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.

## Wednesday, May 28, 2008

### Parsons on math and logic

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## 5 comments:

"one of the reasons for preferring first-order logic was that it was ontologically innocent... Parsons did not think this was a good reason because,for example, first-order logic requires models for its semantics and models are sets."

This seems wrong on two counts. I don't take myself to be disagreeing with you since you suggest that this seems wrong. Here are the two ways in which I would argue it is wrong:

1) FOL is not ontologically innocent because it includes Ex(x=x) as a theorem. It is a logical truth that something exists. This is probably a different sense of innocence than the one Quine had in mind, but one worth noting.

2) In what sense does FOL 'require' sets? Well, if we are going to give a model theoretic semantics for the logic, then we have to use sets by definition. But the logic presumably exists whether or not we study it by way of model theory. One would have to be careful about the modality of this 'requirement' cited by Parsons, but it strikes me as wrong.

This brings me to the broader topic. You end up discussing logic and mathematics, to what extent the two are independent and who might think they are. Consider three senses of 'dependence'.

Dependent1: the discipline of logic is a sub-discipline of mathematics (much as topology or set theory are sub-disciplines of mathematics). That seems obviously false because philosophers also study and contribute to logic.

Dependent2: the subject matter of logic, the entities and properties which it studies, are mathematical in nature. This also strikes me as wrong, but much harder to argue. The fundamental property studied by logic is that of validity. It seems to me like this is a general non-mathematical notion. As for the putative 'entitites' whether logic studies the relations of sentences or propositions, either way those are not strictly mathematical in nature.

Dependent3: the only domain of discourse governed by logic is mathematical discourse, and so only the needs and judgments of mathematics matter in formulating logical laws. Now this very well might be true. Some have suggested to me in conversation that the logician is making a mistake when he/she thinks that natural language as a whole (or even the substantial, regimented portion of that language used for theoretical inquiry) can be regimented under canons of logic. Perhaps the only discourse which has a determinate logic is mathematics. This would surprise me, but I don't know what else to say about it at the moment.

In any case, just to be clear about my two cents: it seems that the 'dependence' of logic on mathematics is at most methodological. We use mathematical tools to be clear and precise about our logical theory. But metaphysicians do that too, and you wouldn't accuse metaphysics of being nothing more than mathematics.

One thing people have said about the difference between logic and math: logic is purely formal (whatever that means) and mathematics is not.

And I thought the same thing as Colin in his 2): a logic need not be identified with its model theory. It seems possible that metalogic could be mathematics, without logic being so.

Oops - I just saw your post with quotes from MacFarlane's dissertation, so you've already thought of what I mentioned in my previous comment.

Colin,

I think Parsons mentioned the point in (1). Your way of putting (2) expresses part of the thought I was grasping at. I should track down the Parsons paper so I can see what his response to that is. I'm not sure if he is thinking of something along the lines of: we should identify logic with its semantics. That strikes me as in desperate need of justification, so I don't know that it helps Parsons (or my version of him).

I want to comment on the three senses of dependence you provided, but that will have to wait a little while. I do want to comment on the point about logic depending on math only methodologically. I'm not sure that that is true. The development of logic in the last century wouldn't be possible without mathematical tools. It seems fair to say that math was essentially intertwined with logic and continues to be in many areas: model theory, recursion theory, algebraic logic in its sundry forms, etc. It seems like there is no thinking about logic beyond a fairly introductory level without invoking a fair number of mathematical notions. Ken Manders makes a point along these lines with respect to physics and math (e.g. in his review of Field's Science without Numbers), so I will co-opt it here. The ways of thinking about logic are mathematical and we lose out on large chunks of logic if we try to approach in it in a demathematized way. (This is excellent since it provides some motivation for writing about Manders's stuff.)

I'm not sure how much I share the modus tollens intuition in your final paragraph. I've flirted with the idea that lots of metaphysics is (or is trying to be) model theory, albeit somewhat informally presented. I've kept this idea quiet since I haven't really done much to verify this or make the point precise. I don't think I'd go as far as saying that metaphysics is nothing but math though; I'm not really sure what I'd say. Along these lines, Mark Wilson has given some talks in which he argues that some work by metaphysicians encroaches on research by applied mathematicians. I'll have to see if I can track down those references.

Greg,

I will, hopefully, have some things to say on the topic of the formality of logic. I just finished printing off MacFarlane's dissertation.

Looking back, the last paragraph of my response to Colin is more wild-eyed than I had expected. More importantly, I don't think I characterized Wilson's position quite correctly. While Wilson did say that metaphysicians in some instances were trying to settle issues in applied math, I don't think he goes on to infer that they are doing math. I think he'd advise them to use more mathematical tools in their analyses than they do. I don't think his position ends up being an example that supports my nutty idea like I had thought.

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