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Wikipedia says about Truth Functions:

Classical propositional logic is a truth-functional propositional logic, in that every statement has exactly one truth value which is either true or false, and every logical connective is truth functional (with a correspondent truth table), thus every compound statement is a truth function. On the contrary, modal logic is non-truth-functional.

On Logical Connectives they also say:

Semantics of a logical connective is often, but not always, presented as a truth function.

I was under the assumption that logic formulas always return a true/false value. Wondering if one could show some examples of logical connectives that are not truth-functional. Looking at the Modal Logic modalities, all the logical statements read to me as true/false statements:

The traditional alethic modalities, or modalities of truth, include possibility ("Possibly, p", "It is possible that p"), necessity ("Necessarily, p", "It is necessary that p"), and impossibility ("Impossibly, p", "It is impossible that p"). Other modalities that have been formalized in modal logic include temporal modalities, or modalities of time (notably, "It was the case that p", "It has always been that p", "It will be that p", "It will always be that p"), deontic modalities (notably, "It is obligatory that p", and "It is permissible that p"), epistemic modalities, or modalities of knowledge ("It is known that p") and doxastic modalities, or modalities of belief ("It is believed that p").

For example, "It is obligatory that p". To me this says "It is obligatory that p is true", ah, but maybe the return value of this "function" would be "obligatory" instead of true. Wondering if that's what they mean, and if one could explain a little more on what kinds of non-truth-functional logical connectives exist. Trying another example, "It is possible that p", to me that says "It is true that it is possible that p". This is making me wonder if logical connectives are about conclusions rather than true/false statements, but I can't think of an example where the conclusion would be something like "hello world" instead of true/false. Wondering also if Universal Quantification falls under this category somehow.

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  • $\begingroup$ Boolean valued models still have a truth function -- but this truth function typically takes many different values and not just 'true' and 'false'. $\endgroup$ – Stefan Mesken Jun 5 '18 at 16:12
  • $\begingroup$ That makes sense, where it takes as input non true/false values. (If I'm understanding correctly). $\endgroup$ – Lance Pollard Jun 5 '18 at 16:18
  • $\begingroup$ If "it is necessary that P" was truth functional, it would have to give the same response for every true P. But some true P's are necessarily true and others are not - that is the entire point of the "necessarily true" modal operator. "Every blue dog is a dog" is necessarily true, but there is no reason to think "my favorite color is blue" (assuming it's true) is necessarily true. $\endgroup$ – Carl Mummert Jun 5 '18 at 23:54
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Although the statements of modal logic give the appearance of being truth functional, when you get down into the details of what can or cannot be proven using the axioms of the various systems, for instance the Lewis systems, there is a mismatch between what can be established using the axioms and what truth-functional methods would yield.

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  • $\begingroup$ Wondering what the Lewis systems have instead of truth values. $\endgroup$ – Lance Pollard Jun 5 '18 at 19:56
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    $\begingroup$ The Lewis systems derive theorems of modal logic from axioms using a deductive approach, so whether a given formula is considered valid depends on whether it can be derived from the axioms or other theorems. This is a more difficult and cumbersome approach from the truth-functional methods of classical 2-valued logic. $\endgroup$ – Confutus Jun 5 '18 at 21:18

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