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Consider the truth table for logical implication.

P...........Q.............P->Q
T...........T.............. T
T...........F...............F
F...........T...............T
F...........F...............T

Notice that for a false statement P, the last two rows of the truth table, both Q and ~Q follow. No matter what Q is, it's truth follows from false statement P, as the third row shows. We can therefore take Q to be "P is true." From here it follows that a false statement P implies it's own truth, as the third row shows.

Do false statements really imply their own truth? Do they really imply contradictions? Are false statements also true?

June 9, 2011

 Response from Allen Stairs on June 10, 2011

Imagine that someone finds it useful to define a new term -- "mimp," say. The newly-defined term is a conjunction, i.e., it's used to link sentences together, and it works this way: "P mimp Q" is false when "P" is true and "Q" is false. Otherwise it's true. With this definition in hand, consider the sentence "New York city has fewer than 150,000 resident mimp the next US president will come from New York." Give our definition, this is true. Our definition of "mimp" guarantees that whenever "P" is false, "P mimp Q" is true.

Looking at "→" this way may help with your puzzle. The symbol "→" (alternatively "⊃" ) is one that logicians found useful to define, and its definition is given by the rule above. Whether it matches any connective in natural language is open to doubt, and in particular, it does not mean what we mean by the phrase "logically implies." After all, "New York City has fewer than 150,000 residents" does not logically imply that the next US President will be from New York. It would be perfectly possible for New York City to have a smallish population and for the next President nonetheless to come from some other place altogether. However, "P → Q" is true given how we've defined "→."

But doesn't "→" represent implication? Yes and no. It does not represent logical implication. It represents something that logicians call material implication. They use that term because "→" has some interesting things in common with English "if...then," which we often describe as a kind of implication. But this sense of "implication" is not the sense in which one thing actually follows from another. After all, suppose that as a matter of fact, if Mary goes to Boston, then John will go Detroit. That might be true and yet it would be misleading to say that "Mary goes to Boston" implies "John will go to Detroit." Using the word "implies" makes it sound as though there is some connection of logic between these two possibilities, when clearly there isn't. The word "material" in "material implication" is meant to cancel the suggestion that we're talking about logical implication. Instead, we're talking about what may be no more than a merely factitious connection: as it happens, the case where Mary goes to Boston and John doesn't go to Detroit won't arise. So we can say that a false statement materially implies any statement at all, but this isn't very shocking when we keep in mind that we aren't talking about logical implication.

All this said there's one more point that needs to be addressed. You noted that we could start with a false statement "P" and let "Q" stand for '"P" is true.' We'd still have "P → Q." This seemed to disturb you because it seems to say that this false statement implies its own truth. And that, rightly understood is correct even if we mean logical implication. It's trivial that every statement logically implies itself. After all, a statement "P" logically implies a statement "Q" when it's impossible for "P" to be true and "Q" simultaneously false. If we put "P" in place of "Q," this amounts to saying that "P" can't be true and false at the same time. But it's also trivial that "P" implies "'P' is true." So "New York has fewer than 150,000 residents" logically implies "It's true that New York has fewer than 150,00 residents." If the first statement is true, then second one is guaranteed to be true as well. However, there's no danger that New York's population will be decimated by a mysterious force of logical implication. Even though the first statement logically implies the second, both statements are false. The fact that one statement is implied by another doesn't make it true, and that saves us from paradox.

 Response from William Rapaport on June 10, 2011

One branch of logic that deals with an alternative to material implication, and that has applications in artificial intelligence, is called "relevance logic". For more information on it, take a look at:

Anderson, Alan Ross, & Belnap, Nuel D., Jr. (1975), Entailment: The Logic of Relevance and Necessity (Princeton, NJ: Princeton University Press) -- especially the introductory chapters that present arguments as to why relevance logic is "better" than classical logic.

Lepore, Ernest (2000), Meaning and Argument: An Introduction to Logic through Language (Malden, MA: Blackwell), §A3 ("Conditionals"), esp. §A3.1.1 ("Paradox of Implication), p. 317, and §A3.1.3 ("Paradox of Implication Revisited"), pp. 319-320.

For a literary discussion of what happens when a computer or robot uses classical logic, see:
Asimov, Isaac (1941), "Liar!", Astounding Science Fiction; reprinted in Isaac Asimov, I, Robot (Garden City, NY: Doubleday), Ch. 5, pp. 99–117.

And for applications to AI, see:

Shapiro, Stuart C., "Relevance Logic in Computer Science", in Anderson, Alan Ross; Belnap, Nuel D., Jr.; & Dunn, J. Michael (eds.) (1992), Entailment: The Logic of Relevance and Necessity, Vol. II (Princeton, NJ: Princeton University Press), § 83, pp. 553-563 and online at: http://www.cse.buffalo.edu/~shapiro/Papers/rincs.pdf

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