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Probably any truth-functionally complete set of operators would suffice to represent the neither-true-nor-false value (and I just told you how to do it, to!).
If the senses of logical constants are individuated in this way by the conditions for their grasp, we can distinguish between truth-functionally equivalent constants with different meanings, like and “ ”, as defined below:
The bar operator and the Z operator provide the essentials of a truth-functionally complete strong Kleene semantics for three-valued logic.
For the former approach, the problem was that the empirical legitimacy of statements obtained via indirect testing also transfered to any expressions that could be truth-functionally conjoined to them (for instance, by the rule of ˜or™-introduction).
And with negations of conditionals and conditionals in antecedents, we saw, the problem is reversed: we assert conditionals which we would not believe if we construed them truth-functionally.
Likewise, the distinction allows Abelard to define negation, and other propositional connectives, purely truth-functionally in terms of content, so that negation, for instance, is treated as follows: not-p is false/true if and only if p is true/false.
The reason why any reductive identification of final ends with any lower-level properties is bound to fail is that the ends possessed by organisms are holistic ends, and no lower-level description of a biological process, however complete it may be, can be equated in meaning with a holistic description of the same process, even if the two descriptions are truth-functionally equivalent.
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