### 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.

## 9 Comments:

I've been having to deal with this issue with my students. Especially when trying to explain why the book considers the connectives to be logical but not the identity sign.

How many people really think of truth as just a filter on an algebra? There's quite a bit of reasoning already hidden under all that machinery. Fortunately, logic is about the (correct) means of deduction, and not about truth in general. After all, every subject (hopefully) has some connection to truth.

Personally, I find the algebraic formulation easier to grok than worrying about truth and such. But maybe I'm just weird :-)

On the other hand, model-theoreticians will probably say that logic is the study of "truth". We don't deal much with formal proofs and deductions.

Recursive theorists, however, will advocate that logic is the study of how to "automate" reasoning.

Anthony: Right, but even model theorists work with some sort of proof system, even if it is just a Hilbert-style axiomatisation :-)

Jon: you are right to a some extent. Results like Craig interpolation may require you to know a

bitabout first-order proof systems. On the other hand, we more often than not just assume that there exists a sound and complete proof system without going into any detail :-)I would not think that non-monotonic logics are concerned with truth, but rather with what should be believed, a different matter.

The definition I've always thought best was: "Logic is the study of formal systems, particularly those formal systems studied by logicians." This allows for the fashions which exist in the discipline (as they do in every academic discipline), and also allows space for the Informal Logic community to define outself outside this definition.

In your list of contributing disciplines, I would also add anthropology. There has been an argument made by some anthropologists that the formal logics studied by logicians have an inherent cultural bias; for example, not all cultures apparently consider modus ponens a valid rule of inference.

If you like MacLane, logic can be thought of as the study of sheaves, or sheaves are the study of logic, or topos theory is the study of both:

Sheaves in Geometry and Logic : A First Introduction to Topos Theory

This nice correlation in topos theory seem to suggest a relation between the study of logic and the study of spaces (see Lambek and Scott, as well).

The conclusion: we logicians need to sit down and learn a healthy dose of homological algebra and commutative ring theory =).

unrelated comment : can you add a xml feed to your blog? i believe it is quite easy to do so within blogspot. thanks for this blog btw!

Just wanted to mention that this question actually was featured as contest question at UNILOG 2005 (in Montreux). There were several highly interesting answers, for instance employing category theoretical characterizations. There's supposed to be proceeding from the conference, but I haven't seen anything yet.

J.-Y. Béziau, R.P. de Freitas, J.P. Viana. What is Classical Propositional Logic? (A Study in Universal Logic), Logica Studies 7, 2001, was proposed as an introduction to the question.

Post a Comment

<< Home