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:

\forall x [Px \to Qx] \wedge Ps \therefore Qs

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

M \wedge N \therefore R ,

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.

13 Comments

  1. It strikes me that the Quine-Duhem thesis, or near to it, fits well here. Stove’s proof seems as significant, and as little taken seriously, as Gödel’s.

  2. Fascinating. I find myself agreeing, using examples that have been posed to me by others before Briggs has, and by those that I’ve posed to myself. My exposure to formal logic was limited to Boolean Algebra. I never took the time to look for more general examples.

  3. Are you still beating your crying babies? Perhaps, if you didn’t beat it, it wouldn’t be crying so darn much. I know, I know, if it didn’t cry, you wouldn’t have to beat it. But, since you are not beating the baby right now, the little guy must have shut up!

    Nonetheless, expect a visit from social services.

    p?q,¬q?¬p

  4. I believe the following argument is doubtful or incorrect:

    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.”

    The reason I say so is that there is an equivocation on the meaning of q between the positive and contrapositive forms. I do not think q is being construed in exactly the same way in the positive q form and negated, not-q form.

    Let me restate the argument:
    If (= whenever) B cries, then we will beat B.
    If it is not the case that we will beat B, then it is not the case that B is crying.

    This is argument is valid, and goes by the name “modus tollendo tollens,” or “modus tollens” for short.

    In the argument above, the positive form of q seems to mean that an event will ensue upon the occurrence of a condition p. In the contraposed form, q sees to mean a habitual or permanent action. I believe this shift in the meaning of q is enough to say that this proposed counterexample to modus tollens is not valid.

    Thank you for your consideration.

  5. If Baby cries, then we will beat him in the very near future.
    Therefore, if we are not beating Baby, he hasn’t cried in the very recent past.

    I agree with Fr. Rickert, the logic seems sound to me, when stated accurately. If A causes B, and B hasn’t happened, then A won’t happen in the future? What law of logic allows one to swap the time order of cause and effect in the contrapositive?

    Got any other (more convincing) examples?

  6. My understanding is that the first conjecture has to be true for the contrapositive also to be true.
    The first argument is not always true so it is already false before testing the contrapositive.

    Something must be known about the matter to know that the first conjecture holds in all examples such as set and a subset where p is contained within the set of Q. I am using Joy language, I have no idea whether this is right or not.

    That this isn’t confusing is a sign that I don’t understand.
    I thought the first line was “for all p and all Q”

    PDF’s are off limit, I can’t read chapter ten at present. The talking man just plays dumb. He’s learning some life skills.

  7. Fr. Rickert’s analysis hi-lites a very fundamental issue — when approaching a situation from a strictly philosophical perspective, such issues arise.

    However, when approaching them from real-world physics-based perspective, they might come up, but often get addressed and never become an issue. In applying the kind of analysis Briggs touched on (predicate & propositional logic are but two types in a range of such analytical approaches) a modeler will define ‘truth tables’ that establish what is and is not — those are based on what the thing being modeled is and isn’t doing relative to the environment in which it acts.

    Application of such logical approaches is intrinsic to AI. For a topical example see: https://www.csee.umbc.edu/courses/671/fall09/notes/10a.ppt (or, google the following keywords & take your pick from the responses: advantages of predicate logic over propositional logic).

    This is a recurring issue here — philosophy is NOT a good tool for assessing logical issues in reality. It allows one’s ignorance of the real-world factors applicable to some topic to facilitate seemingly logical, perhaps conceptually unassailably logical, sequences of ‘if-then’ analyses to achieve an absurdly ridiculous conclusion

    (e.g.: if one adds a few drops of color to a gallon of paint, then stir clockwise to mix, then, one can un-mix the colors by stirring the same amount counter-clockwise — seems silly, but if the subject is in some way constrained by real-world factors and one is ignorant of them, careful step-by-step application of logic can & does lead to such absurd conclusions, too often going unnoticed;

    an arguably even more complex area is in financial market models — there’s a variety of abrupt external influences that apply such that logical application of ‘if-then’ sorts of logic of the crying baby sort that suddenly become pre-empted by other factors (e.g. automated margin calls or sell orders that stop some particular action); logic applied there from a strict philosophical perspective simply fails [very quickly] unless one factors in the external environmental factors).

    Using philosophy, and remaining dependent on that, leads to debates about this or that approach’s pitfalls/advantages and logical conundrums that don’t occur, much, when real topical experts inject situational reality into an analysis. …but then…”situational reality” is or may be in another words, “science,” and heaven forbid we permit that to intrude and, perhaps, undermine some deeply held sacred belief.

  8. Ken —

    Worth noting that even logicians realize the need for the distinction between validity and soundness.

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