The Connectives

9. Universally Representative and Special Connectives

Chapter 9 Universally Representative and Special Connectives§9.1 UNIVERSALLY REPRESENTATIVE CLASSES OF FORMULAS 9.11 Introduction This chapter introduces and compares two properties a connective may have or lack in a logic: the property of being ‘universally representative’ and the property of being what we shall call ‘special’. It will be sufficient to treat logics as individuated by the set of sequents they contain. The logical frameworks from which our examples are taken are Fmla and Set-Fmla. In the latter case, closure under the structural rules (R), (M) and (T) will be assumed, so that we could equally well treat logics as consequence relations, and say—parallelling the usage introduced at the outset of 3.11—that a connective in the language of such a consequence relation  is universally representative (or, is special) according to . When we have the ‘collection of sequents’ interpretation in mind (for Fmla or Set-Fmla) we will usually say ‘universally representative’ in such-and-such a logic. The extension of these concepts to Set-Set (or to gcr’s) is entirely routine; to avoid clutter, no further mention will be made of this fact. We need to recall the notion of synonymy, as applied when logics are conceived in the above way: formulas are synonymous when the replacement, not necessarily uniform, of one by the other in a sequent belonging to the logic yields a sequent also belonging to the logic. Now the basic notion for the present subsection is that of a class Δ of formulas being universally representative in a logic, which we define to be the case when every formula in the language of that logic is synonymous with some formula in Δ. Our Set-Fmla formula examples will all be congruential logics (i.e., their associated consequence relations – à la 1.22.1, p. 113 – are congruential in the sense of 3.31), for which cases we can alternatively describe Δ as universally representative when every formula is equivalent (in the  sense) to some formula in Δ. And our Fmla examples all feature logics whose languages contain the connective → and in which membership of both A → B and B → A is both necessary and sufficient for the synonymy of 1287 1288 CHAPTER 9. UNIVERSAL AND SPECIAL A with B according to the logic, in which case we will again describe A and B as equivalent, and again have the above alternative description of universal representativeness available. (Sometimes there will also be the connective ↔ and the given relation holds between A and B when A ↔ B is in the logic; these are the ↔-congruential logics of 3.31. Note that strictly speaking we should say in all these cases that the sequents  A → B,  B → A, and  A ↔ B are in the logic: but as usual we disregard the distinction in Fmla between a formula and the corresponding sequent.) Examples 9.11.1(i) The class of all formulas in the language of some logic is universally representative in that logic (since synonymy is reflexive). (ii) For classical logic in Fmla or Set-Fmla, assuming the presence of (e.g.) ∧ and →, the class of formulas containing at least twenty distinct propositional variables is universally representative (since—whether we interpret this counting by types or by tokens—any formula falling short can be fleshed out to an equivalent in which the additional variables occur ‘inessentially’, conjoining the given formula with suitable many formulas pi → pi until the required quota of variables is present). (iii) With the same logic in mind, the class of conjunctive formulas is universally representative, since A is synonymous with A ∧ A. We shall be especially interested in examples of type (iii) here, and introduce the following terminology to facilitate their study. We call an n-ary connective # in the language of some logic universally representative in that logic when the class of formulas of the form #(A1, . . . , An) is a universally representative class in that logic. Thus 9.11.1(iii) shows that the connective ∧ is universally representative in CL: every formula has a conjunctive equivalent. Note that we could equally have chosen A ∧ (p → p) as a conjunctive equivalent of A—or for that matter A ∧ (A → A)—given the assumption there in force as to the language; if  is present, A ∧  would have done as well. Exercise 9.11.2 (i) Assuming the presence of all the connectives ∧, ∨, ¬, →, ↔ in the language of CL (in Set-Fmla, for definiteness): which of the connectives listed are universally...