I am teaching logic, for my first time. It's a tools course for grad students, covering the basic logical tools that are of general applicability (plus three lectures on meta-theory): First Order Logic (FOL), basic (ZFC) set theory, basic (Kolmogorovian) probability theory and modal logic, using Barwise and Etchemendy. I've really nervous about the FOL portion, because I've never taken, TA'ed or taught a class on FOL. (There is a joke about how two Jesuits, who taught in a high school, teachers were talking. One says: "Do you know any chemistry?" The other says: "No, I haven't even taught it.") As a mathematics/physics undergraduate, I took a model theory class (with John Bell and William Demopoulos) and did an independent study on topos theory (with John Bell). I then did independent study in category theory and topos theory as a math grad student, and as a philosophy grad student I took an oral exam in logic (covering through the Goedel theorems). It comes by nature to me to approach logic always as yet another branch of mathematics, with languages being an algebraic structure like groups and fields, which can be given a set-theoretic account: "A first-order language is a 12-tuple <*L*,.,*S*,*C*,*P*,*V*,*a*,*e*,*m*,*o*,*c*,*n*> such that..." This, however, wouldn't be a pedagogically very good approach for teaching students, and it isn't the approach of Barwise and Etchemendy. (And now that I am no longer a mathematician, I worry that the lucidity of the above approach is illusory, due to an illusion of thinking one knows what one is talking about when one is talking about sets.)

I've found Barwise and Etchemendy extremely difficult to understand. They are apt to say something like that **john** (I am using bold to render their sans-serif) is an individual constant, **Sits** is a predicate, and **Sits**" followed by a parenthesis followed by **john** followed by a parenthesis is an atomic sentence. And then I get really puzzled. Are the individual constants particular token inscriptions? It seems they are, since they can stand in spatiotemporal relations ("followed by") while types can't. On the other hand, they talk of the same individual constant reappearing in multiple sentences, and that doesn't work if it is a token (I believe multilocation is possible, but, except perhaps for elementary particles, it takes a miracle). So it's a type, but a type that stands in spatiotemporal relations. Weird. And what does an inscription like **Sits(john)** in the book refer to? Does it refer to the token? No. It refers to some type. But it's made up of several types: **Sits**, **(**, **john**, **)**. This is really confusing.

I think I finally have two consistent interpretations of the text worked out. But it was hard to get there.

## 9 comments:

Hey Alex - this is totally not relevant to this post, but I was curious about your reaction to this story: http://www.telegraph.co.uk/news/worldnews/europe/vaticancityandholysee/4247756/Vatican-reveals-secrets-of-worst-sins.html. In particular, doesn't it strike you as at least a little odd that genocide should be considered less problematic in some sense than a priest having sex? I'm trying to work out a way for that not to be the case - for, in other words, my immediate interpretation of this to be mistaken - but I'm having a pretty hard time of it.

Alex,

I have never before heard of anyone getting hives over type/token distinctions in understanding logic. It's almost funny, coming from someone as logically sophisticated as you. Not that there's not a problem, just that it doesn't get most people.

Don't the same problems come up in language generally? What do we want to say about "John sits"? I think that if there is a problem it is a problem with the type/token distinction. What we need is Wetzel's "occurrences."

BTW, it's been a while now, but I have taught most of the material in LPL, so if you ever want to discuss it I can.

Alex,

"First time caller, long time listener...."

What would we say of the type-sentence "John sits"? Wouldn't we say that "sits" follows (or comes after, or whatever) "John"? And can't we mean by this just that, for any token of that type, "sits" will literally (i.e., spatially, temporally, or both) follow "John"?

N.N.:

I think this is almost right. But we need a modal operator, for it might be that a given sentence never will have a token. So we need a definition somewhat like this:

B follows A in C (where A, B and C are types) iff necessarily (every token of C contains a token of A that is followed by a token of B).

An interesting question is what the modality of the "necessarily" is. It can't be the logical consequence necessity that we are trying to understand by creating FOL. I think the only plausible thing for it to be is metaphysical necessity. And then, this is interesting: it means that this whole approach to the development of logic depends on the notion of metaphysical necessity. Which is fine, of course, but still interesting.

Heath:

I think some of my worries are due to the fact that I have rather nominalistic intuitions, and so types are mysterious to me.

Larry:

I think the claim that these sins are more heinous than others is not a claim that the Vatican makes, but a claim of the article's authors.

There is sometimes good reason for a law (of the Church or of the state) to treat a lesser offense more harshly than a greater offense, if the lesser offense is more common, or easier to get away with, or not punished by other laws, etc. For instance, we do not punish certain kinds of betrayals of a friend's confidence at all, but we punish theft of very small amounts, even though the former may well be a greater offense.

Of course, one has to be sure here that the lesser offense is not excessively punished, but that is not the case in the examples in the story.

Well, it's a bit silly to say here that these aren't punished by other laws: the Church is the body that has control over this, and they can do essentially whatever they please to punish people (unlike, say, non-totalitarian governments, which have relatively strict limits in the way they enforce laws). I also find it relatively hard to believe that attempted assassinations of the Pope are easier to get away with or more common than, again, genocide. But you bring up a good point: the apparent lack of moral reasoning on the Church's part here leaves something to be desired.

Heath:

I think part of my puzzlement was due to my implicitly thinking of formal logic as a branch of mathematics. But if it's a branch of mathematics, then one can't treat of tokens, at least not of spatiotemporal tokens, because mathematics only concerns mathematical objects, and no mathematical objects are in spacetime (theoretical physics is not mathematics). So for me the natural approach to a logical language is to think of it as a mathematical structure of a certain sort--a structure that includes, say, a concatenation operator that joins strings into a "larger" string, and so on. This is all very nice, and then you can prove all kinds of cool theorems, like the incompleteness theorem, and so on.

If that is how one thinks of a formal logical language, then one will deny that a formal logical language is a language in the same sense in which English is a language. For, a formal logical language is a set (or maybe some other mathematical object) while ordinary languages are not sets but social practices. (I suppose one could claim that sets are social practices, but that is very implausible.)

Moreover, on the abstract approach, one will have little room for the notion of a token. There are just the mathematical entities, and that's all. Of course, there are inscriptions in the metalanguage that refer to entities in the mathematical structure, and if one wants to, one can call these inscriptions "tokens". But they are inscriptions in the metalanguage, so they're not really tokens of the object language.

But B and E treat FOL as a language in the same sense as English is a language. So, their FOL is a social practice and its tokens are spatiotemporal objects (since it is essential to the nature of languages like English that they are social practices and their tokens are spatiotemporal entities). Thus, they are not primarily doing mathematics. The primary subject of their discussion is not a mathematical structure but a social practice. (A social practice brought into existence by its being described. Or perhaps they are studying a family of metaphysically possible social practices.)

It is kind of like a physics textbook. Physics textbooks can be really confusing in an even worse way. They might give an equation like F=ma, and say that F is a force, m is a mass and a an acceleration. But while this stuff didn't seem to confuse me when I took physics classes (though the lack of mathematical rigor in physics classes did bother me a lot eventually--though that wasn't what should have bothered me), now it's really confusing. A mass is a physical feature. But how can one multiply a physical feature by anything? (Can one multiply a mountain, or take the square root of a noise?) What they may really mean is that given an appropriate representational system ("a system of units"), if F is a number representing the force in this system, and m is a number representing the mass in this system, and a is a number representing the acceleration in this system, then F=ma. Now isn't that a lot clearer? (Well, maybe not a lot clearer. For we don't really know what a representational system is, etc.)

Alex,

I will confess you have thought about this several levels deeper than ever occurred to me. FWIW, I came to logic through computer science, so it was very natural to me to think of the operators as functions returning truth-values, for example. It was also natural for me to think of FOL as abstract in a sense, and governed by formal rules, but also something you could write down--like a computer language. B&E give some indication that this is their starting point too. Again, FWIW.

Also, over at PEA Soup I have a post on teaching logic, if you'd like to check it out.

As a matter of curiosity, what about the mathematical approach do you think creates the major obstacle for teaching students? I'm curious because I've come (since grad school) to think that we philosophers often give ourselves too many liberties in logical matters, and should be doing more to connect, explicitly, what we are doing (in our standard uses) with what is done in mathematics and computer science. But it's difficult to see how we could do this, beyond the small handful who do it already, unless either we start getting massive influxes of people crossing over from more mathematical fields, or it becomes more of a standard part of the graduate curriculum. Do you think it's a matter of background, or distance between mathematical abstraction and philosophical application, or differences in terminology, or something else?

Well, the mathematical approach presupposes set theory, and that makes it harder to teach. But more seriously, it raises a circularity problem. One of the things we want to use logic for is to analyze the language of set theory, and to analyze the possibilities for alternate set theories, and so on. But if we use set theory to construct logic, then there is a danger of circularity. I am not saying there actually is circularity here. But one has to be at least rather careful.

Post a Comment