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2 Bivalence is deepest 3 Why these distinctions might matter 4 External links |
For any proposition P, at a given time, in a given respect, there are three related laws:
It is possible to state the laws of non-contradiction and the excluded middle in the formal manner of the traditional propositional calculus:
It is, however, not possible to state the principle of bivalence in such a way, as the traditional propositional calculus just assumes sentences are true or false.
These different principles are closely related, but there are certain cases where we might wish to affirm that they do not all go together. Specifically, the link between bivalence and the law of excluded middle is sometimes challenged.
A famous example is the contingent sea battle case found in Aristotle's work, De Interpretatione, chapter 9:
In fuzzy logic or other multi-valued logics dealing with vagueness, again it may be the case that (P or not-P) is true but P is not considered true or false. For example, suppose P is:
The laws
Bivalence is deepest
In fact, with the law of bivalence taken for granted, the two other laws can be derived as theorems, using the rules of propositional calculus.Why these distinctions might matter
Future contingents
The law of the excluded middle clearly holds:
However, some philosophers wish to claim that P is neither true nor false today, since the matter has not been decided yet. So, they would say that the principle of bivalence does not hold in such a case: P is neither true nor false (yet). This view is controversial, however.Vagueness
In this case, in multi-valued logics, this proposition may be given a truth value in between "true" or "false" (if Joe has a bit of hair, but not much). Nevertheless, under some such logics, the following statement may be considered true:
So such logics reject bivalence, but maintain the law of the excluded middle.External links