About a month ago Charles Parsons gave a very difficult talk at Pitt on the consequences of the entanglement of logic and math. The details of it were somewhat obscure at the time and I haven’t yet been able to track down the paper on which the talk was based, so I am not going to go into the details all that much. The point of the talk, which became clearer after reflection, was that there isn’t a clear distinction between logic and math, or more particularly between logical notions and mathematical ones.

Part of the talk dealt with some of Quine’s reasons for rejecting second-order logic. Parsons said that one of the reasons for preferring first-order logic was that it was ontologically innocent, not entailing commitment to lots of entities, whereas second-order logic had ontological commitments. Parsons did not think this was a good reason because,for example, first-order logic requires models for its semantics and models are sets. There is an ontological commitment in first-order logic then, namely to at least as much set theory is needed for the relevant model theory. This strikes me as an odd objection to Quine’s view of logic, but I am having a hard time placing my finger on what exactly is odd. I don’t remember Quine talking much about model theory in his discussions of first-order logic although I do not know if he proposed a different semantics for it.

There was another example discussed, Etchemendy’s criticism that Tarski’s notion of logical consequence depends on background set theoretic assumptions that are not themselves logical. The criticism is, if I recall correctly, that whether certain sentences count as logical truths according to Tarski’s definition depends on the background universe of sets being of a certain size. For example, that there are only finitely many objects or that there are at least three objects will be logical truths if all the models either have only finitely many objects or have at least three objects in their domains. Etchemendy thinks, perhaps rightly, that these sentences should not count as logical truths under any definition since their status as logical truths depends on how things are with some mathematical objects, namely the background set theory, which is not purely logical.
That is a rough sketch of the criticism, but it gets to the part of the talk that I have been thinking about the most: to what extent is there a notion of logicality apart from the mathematical? Certainly since Frege, possibly since Boole, logic and math have been intertwined. Why would one think that there is a notion of logic that stands apart from all of math? I suppose that the tradition from Aristotle to the early 19th century viewed logic as mostly independent of math. It was certainly not as highly mathematized. THe connections with math start to emerge with Boole’s algebraic treatment of math and Frege’s work. This could probably be extended to include Peirce and Schroeder as well, but I am not that familiar with their work. I said above that this connections between math and logic emerged with developments in the 19th century and later. One might object that the connections didn’t emerge so much as the notion of what is logical changed, from a neo-Aristotelian/Kantian one to a modern one that is more mathematical. This is a response I’d like to resist, but I don’t know what a good response to it would be. It would take some detailed research looking in the development of logic during the 19th century to tell how continuous the changes were.

Why would one expect that there isn’t a notion of logic apart from mathematical notions? One reason I came up with is that logic is supposed to be general, applying to both finite cases and infinite ones. Logic should specify the consequences of a finite bunch of axioms as well as an infinite set of them. If a notion is to apply to the infinite case, it will have to employ some, possibly heavy, mathematical resources. This might be seen as question begging, that the proper domain of logic, the paradigm, is the argument from finitely many premises, so the consequences of an infinite set of axioms is besides the point as one is already having to invoke sets in even setting up the problem. However, if cardinality quantifiers , e.g. there are countably many, are logical then one would be hard pressed to come up with a notion of logic that does not invoke some decent amount of mathematics, it would seem.

This is not an entirely philosophical reason, but all the logic texts that I am familiar with are all mathematical logic texts. Are there any logic texts in widespread use that one could describe well as non-mathematical? It would be good to look at those to see to what extent they lay out a conception of logic distinct from what one finds in other texts.

One thing about Parsons talk I was wondering about was who were the people that held that there was a notion of logic completely independent of mathematics? Parsons seemed to accuse Etchemendy of this. I haven’t gone back through his book to check to what extent this was right. Stephen Reed says some things in his Thinking about Logic that sounded like he thought there were logical notions distinct from mathematics. I’m not sure who else from the latter part of the 20th century would count. To clarify this it would probably be useful to put a finer point on what it is to be completely independent or distinct from mathematical notions. I’m not sure at the moment how to spell this out more. There are definitions of logicality articulated in mathematical terms, but it isn’t clear that any of them capture the full notion of logicality. However, I don’t think I’ve seen any definitions of logic given in non-mathematical terms that capture the full range of the logical either. This seems to be something that needs to be cleared up in order to proceed but I don’t have any suggestions at present.