Does bivalence assume truth values to preserve？ 第1页

In classical (propositional) logic, truth is preserved in the following sense: given a set of propositions ({p, q, r} for example), a "valuation", or a "truth assignment" is a function that takes each of the propositional variable p, q, r to a unique truth value (either "True" or "False") such that a propositional variable is assigned True if and only if its negation is assigned False. Note that the principle of bivalence is a semantic notion (notion about truth and falsehood in a given interpretation). A closely related syntactic (concerned with forms of sentences) notion is the law of excluded middle, which says that sentences of the form "p or not p" are always a valid inference from any set of premise.

Now back to your question. Classical logic "assumes" truth to preserve insofar as truth assignments are the only factor that determines the truth value of an atomic sentence (sentence with no logical connectives like "not", "and", "if", "or"). Note that classical logic is insensitive to tense (both the grammatical notion and the metaphysical notion of time) such that there's no difference whatsoever between "he was alive", "he is alive", and "he will be alive" unless they are treated as atomic. The consequence of this (plus the principle of bivalence) is that statements about the future are either true or false, which, it seems to some people, gives classical logic a fatalist flavor. Indeed it seems counterintuitive to say "either it's true that it'll rain tomorrow or it's false that it'll rain tomorrow" instead of "it's unknown whether it'll rain tomorrow." A closely related consequence of ignoring tense is contrapositive inferences (from "if p the q" inferred "if not q the not p" and vice versa) of the following kind when it comes to counterfactuals: premise: if Kant didn't die in 1804, then he still wouldn't be alive today; conclusion: if Kant is alive today, then he would have died in 1804.

So citing time as grounds for rejecting bivalence appeals to the kind of consideration briefly sketched above. The argument is roughly this: if we want logic to capture our intuition that some propositions about the future are undetermined with regard to their truth value at the time of utterance, we must introduce a third truth value ("unknown" or "possible") so that "it'll rain tomorrow" uttered on Jan. 1, 2017 has the truth value "unknown" on Jan. 1, 2017 but has the truth value "True" on Jan. 2, 2017 (supposing that it rained on that day).

Note that truth assignment is no longer the only factor that determines the truth value of an atomic sentence. What we need is a truth assignment relative to some time. This makes the resulting logic no longer classical. But truth is still "preserved" in the following sense: while a poposition can have different truth values at different times, for any given time a proposition has only one unique truth value. The kind of time-sensitive logic is often grouped under the umbrellla term "temporal logic". SEP has a nice article on that topic.