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Some philosophers argue against truth bivalency, and say that not every statement must be true or false, but some statements can be untrue without being false, or truth-ambiguous, or both true and false (I can link to articles proposing these positions, though I've personally never understood that last one.) This comes up most often in contexts of the liar paradox and its many iterations.

Would such a position throw a wrench in making truth tables? Would it only be possible to make three-tiered truth tables? I know that the consequences of rejecting bivalency for the law of the excluded middle is discussed, but I haven't seen anyone talk about (truth-false) truth tables - perhaps because nobody uses them seriously, or because the answer is obvious and I just haven't figured it out.

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  • Aristotle himself realized there could be exceptions to his Law of the Excluded Middle. He gave an example: "Last week there was a naval battle on the surface of the Mediterranean sea."
    – maurice
    Dec 13, 2017 at 23:03

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No, not particularly; even given the debate on other kinds of logic that you allude to bivalent logic will still remain important; hence truth-tables will also remain important. Even in the context of multivalued logics one can still find truth tables useful.

For example, this wiki-page gives truth tables for two ternary (three-valued) logics - Kleenes & Łukasiewicz; the extra value is named unknown; a variety of this logic is used in commercial databases that are based on SQL; it could also be described usefully as indeterminate which leads onto the logic of paradox in which this new value is named both.

It turns out that Łukasiewiczs logic can be generalised to any number of truth-values; of course the value of truth-tables here quickly vanish, at least for a human, whereas they are still important for computers where truth-table lookups are usually the most efficient to parse a calculation.

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