Logical Constants as Punctuation Marks

This paper presents a proof-theoretical approach to the question "What is a logical constant?" This approach starts with the assumption that logic is the science of formal deductions, and that basic formal deductions are structural deductions, i.e. deductions independent of any constant of the language to which the premises and conclusions belong. Logical constants, on which the remaining formal deductions are dependent, may be said to serve as "punctuation marks" for some structural features of deductions; this punctuation function, exhibited in equivalences which amount to analyses of logical constants, is taken as a criterion for being a logical constant. The paper presents an account of philosophical analysis which covers the proposed analyses of logical constants. Some related assumptions concerning logic are also considered. In particular, since a logical system is completely determined by its structural deductions, alternative logical systems arise by changing structural deductions while having constants with the same punctuation function. Some other approaches to the question "What is a logical constant?", grammatical, model-theoretical, and proof-theoretical, are briefly considered. / Introduction It is clear that an answer to the question "What is a logical constant?" would provide us with the means to answer the question "Where are the limits of logic?" Since the latter question is obviously very close to the ques*This paper is based on the philosophical part of my doctoral thesis [9], and on lectures delivered at the Mathematical Institute in Belgrade, the University of Konstanz, and the University of Notre Dame. I would like to acknowledge my debt to Professor Michael Dummett, who supervised my work on [9]. I am also indebted to Dr. Peter SchroederHeister, who invited me to Konstanz, and encouraged me in a number of discussions to try to explicate the ideas propounded here. Professor Michael Detlefsen has been very kind to invite me to Notre Dame, and to show an interest in my work. I would like to thank Professor Detlefsen and Dr. Michael Kremer, also from the University of Notre Dame, for reading and discussing this paper. I am also grateful to them for correcting some solecisms in my English. Received June 8, 1987 LOGICAL CONSTANTS AS PUNCTUATION MARKS 363 tion "What is the subject matter of logic?" one could legitimately assume that our question "What is a logical constant?" is among the central questions of the philosophy of logic. Apart from its intrinsic philosophical interest, the problem of the demarcation of logic is obviously of crucial importance to the logicist program in the foundations of mathematics. However, no definite criterion for this demarcation seems to have come out of the work of the logicists, with which modern logic started. On the other hand, one of the main reasons why logicism was abandoned was that at some point it was felt that the limits of logic must have been transgressed in the logicist reconstruction of mathematics. The problem of the demarcation of logic is also in the background of the discussion concerning the status of second-order logic, which started with Quine's attack upon second-order logic (see [6]). In spite of all that, it doesn't seem that logicians, even those who are philosophically inclined, are trying very hard to answer the question "What is a logical constant?" A much more characteristic attitude in modern logic is that of a certain skepticism as to whether the distinction between logical and nonlogical expressions can be clearly drawn. Most logicians, like so many followers of Protagoras, are content with just listing what they take as logical constants. A clear exponent of this skepticism, and probably one of those who made it the accepted position, is Tarski, in his famous paper on the notion of logical consequence [40]. It is interesting that Bolzano anticipated Tarski not only in his definition of the semantical notion of logical consequence, but also in the belief that it is doubtful that a criterion can be found for drawing the distinction between logical and nonlogical expressions (see [5], Section 148, and [23], p. 366). Occasionally, however, attempts do arise to find this criterion. Without trying to ascertain their merits, let us just mention some of these attempts. In [36] Quine proposed to distinguish logical constants from other expressions (his terms are respectively "particles" and "lexicon") by saying that the grammatical categories of the latter are "infinite and indefinite" (see pp. 2830, 59). In [43] Wang examined some proposals, including a grammatical one, linked with Quine's, but in general he was in a rather skeptical mood (see pp. 143-165). It is interesting that in a lecture from 1966 [42] Tarski seems to have abandoned his skepticism up to a point, and to have found a criterion for the demarcation of logic by elaborating ideas suggested by Klein's Erίange Programm, and some model-theoretical results of Lindenbaum and Tarski [25]. Chief among these results is that " . . . every relation between objects (individuals, classes, relations, etc.) which can be expressed by purely logical means is invariant with respect to every one-one mapping of the world (i.e., the class of all individuals) onto itself . . ." ([41], p. 385; a reference to Tarski's lecture of 1966, and a critique of his views, can be found in [28]). The proposal for the demarcation of logical constants found in [14] (pp. 21-22), which is in principle grammatical (essentially, it takes as logical constants functors from sentences and predicates to sentences and singular terms), also has a footnote dealing with identity, in which a view very similar to Tarski's conception of 1966 is propounded. Finally, Lindstrδm in [26] and [27] suggested a rather technical model-theoretical criterion for the demarcation of logic, by showing that the Lδwenheim-Skolem Theorem together with the Compactness Theorem for de-