# Benson Mates wins!

Benson Mates starts the introductory chapter of his classic Elementary Logic as follows:

This chapter is designed to give an informal and intuitive account of the matters with which logic is primarily concerned. Some such introduction is surely required; otherwise, the beginner is likely to feel that he does not get the point of the formal developments later introduced.

Exactly! And Mates stresses that smoothing the way to a later grasp of technicalities doesn’t mean (at this stage) nailing everything down  in way that will in all respects survive later fine-grained philosophical scrutiny — again, we need a tolerably relaxed preamble  to help get the show on the road.

I’ll say something just a bit more detailed about three of the “pre-formal preambles” I picked out in the last couple of posts, from Mates, from Bergman, Moor and Nelson, and from my namesake Nick Smith.

Mates’s opening chapter, it seems to me, still works the best, covering some of the right things, at about the right level, with the right caveats (except, perhaps, that his §4 — where he pauses to, among other things, cast doubt on the notions of a proposition, statement, thought, and judgement — rather over-eggs the pudding). In §1 Mates defines an argument as valid [he says “sound”] if and only if it is not possible for its premisses to be true and conclusion false. He then elucidates the relevant notion of what is possible in terms of what is conceivable, and then explicates that to mean there are no lurking contradictions — which in turn is explicated in terms of no contradiction being derivable (so we have gone round in a circle, but Mates says why it is an illuminating one). In §2, the validity of an argument (with finitely many premisses) is related to the necessity of a related conditional. In §4 we meet the idea of a form of argument, and we get the idea of a logically necessary truth as being one that is necessary by virtue of its logical form, and a corresponding notion of logically valid argument. But Mates is clear that the question of which words should be considered logical, so what belongs to logical form, involves as he puts it, “a certain amount of arbitrariness”. §5 then notes just a few of the vagaries of ordinary language which mean that, once we want to start talking about patterns or forms of arguments, some degree of formalisation is more of less inevitable (“it is clear for the natural language that there are few, if any, matrices that literally have only necessary truths as substitution-instances). All this is done very, very clearly.

Turning to the later edition of The Logic Book,  Bergman, Moor and Nelson start by defining an argument as follows:

An argument  is a set of two or more sentences, one of which is designated as the conclusion and the others as the premises.

They immediately tell us that sets are abstract objects (and introduce the brace notation).  But what work is this talk of sets really doing? It’s unnecessary — the classes here are virtual classes — and I’d say best avoided (like much pointless talk of sets).

Then we are told

An argument is logically valid  if and only if it is not possible for all the premises to be true and the conclusion false.

So we don’t get the distinction we get in Mates, between truth-preserving arguments generally and those arguments which might be said to be purely logically valid. Indeed, we don’t get anything general about form at this early stage in The Logic Book (and a search seems to reveal that the authors oddly eschew all use of the phrase).

We next get something about logical necessity and logically consistent sets of sentences; and then a section which points out that the given definition of validity means an argument is valid if it has a necessary conclusion or has contradictory premisses. Now, I too had such a section towards the end of my longer preamble in IFL1: I think that was a  mistake. Of course, the point has to be made somewhere; but it now seems to me to be better discussed later (e.g. in the context of talking about tautological validity, when we consider whether all tautological valid are valid in the intuitive sense we were after in the preamble and tried to capture with the informal “necessary truth-preservation” definition). Certainly, this is not a point to be made very near the outset to students who still need to be won over!

Bergman, Moor and Nelson are lucid enough (though the prose can be a bit plonking). But having nothing to say here about any notion of logical form — if only to criticise it — means that I’m not going to be recommending their chapter as parallel reading for IFL2.

Turning to Logic: The Laws of Truth, Nick Smith writes with enviable accessibility. I do have a worry about his initial claim in §1.1 that logic is “the science of truth” (he quotes Frege to the effect that logic has the same relation to truth as physics has to weight or heat — which will puzzle those students who are in another course being sold a deflationary theory of truth!): But let that pass, as in context the message is that logic isn’t about the psychological process of reasoning but about relations of logical consequence between the propositions we reason with.

§1.2 is about the notion of a proposition as the bearer of truth. But there are six and a half pages on this, and I’m not sure that it’s best policy to pause on such matters which are in fact going to be side-stepped when we adopt logically perfect languages to play with. §1.3 defines arguments in the usual way (but still, as with Mates, only one inference step is allowed, which ought to strike students as a strange regimentation of the notion of an argument!). §1.4 like Mates gives us various formulations of the idea of being a necessarily-truth-preserving argument. But Smith reserves the word “valid” for those arguments which are necessarily truth-preserving and where the “form or structure” of the argument guarantees that it is necessarily truth-preserving. So Smith (like Mates but unlike the authors of The Logic Book) makes the distinction between the two notions,  but his terminology is minority usage, I think. However, Smith says surprisingly little about the notion of form here. Thus “John is Susan’s brother; hence Susan is John’s sister”  is supposed not to be valid — but isn’t it an instance of the form “X is Y’s brother; hence Y is X’s sister” which guarantees truth-preservation? Or if that sort of “form” doesn’t count, why not? Smith doesn’t really tell us: at least Mates indicates there is an issue here. §1.6 is about soundness. And then Smith’s  introductory chapter starts talking about propositional connectives.

As I said, Smith writes very well. But I still, in sum, prefer the way that Benson Mates covers the same ground.

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### 5 Responses to Benson Mates wins!

1. Jon Awbrey says:

That is a nice preamble. It is good to grasp the purpose of logic before attempting to grapple with the formalisms.

C.S. Peirce • On the Definition of Logic

Put another way, he identifies logic as the normative science of representation, concerned with the question of what makes a representation good.

Thus, he places logic within the larger scientific enterprise of forming good representations of reality.

2. Rowsety Moid says:

I’m glad you brought up the brother-sister example, because it can be tricky to see what the logical form of a statement is until you know what logical forms are available; and then there’s the question of why those particular logical forms were chosen rather than some other ones.

This also affects the technique of showing an argument is invalid by using an obviously invalid argument of the same form; and there especially I think a word of warning is needed, because arguments in the wild often rely on implicit premises or background facts that can mean the argument actually is correct even though it appears to have an invalid form.

3. David Auerbach says:

I’ve switched books many many times in 40 years of teaching the baby logic course to very average students. (Last used was a book by Peter Smith. Also have used the Logic Book, Hodges, Jeffrey, and several others. I dimly remember using Leblanc and Wisdom.) What has changed less is what I do: Symbolization and Trees. (A experiments in natural deduction ended up badly.) I try, in the initial classes, to get them to identify form. (And, in introducing trees, to discover trees as a bookkeeping technique for a search strategy. Knight/knave problems are good for that.) About 14 minutes into this (https://mediasite.online.ncsu.edu/online/Play/ee535a1cb0f045a893342c5adc0a6c2d1d?catalog=24a22a6f-4be8-4eee-9c15-f2d37e313ebd ) you can see what I do. I’ve recorded this class 3 times and it’s never been the best versions. And no money for editing or anything fancy.

• Rowsety Moid says:

Could you say something about the problems you encountered when using natural deduction and what sort of natural deduction system you used?

• David Auerbach says:

I get a broad swath of students, often students satisfying a math requirement but avoid a course title with ‘math’ in it. So many of the students have no prior facility with, let’s call it, symbol use. (Grades end up fairly bimodal.) And it takes me a while to coax them out of looking for simple rules of symbolization (“put in a horseshoe whenever you see an if”). Since my major goal for the course is to teach them symbolization into first-order logic with identity, facility with proving things takes second place. Trees are quicker and there isn’t enough time. Now if I had had the luxury of a grader…
For systems, I’ve used (a while ago) Layman, more recently The Logic Book.
In the intermediate course (which is smaller and aimed at philosophy majors, math majors and a smattering of computer science majors) I’ve done ND much more often. But that course has its own special problems.