northanger (northanger) wrote,
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vacuous truth & intuitionistic logic

Vacuous truth @ wikipedia

AQ 297 = VACUOUS TRUTH = THE CHAOS RAMPANT = HERMETIC MARRIAGE

AQ 408 = INTUITIONISTIC LOGIC = SPINOZA'S PARALLELISM

Informally, a logical statement is vacuously true if it is true but doesn't say anything; examples are statements of the form "everything with property A also has property B", where there is nothing with property A. It is tempting to dismiss this concept as vacuous or silly. It does, however, have useful applications. One example is the empty product -- the fact that the result of multiplying no numbers at all is 1 -- which is useful in a variety of mathematical fields including probability theory, combinatorics, and power series. The eminent mathematician Gian-Carlo Rota speaking before an audience of perhaps 1000 mathematicians in Baltimore in January 1998, stated that physicists in particular like to dismiss this idea as ivory-towerish (Rota's rhetorical style often used exaggeration and hyperbole), and mentioned as an example that the elementary symmetric polynomial in no variables at all is 1. Then he went on to say that this led to the remarkable discovery that the Euler characteristic is actually one of the finitely additive "measures" treated in Hadwiger's theorem, so that "pure" mathematicians who attach importance to this kind of "vacuity" have the last laugh.

Vacuous truth should be compared to tautology, with which it is sometimes conflated.

In logic, a tautology is a statement which is true by its own definition, and is therefore fundamentally uninformative. Logical tautologies use circular reasoning within an argument or statement. A logical tautology is a statement that is true regardless of the truth values of its parts. For example, the statement "All crows are either black, or they are not black," is a tautology, because it is true no matter what color crows are. As a humorous example, tautology is famously defined as "that which is tautological". (That definition is, of course, tautological.) In a more realistic example, if a biologist were to define "fit" in the phrase "survival of the fittest" as "more likely to survive", a tautology would have been formed.

Vacuous truth is usually applied in classical logic, which in particular is two-valued ... However, vacuous truth also appears in, for example, intuitionistic logic.

Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. Roughly speaking, "intuitionism" holds that logic and mathematics are "constructive" mental activities. That is, they are not analytic activities wherein deep properties of existence are revealed and applied. Instead, logic and mathematics are the application of internally consistent methods to realize more complex mental constructs (really, a kind of game). In a stricter sense, intuitionistic logic can be investigated as a very concrete and formal kind of mathematical logic. While it may be argued whether such a formal calculus really captures the philosophical aspects of intuitionism, it has properties which are also quite useful from a practical point of view.

In intuitionistic logic, epistemologically unclear steps in proofs are forbidden. In classical logic, a formula — say, A — asserts that A is true in an abstract sense. In intuitionistic logic, a formula is only considered to be true if it can be proved. As an example of this difference, law of excluded middle, while classically valid, is not intuitionistically valid, because, in a logical calculus that allows it, it's possible to argue P ∨ ¬P without knowing which one specifically is the case.

This is fine if one assumes that the law of the excluded middle is some kind of subtle truth inherent in the nature of being; but if the validity of a mental construct is entirely dependent upon its coherence with its context (the mind), then epistemological opacity is, in effect, cheating. In intuitionistic logic, it is not permitted to assert a disjunction such as P ∨ ¬P without also being able to say specifically which one is true. More generally, the formula P ∨ ¬P is not a theorem of intuitionistic logic as it is of classical logic. In classical logic, P ∨ ¬P means that one of P or ¬P is true; in intuitionistic logic, P ∨ ¬P means that one of P or ¬P can be proved, which is a much stronger statement, and which might not always be the case.

Intuitionistic logic substitutes justification for truth in its logical calculus. Instead of a deterministic, bivalent truth assignment scheme, it allows for a third, indeterminate truth value. A proposition may be provably justified, or provably not justified, or undetermined. The logical calculus preserves justification, rather than truth, across transformations yielding derived propositions.

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