Ramble, Part 56: If...

[stoutfellow]

(Has it really been a month and a half? Sorry, sorry....)

The symbolic logic devised by Boole and his immediate successors treated the AND, OR, and NOT operators algebraically. It took something like half a century before Charles Sanders Pierce and Gottlob Frege, working independently, added the fourth major operator, IF...THEN..., to the inventory.

It's a bit of a puzzle, why it took so long. Boole had noticed that his algebraic logic could be interpreted in two different ways, as describing propositions or classes, and IF...THEN... doesn't fit this quite so well; this may have had something to do with it. Also, the algebraic properties of the operator are quite different from those of AND, OR, and NOT (which also resembled the more familiar properties of the arithmetic operations); perhaps this played a role. But I think that there's a more fundamental problem.

To a greater extent than the other operators, ordinary-language "if A, then B" demands a relationship between the propositions A and B; we expect that knowing A would provide grounds for believing B. The Boolean idea was that operators should be "truth-functional", that is, that the truth or falsity of the compound statement should depend only on the truth or falsity of the components, not on their actual meanings, and this does not sit well with the ordinary-language use. A good deal of work has been expended on resolving this problem, with a fair degree of success.

Let's start by working out how to approximate the ordinary-language use with a truth-functional operator. I think it works best to start with an "if A, then B" statement which is incontrovertibly true, and see what choices of truth-values for A and B are compatible with it and which are not. Let's go with this one:

If the animal behind the door is a tiger, then it has stripes.¹

Statement A here is "The animal behind the door is a tiger"; statement B is "The animal behind the door has stripes". There are a number of possibilities. First, the animal may in fact be a tiger; statements A and B are then both true. (This tells us that our truth-functional operator must give IF T THEN T the value T.) Second, the animal may be a lion; A and B are both false. (IF F THEN F is T.) Third, it could be a zebra; A is false, but B is true. (IF F THEN T is T.) The one combination that is ruled out is the possibility that A is true, but B is false. (IF T THEN F is F.) This completely specifies our truth-functional IF...THEN... operator.

This operator is not a very good approximation of ordinary-language "if...then...", primarily because it requires us to accept as true a lot of statements that strike the ear as, at best, non sequiturs: "If the Sun is a G2 star, then George Washington was the first President of the United States." (Also, perhaps disturbingly: "If the Sun is an F0 star, then George Washington was the first President of the United States.") For this reason, some logicians, such as C. I. Lewis, have distinguished between "material implication" (the truth-functional operator) and "strict implication", which fits the ordinary-language use better. The latter is much harder to manipulate mathematically, however.

Mathematicians don't have too much trouble with this, because our "if...then..." statements generally don't appear in their naked form, but rather embedded in universal statements: "In every situation (of some desired type), if A, then B". This means that, in the situations of interest, any of the three allowable combinations (A=T, B=T; A=F, B=T; A=F, B=F) can occur, but the fourth (A=T, B=F) cannot. This fits the natural-language use much better; after all, where do our notions of causation come from, except by repeated observation of situations and noticing that, in every situation where A holds, B does also?

Modern logicians resolve the problem in a similar way. Lewis' "strict implication" is a statement that in all (or some acceptable subset of all) possible worlds, whenever A is true, so is B. There's a fairly complex systematic way of dealing with this, with which, alas, I am only slightly familiar; if you want to look further, look up the name Saul Kripke.

The last two paragraphs point towards another major development in logic, the study of "quantifiers". We'll take that up next time.

[1] Don't give me any crap about albinos, OK?

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Comments

[sunlizzard.livejournal.com]

Wheee! I think this is a first with the Rambles: I actually get most of this one. It was just this sort of clear, clean logic that made (admittedly very, very basic!) computer programming so interesting to write. And I've never forgotten how handy it is, particularly the if/then statement, when applied to innumerable other instances--good logic is good logic, period.

[stoutfellow.livejournal.com]

Glad to hear it! A lot of the nineteenth-century stuff is hard to describe without getting very technical (and the twentieth century, even more so), but most of what I want to talk about from here on has to do with foundational issues - logic, set theory, and so on. I'm hoping I can keep that accessible.

[sunlizzard.livejournal.com]

Kewl. That'll be fun. I do always read them, but I nearly might as well have been scanning Sanscrit for as much of it as "sticks" in my brain! Just not math-cognizant, me.