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Material ImplicationDecades ago in a philosophy class I learned the simplest system of formal (mathematical or symbolic) logic. Most of what we learned was purely formal -- the construction of proofs and the symbolic expression of such complicated statements as "If all x are B and some y are C, and if there exists a z such that for every y...." etc., etc. I did OK on the manipulation of the symbols, but I never felt quite right about the central concept of the system, which was material implication. Material implication (symbolized, for example, "A --> B" for "A implies B") was supposed to symbolize ordinary-language implication of the type "All crows are black" / "If it's a crow, it's black". However, what "A-->B" actually was defined as was simply "either not-A, or B, or both". This definition of implication is expressed "A --> B = ~A v B df." My thesis is that material implication
doesn't have much to do with ordinary-language implication, to the point that
calling it "implication" is misleading. My speculation is that the craving for
formalization and mathematical-seeming notation caused logicians to accept a
defective system rather than not have any system at all. Why material implication
is still taught to beginners, I don't know. Is Material Implication Really Implication? The problem is that material implication is not implication in any ordinary-language understanding of the term. The negation of material implication does make sense: ~(A-->B) = (A.~B). If there is a crow which is not black, then it is not true that all crows are black. But any of the following can equal "all crows are black": a black crow, a white owl, or a black owl. This does not make any sense at all. So we apparently have a good definition of the disproof of implication, rather than a definition of implication. (Perhaps there was influence of Popper's definition of "true" as, approximately, "not yet proven false".)
One explanation of this I have been given is that "material implication" is implication all right, but that it's a scientific technical concept to be distinguished from the naive and inexact ordinary-language notion of implication. This is self-serving and misleading, however. Material implication is not a precise formalization of the imprecise ordinary language term "implication". It is a precise way of expressing a quite different idea, which is the possibility of ordinary-language implication. Thus "(A-->B)= (~A v B)" is a good expression of the ordinary-language idea that "It is possible that all crows are black". This is much weaker than ordinary-language implication, but it is something meaningful. The useful part of material implication is its negation "~(A-->B) = (A.~B)", which really does mean "It is not possible that all crows are black", and represents the Popperian negative fact, i.e. a non-black crow. "A-->B" is called "implication", but it really only means the possibility of implication.
Ordinary Language Interpretations of Material Implication
In quantum theory and relativity, formal
mathematical expressions of basic concepts are often
counterintuitive and impossible to express in ordinary language.
Logic teachers make it seem that material implication is another
such case, but it isn't. All expressions in material implication can
be expressed in ordinary language. The real problem is that material
implication is a poor formalization of the ordinary-language idea of
implication.
The biggest debate about material implication was counterfactual conditionals. For example, in English we can say things like "If the Japanese had won the Battle of Midway, WWII would have lasted much longer" or "If Jimmy Carter had kept inflation under 10%, he would have been re-elected". Whether true or not, these are meaningful statements that people might actually say. And while they are valid implications in material implication, that means nothing, since in material implication ALL counterfactuals are valid implications. You could just as well have said "If the Yankees had won the World Series in 1940, WWII would have lasted much longer" or "If Bob Dylan hadn't released 'Slow Train Coming', Carter would have been re-elected". All these are symbolized "~A.~B", which is a valid implication. This form of implication is often used in ordinary speech in a joking way -- "If George W. Bush deserved to be President, then I deserve to play in the NBA". While we can say that these statements do have a formal, hypothetical validity, their existence is another reason why material implication is not a usable formalization of ordinary-language implication. The usual solution is to declare this to be a pathological exception and make up an ad hoc rule against counterfactuals to avoid it, but this is cheating. For example, if I say today that "If the Yankees win the World Series (A), the Democrats will win the next election (B)", does that implication become valid once the Yankees lose? (For in material implication, that statement is cannot be false unless A.~B). It really seems that material implication is just plain not usable as a formalization of a large class of real-world phenomena, namely all those having to deal with historical events. There have also been attempts to invent other formalizations of implication which are usable in real-world situations, but my perception is that these formalizations give up in simplicity and power what they gain in plausibility, and that the whole point of formalization is thereby lost.
When the "-->" sign (which is usually a horseshoe on its side and open to the left, and counts as one sign even though I write it here with three) was introduced, it was explained that "A --> B" was simply an ink-saving abbreviation for "~A v B". This is not true, however, as the charts below show. The addition of the new --> sign doesn't save any ink and is unnecessary. The form "A --> B" is in fact briefer than "~A v B" (if you count "-->" as one sign), but if you work it out for the rest of the paradigm you come out even. Thus, you have added an additional symbol to your system, supposedly for brevity's sake, which does not make anything briefer.
With the "-->" sign, you have 16 signs; without the "-->" sign, you also have 16 signs. On the whole, there is no saving of ink. Only if statements of the positive form "All crows are black" are privileged is there a saving. This is, in fact, the form commonly used by logicians in their ordinary-language examples, but formally it should be regarded as equivalent to the other forms. The "-->" sign saves nothing in the formalization, but merely adds a non-functioning fifth wheel to the sign set in order to pretend that material implication is a formalization of ordinary-language implication. In the material-implication system without the "-->" sign, the expression of two types of statements are actually shorter, balancing out the two which are shorter with the "-->" sign: "All non-crows (~A) are black (B)" and "No non-crows (~A), are black (~B)" (or "Only crows are black"). So it seems the only justifications for adding a new sign to the system were rhetorical: either to make it more plausible to present primitive implication as a representation of ordinary-language implication, or as a convenience for formal users, rising from the fact that implications of the positive form "A --> B" are empirically more often found in practice. My conclusion is that the "-->" sign plays no role in the system itself, but is just a re-labelling, intended to convince the reader that material implication is, in fact, what we normally think of as implication. But really, "~A v B" is just "~A v B". It's not "A-->B".
In my discussions of this,
I've been assured that material implication has its uses in computer
programming, and also elsewhere in formal logic. This
isn't really surprising; certainly in some circumstances
"~A v B" is just what you need to know, and it might be
handy to be able to abbreviate it as "A --> B". I've also
had it pointed out to me that the whole project of formalization
has been very successful and valuable, leading ultimately to
computers, robotics, artificial intelligence, etc. I agree with this, and as an
early attempt in that direction I'm sure that material implication was seminal
and pioneering. This has been a tedious exposition, but most of logic is like that. The difference is that my plodding exposition, rather than developing logic itself in increasingly more meticulous detail, has shifted the focus to logic's relation to the larger system (informally called "reality") of which logic is part. The sign specifically used to represent "material implication" (usually a horseshoe with the open end to the left, though I have to use "-->") does not represent implication. What it represents is formal logic's elusive object of desire: the desire to represent implication. Update: From the formalist point of view, E. W. Dijkstra argues that formalization is a way of escaping from verbal language. A formalized statement should not be thought of as a restatement of a verbal statement at all, and it creates confusion to think of it that way. (Thanks to Max Strini).
Read More This is the #4 Google hit for "material implication" at the moment. The others in the top five are these: http://personal.bgsu.edu/~roberth/m-imp.html http://philosophy.lander.edu/logic/conditional.html http://www.earlham.edu/~peters/courses/log/mat-imp.htm
http://en.wikipedia.org/wiki/Material_implication
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