z′ = {n: ∼φn is provable in T′1} would separate the sets X1 and X2.

Solution of a problem of Tarski

We presuppose the terminology of [1], and we give a negative answer to the following problem ([1], p. 19): Does every essentially undecidable axiomatizable theory have an essentially undecidable finitely axiomatizable subtheory?We use the following theorem of Kleene ([2], p. 311). There exist two recursively enumerable sets α and β such that (1) α and β are disjoint (2) there is no recursive set η for which α ⊂ η, β ⊂ η′. By the definition of recursive enumerability, there are recursive predicates Φ and Ψ for whichWe now specify a theory T which will afford a counter-example to the given problem of Tarski. The only non-logical constants of T are two binary predicates P and Q, one unary operation symbol S, and one individual constant 0. As in ([1], p. 52) we defineThe only non-logical axioms of T are the formulae P(Δm, Δn) for all pairs of integers m, n satisfying Δ(m, n); the formulae Q(Δm, Δn) for all pairs of integers m, n satisfying Ψ(m, n); and the formulaT is consistent, since it has a model. It remains to show that (1) every consistent extension of T is undecidable (2) if T1 is a finitely axiomatizable subtheory of T, there exists a consistent and decidable extension of T1 which has the same constants as T1.