Valuations, combinatorial vs realizable
This is request for references on an issue in very elementary logic!
To set the scene, suppose we take the atoms of a formal language for propositional logic to be interpreted. Yes, yes, I know that different authors take different official lines about how to treat their ‘P’s and ‘Q’s — hence the ‘suppose’! We are considering the line where a formal language is indeed taken to be a language, with meaningful wffs, so inferences in the language really are genuine inferences, etc.
Perhaps then the glossary for a particular PL language reads
P: Water is H2O,
Q: Jill is married,
R: Jill is single.
So now consider, then, writing down a truth-table for a wff built from these atoms, as it might be ‘(P ∧ (Q ∨ R))’. We of course standardly consider all combinatorially possible assignments of values to the three propositional atoms, giving us an eight-line table. But we might now remark that (according to most) there is no possible world at which ‘P’ is false. And (according to everyone, assuming it is the same Jill, etc.) there is no possible world at which ‘Q’ and ‘R’ take the same value. Hence, of the combinatorially possible assignments of values to these three interpreted atoms, in fact only two (on the majority view) correspond to a possible world. In a word, perhaps, in this case only two of the eight combinatorially possible valuations are realizable possible valuations (meaning realizable-at-some-possible-world). But there must be a better word!
Looking ahead, we define the tautological validity of a PL inference in terms of truth-preservation on all combinatorially possible valuations of the relevant atoms; whereas plain deductive validity is a matter of truth-preservation with respect to any possible world, i.e. on any valuation-realizable-at-some-possible-world — which is why tautological validity implies validity for PL inferences, but not vice versa. (If, as some do, you prefer to build ‘in virtue of logical form’ into your official definition of validity, then replace talk of plain validity here with talk of necessary preservation of truth.)
OK, having set the scene, here’s the request. The point that combinatorially possible assignments of truth-value for an interpreted PL language may in some cases (depending on the intepretations of the atoms) not correspond to possible worlds, is an entirely elementary one. But which elementary texts (or sets of detailed online notes) make the point particularly clearly? At some point a couple of months ago, I did read a text — online I think — which handled this particularly clearly, and used a better word than “realizable”. But like an idiot I didn’t take notes at the time. So any suggestions/pointers?
(Full disclosure: This is one of the many issues that I want to handle better in IFL2 than in IFL1, and so I’d really like to check my draft treatment against versions elsewhere — and also like to see how others who are clear in the propositional case handle the analogous distinction when it comes to predicate logic.)
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