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The Bounds of Logic Part 2 To Ik a Logical Term 37 Chapter 3 To Be a Logical Ternl logic; at the same time, the principles developed by Tarski in the 1930s are still the principles underlying logic in the early 1990s. My interest in Tarski is, Ileedless to say, not historical. I am interested in the modern conception of logic as it evolved out of Tarski's early work in semantics. The Task of Logic and the Origins of Semantics In "The Concept of Truth in Formalized Languages" (1933). "On the Concept of Logical Consequence" (1936a), and "The Establishment of Scientific Semantics" (1936b), Tarski describes the semantic project as comprising two tasks: Since the discovery ofgeneralized quantifiers by A. Mostowski (1957), the I. Definition of the gelleral concept of truth for formalized languages question "What is a logical term?" has taken on a significance it did not 2. Dclinition of the logical concepts of truth, consequence, consistency. have before. Are Mostowski's quantifiers Hlogical" quantifiers'! Do they etc. , differ in any significant way from the standard existential and universal The main purpose of (I) is to secure meta logic against semantic para­ I quantifiers? What logical operators, if any, has he left out? What. ill doxes. Tarski worried lest the ullcritical usc of semantic concepts prior to I all, are the first- and second-level predicates and rcla tions that can be his work concealed an inconsistency: a hidden fallacy would undermine construed as logical? the cntire venturc. Be therefore sought precise, materially, as well as One way in which I do not want to ask the question is, "What, ill Ihe formally, correct definitions of "truth" and related notions to serve as a I nature ofthings, makes a property or a relation logical'!" On this road lie hedge against paradox. This aspect of Tarski's work is well known. In t the controversies regarding necessity and apriority, and these, I bclieve, "Model Theory before 1945" R. Vaught (1974) puts Tarski's enterprise in should be left aside. Although some understanding of the modalities is a slightly different light: I essential for our enterprise, only their most general features come into [During the late I920s] Tarski had become dissatisfied with the notion of truth as play. A detailed study ofcomplex and intricate modal and epistemic issues it was heing lIsed. Since the notion "0 is true in '11" is highly intuitive (and perfectly would just divert our attention and is of little use here. But if "the nature clear for allY definite 0), it had heen possible to go even as far as the completeness of things" is not our measure, what is? What should our starting point be? theorem hy treating truth (consciously or unconsciously) essentially as an unde­ What strategy shall we decide upon? lined notion ---one with many ohvious properties .... But no one had made an analysis of truth, not even of exactly what is involved in treating it in the way just A promising approach is suggested by L. Tharp in "Which Logic Is the mcntioncd. At it time when it was quite well understood that 'all of mathematics' Right Logic'!" (1975). Tharp poses the question, What properties should a could he done, say, in ZF, with only the primitive notion E, this meant that the system oflogic have? In particular, is standard first-order logic the "right" theory of models (and hence much of meta logic) was indeed not part of mathe­ logic? To answer questions of this kind, he observes, it is crucial to have a matics. It sccms clear that this whole state of affairs was bound to cause a lack of dear idea about "the role logic is expected to play." 1 Tharp's point is sure-rootedness in metalogic.... [Tarski's] major contribution was to show that the notion "a is trlle in \11" can simply be defined inside of ordinary mathematics, worth taking, and it provides the clue we are searching for. If we identify for example, in ZF. 2 a central role of logic and, relative to that role, ask what expressions can function as logical terms, we will have found a perspective that makes our On both accounts the motivation for (I) has to do with the adequacy of question answerable, and significantly answerable at that. thc system designed to carry out the logical project, not with the logical The most suggestive discussion of the logical enterprise that I know of project itself. The goal of logic is notlhe mathematical definition of "true appears in A. Tarski's early papers on the foundations of semantics. scntcnce," and (I) is therefore a secondary, albeit crucially important, task Tarski's papers reveal the forces at work during the inception of modern ofTarski an logic. (2), on the other hand, does reflect Tarski's vision of the 39 Chapter 3 38 To Be a Logical Term role of logic. In paper after paper throughout the early 1930s Tarski In every deductive theory (apart from certain theories ofa particularly elementary 3 nature), however much we supplement the ordinary rules of inference by new described the logical project as follows: The goal is to develop and study purely structural rules, it is possible to construct sentences which follow, in the deductive systems. Given a formal system 2) with languagc L and a uSlIal sense, fwm the theorems of this theory, but which nevertheless cannot be 4 definition of "meaningful," i.e., "well-formed," sentcm:e for I"~ a (dosed) proved in this thcory on the basis of the accepted rules of inference. deductive system in !t:' is the set of all logical consequences of somc set X Tarski's conclusion was that proof theory provides only a partial ac­ of meaningful sentences of L "Logical consequcnce" was defined proof­ count of the logical concepts. A new method is called for that will permit theoretically in terms of logical axioms and rules of inferencc: if.w and .elf a more comprehensive systematization of the intuitive content of these are the sets of logical axioms and rules of inference of !.t', respectively, the set of logical cOllsequences of X ill !.t' is the smallcst set of well-formcd concepts. The intuitions underlying our informal notion of logical consequence sentences of L that includes X and and is closed undcr the rules in .w (and derivative concepts) are anchored, according to Tarski, in certain 91. In contemporary terminology, a deductive system is a jtJr/llul th('ory relationships between linguistic items and objects in (configurations of) within a logical framework !i:'. (Note that the logical framework itself can the world. The discipline that studies relationships of this kind is called be viewed as a deductive system, namely by taking X to be the set of logical axioms.) The task of logic, in this picture, is performcd in two steps: (a) semantics: We ... understand by semantics the totality of considerations concerning those the construction of a logical framework for formal (formalized) theories; concepts which, roughly speaking, ex.press certain connex.ions between the ex.pres­ (b) the investigation of the logical properties consistency, complctcncss, sions of a language and the objects and states of affairs referred to by these axiomatizability, etc.-~of formal theories relative to the logkal framc­ exprcssions. ~ work constructed in step (a). The concept of logical CO/l.\'('qll('llCl' (togcther The precise formulation of the intuitive content of the logical concepts is with that of a well-formed formula) is the key concept of Tarskil.111 logic. thcrcfore a job for semantics. (Although the relation between the set of Once the definition of "logical consequence" is given, we can easily scntences "/'11" and the universal quantification "('tx)Px," where x ranges obtain not only the notion of a deductive systcm but also those of a over the natural numbers and "n" stands for a name of a natural num­ logically true sentence; logically equivalent sets of sentences; an axiom ber, is not logical consequence, we will be able to characterize it ac­ system ofa set ofsentences; and axiomatizability, completeness, and con­ curately within the framework of Tarskian semantics, e.g., in terms of (JJ sistency of a set of sentences. The study of the conditions under which various formal theories possess these properties forms the subject matter complctellcss.) of metalogic. 2 The Semantic Definition of "Logical Consequence" and the Emergence Whence semantics? Prior to Tarski's "On the Concept of Logical Con­ sequence" the definitions of the logical concepts were proof-theoretical. of Models The need for semantic definitions of the same concepts arose when Tarski Tarski describes the intuitive content of the concept "logical consequence" realized that there was a serious gap between the proof-theoretic defini­ tions and the intuitive concepts they were intended to capture: many as follows: intuitive consequences of deductive systems could not be detected by the Certain considerations of an intuitive nature will form our starting-point. Consider any class K of sentences and a sentence X which follows from the sentences of this standard system of proof. Thus the sentence "For every natural number class. From an intuitive standpoint it can never happen that both the class K seems to follow, in some important sense, from the set of sentences n, Pn" consists only of true sentences and the sentence X is false. Moreover, ... we are "Pn," where Il is a natural number, but there is no way to express this f~\ct (.:oncerned here with thc concept of logical, Le.formal, consequence, and thus with by the proof method for standard first-order logic. This situation, Tarski a relation which is to be uniqucly dctermined by the form of the sentences between which it holds...
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