Formal Logic And Probability

One of the arguments is that probability does not extend predicate logic, but does extend propositional logic. The concern is that because predicate logic is formal and propositional logic is not, or not to the same extent, that probability is therefore of limited use. This is explained (in classical terms) on David Chapman’s site. I don’t have the space here to rebut everything there with which I disagree (there is no such thing as unconditional probabaility for one; believing there is accounts for most of what’s wrong): that would take a book. Which I just happen to have written: Uncertainty: The Soul of Modeling, Probability & Statistics. But we can do a few things here.

The biggest weakness of predicate logic is that, pretty and mathematical as it is, any time you want to use or apply an argument in predicate logic to some real world proposition, you need to “lapse” into propositional logic, which is to say plain English (or whatever language you use for understanding). For instance, in predicate logic you can write:

which is peachy and correct (some might use different symbols) and formal. It’s the formality that makes it mathematical and which allows it to be manipulated algorithmically. But it is also what kills its usefulness in actual applications.

In the equation, P and Q are predicates and x is a variable. The formality means that we can stick any predicate and any variable into the formula and it should work. One predicate (for P) might be “is a man”, and another (for Q) “is mortal”; a reasonable variable is s = “the man Socrates”. Both can be inserted into the formula to produce, finally in English, “All men are mortal and Socrates is a man, therefore Socrates is mortal.”

Formalists will say the conclusion is true because of the schema or form of the predicate-logic formula. The symbols—not the words—are purely formal objects which slide through a rigorously constructed pipeline to the conclusion, just like the quadratic formula provides solutions to quadratic equations (keep this example in mind). In propositional logic, plain common sense will say the conclusion is true because the conclusion shares in the essence of the premises. Yet formalists will complain and say that the propositional logic version of the argument amounts to

,

where M is the proposition “All men are mortal”, N = “Socrates is a man”, and R = “Socrates is mortal.” This, they say, isn’t formal, because the propositions in this formula aren’t “about” anything. They’re floating symbols, so, of course, R doesn’t follow from the conjunction of M and N. How could it? One cannot stick in just any old propositions for M, N, and R and have any hope the argument will produce a true conclusion. The schema itself, formalists say, is invalid (yet it produces the odd true argument).

Aristotelian logic, on the other hand, takes the argument as a syllogism and, partly by virtue of its syllogistic form, and partly from the plain understanding of the words and grammar, sees the conclusion as valid. The argument is also considered sound because of the understanding of intension (this is not a misspelling) of the terms.

Why the hunger for formality? Well, that’s what math is all about and, as such, there is nothing wrong the goal. But to say all logic should be formal is to claim all thought can be quantified or made into mathematics somehow. And that is the goal of many; think of certain forms of artificial intelligence. There is no proof of that claim; there is only the assurance or hope that it can be so.

But there is bad news for formalists. In 2009, David Stove proved logic is not formal (I’m quoting from my own article, which in turn draws quotations from Stove’s Rationality of Induction; see the original for details).

An argument is formal “if it employs at least one individual variable, or predicate variable, or propositional variable, and places no restriction on the values that that variable can take” (emphasis mine). Stove claims that “few or no such things” can be found.

Here is an example of formality: the rule of transposition. “If p then q” entails “If not-q then not-p” for all p and for all q.

This is formal in the sense that we have the variables p and q for which we can substitute actual instances, but for which there are no restrictions. If Stove is right, then we should be able to find an example of formal transposition that fails.

First a common example that works: let p = “there is fire” and q = “there is oxygen”, then

    “If p then q” == “If there is fire there is oxygen”.

And by transposition, not-q = “there is no oxygen” and not-p = “there is no fire” then

    “If not-q then not-p” == “If there is no oxygen then there is no fire.”

For an example in which formal transposition fails, let p = “Baby cries” and q = “we beat him”, thus

    “If p then q” == “If Baby cries then we beat him”.

But then by transposition, not-q = “We do not beat Baby”, not-p = “he does not cry”, thus

    “If not-q then not-p” == “If we do not beat Baby then he does not cry.”

which is obviously false. (Stove credits Vic Dudman with this example.)

So we have found an instance of formal transposition that fails. Which means logic cannot be “formal” in Stove’s sense. It also means that all theorems that use transposition in their proofs will have instances in which those theorems are false if restrictions are not placed on its variables. (It’s worse, because transposition is logically equivalent to several other logical rules; we won’t go into that now.)

It is Stove’s contention that all logical forms will have an example where it goes bad, like with transposition.

Now, as I said, some form of Aristotelian logic or of something more propositionally informal and fundamental must take place when we assent that the proposition “Socrates is mortal” follows from the other propositions. It is not the schema that makes something true. Schemas have no power! Things are not made true by mathematical or logical form (this “form” is not the same as the Aristotelian “form”, of course: for “form” here, read “formula”). They are caused to be true by something, all right, but a schema has no causal power.

Go back to the quadratic equation example. It, like all mathematical theorems, has a proved formal structure. But is not purely formal (in Stove’s sense). The quadratic formula has restrictions. You cannot input matrices into it, for example. The pure formality doesn’t exist because of these restrictions.

As said above, when applying predicate calculus to a real-world problem, we always must lapse into propositional logic or plain English. This falling back, as it were, always brings with it restrictions, which is why ordinary discussions aren’t purely formal. The real problem lies in attempting to formalize what ultimately cannot be formalized.