### NCB: What is logic?

Hi, welcome back to NCB. This post is not about any specific talk. Rather, it is about a question I have been wondering about recently, after seeing philosophers going on about it at various conferences.

So, I am pretty happy to tell people that I work in logic (despite the odd looks I get). But when they ask for more details of what logic actually is, I flounder. There does not seem to be an all-encompassing answer. So, what is logic anyway?

If you're an old timer, you may say that logic is the study of truth. This is usually modelled algebraically by looking at something like (A,F), where A is some sort of (universal) algebra and F is a filter on A. For instance, for propositional logic we can take A to be a boolean algebra. For first order logic, we may like to use a polyadic algebra. The algebra A is essentially the term algebra of well formed formulae of our logic. The filter F is the set of "true" statements of our logic. Dually, we may like to look at the collection of all "false" well formed formulae, which forms an ideal I. We can rephrase logical statements in a nice way using algebra. For instance, sticking with boolean algebras, the associated logic is consistent iff I is a proper ideal. If (A,I) is consistent, then it is complete iff I is maximal in A. From this, it follows that (A,I) is consistent and complete if and only if the quotient A/I is the unique boolean algebra on two elements. For more on this view of logic, have a look at the following beautiful paper:

*The basic concepts of algebraic logic*. Paul Halmos. American Mathematical Monthly vol. 63 (1956) pp 363--387.

Ok, very well, so there is some nice algebra associated with the view that a logic ought to be equated with the concept of truth. But this does not seem to capture what is usually meant by logic. For instance, generations of students have had it hammered into their heads that logic is the study of reasoning. So how do we know that a given statement is true? Is it handed to us by some higher power?

The spiffy modern logician views logic as the study of consequence or deduction. That is, logic is more about "correct" forms of reasoning than about truth. Immediately we run into a problem.

Given that there is some collection of true statements, is there more than onecorrectway of deriving those statements? That is, is there only one logic or a multitude of logics?

If you answered yes to the first question, then you are a logical pluralist. If you answered no, then you are a logical monist. Now, for a mathematician, pluralism is perfectly reasonable. For instance, say I look at a category whose objects are smooth manifolds. Then what are the "correct" maps? Continuous? Differentiable? Smooth? Truth is, the choice of which class of maps you work with depends on what you are trying to capture. So why should it not be that way in logic?

The pluralism position has been parodied by saying something to the effect of "If you give me a piece of an argument, I'll give you a logic wherein it is valid". I highly doubt that any pluralist seriously thinks of things this way.

So, what is the answer to the subject of this post? Well, it seems reasonable to take logic to be the study of methods of deduction. If you're a philosopher, you'd probably want to change that to "

*...correct*methods of deduction". But such a statement seems presumptious. Taking logic to be the study of deduction, it incorporates aspects of mathematics, computer science, philosophy and linguistics and should not be seen as lying completely in any one of these.