Back to those choice-points noted in the first of these three posts: one formal language for FOL or many? Tarski or quasi-substitutional semantics? use of symbols as parameters as opposed to names or variables to be syntactically marked?
In the second post, I gave a Fregean reason for inclining to syntactically marking parameters (the thought being we should syntactically mark important differences in semantic role).
I certainly don’t want to go with van Dalen and use free variables for this role, for the very basic reason that I simply want to avoid using free variables in an elementary text. Back to Begriffsschrift! By my Fregean lights, a formal language quantifier should be thought as ‘\((\forall x)\ldots x\ldots x\ldots\)’, the whole being a slot-filler operating on a gappy predicate. The artifice of parsing a quantified wff as ‘\((\forall x)\)’ followed by a complete wff which has a free variable ‘\(x\)’ when stand-alone just obscures Frege’s insight into the nature of quantification. (And before you object that Frege has italic letter variables that look like free variables, remember that they aren’t: the italic letter wffs are just introduced as abbreviations for corresponding (universally) quantified wffs where the quantifier is given maximal scope — so for Frege, the ‘free’ variable wffs are explained in terms of quantified wffs, rather than vice versa.)
It is at least less misleading to use name-expressions from our formal language in the parametric role — relying on the similarities between, so to speak, permanent names and temporary names. But now note that there can be a clash here with the line we take on the first choice-point. If we say that there is one big language of FOL with an indefinite supply of names (only some of which get a fixed interpretation) then fine, that leaves us plenty of names to use a parameters. But suppose we take the many-language line, and think in terms of a language as having a particular fixed number of names (maybe zero, as in the basic language of first-order set theory!). Then there may not be enough names to recruit for parametric use in arbitrarily complex arguments. Compare: at least we don’t run out of variables, which is why van Dalen, and Chiswell/Hodges, who take the many-language line, have to recruit variables to use as parameters.
So I think that many-language logicians who want to use natural deduction with its parametric reasoning have very good reasons for introducing a distinguished class of symbols for parametric use. For on most many-language stories there is a fixed number, maybe zero, of proper names in the language, so we won’t have enough parameters if we stick to names. While the Fregean precept about marking important differences counts strongly against re-using variables-for-quantifiers in a quite different use as temporary names.
(What about Barwise and Etchemendy who I have down in my first post as both taking a many-language view and as using names as parameters? Well, to be honest, I find their position not entirely clear, but I think they conceive of a language as having some distinguished names, with an intended interpretation, though perhaps zero as in the language of set theory, but also having a supply of further names that can be used ad hoc, e.g. to name particular sets as on p. 16, such names also being available for use as parameters.)
But should we go for many languages or one all-purpose-language? It is notable that in my list of texts in the first of these posts, the more mathematical authors go for many languages. That is no accident surely; when in more advanced logical work we regiment mathematical theories, it is very natural to think of these various theories having their own languages, the language of set theory, the language of first-order arithmetic, the language of category theory, and so on. This conforms with how mathematicians tend to speak and think. So this gives us a serious reason to start as we mean to go on, thinking in terms of their being many first-order languages, with their distinct signatures and particular non-logical vocabularies. Another reason for taking the many-languages line from the outset is the sheer inelegance of the one all-purpose-language picture. We build up a language with infinite non-logical vocabularies of every possible arity; we then throw away again almost all the complexity either by making interpretations partial or by in principle interpreting everything and then showing almost the interpretative work is redundant. It is difficult to find that very aesthetically pleasing.
OK. None of all that is decisive. None of the logic books which make different choices are thereby bad books! Still, I think there are reasonably weighty reasons – rather more than mere considerations of taste – to go for many languages, and (as we’ve just seen, not unconnectedly) for giving a language a class of symbols to serve as parameters (‘arbitrary names’ or whatever your favourite label is) which is syntactically distinct from names (proper names, individual constants) and variables. So that’s what I’ll be doing in the newly added natural deduction chapters in the second edition of IFL.
To be continued