Valuations again
As a footnote to my last post, I want to consider a passage in Nick Smith’s Logic: The Laws of Truth (rather unfairly picked out from a number of candidates).
Smith talks of giving a glossary for PL, a list like
A: Antelopes chew the cud,
F: Your best friend is my worst enemy,
N: Albany is capital of New York
(his examples). In such a case, he says (p. 33) that the sentence letter “represents” the proposition expressed using the sentence on the right, and we might wonder what “represents” means here. He also talks (p. 34) of a sentence letter “stand[ing] for” a proposition, and then (p.35) of a formula “express[ing]” a proposition. I’d say that representing, standing for, and expressing are different — but let’s not nag about that. I think it is clear enough that Smith thinks of a glossary for some PL sentences as assigning them at least Fregean senses (i.e. truth-relevant meanings), so that they are meaningful and express propositions. Which is fine by me, so long as we understand talk of propositions in a sufficiently neutral way.
Smith distinguishes between an argument’s being necessarily truth-preserving and its being necessarily truth-preserving in virtue of its form or structure. Some (e.g. me now in IFL2 though not in IFL1) would mark the difference as the difference between being valid and being logically valid. Smith, with about as much warrant from the tradition, reserves “valid” for the second status. But we agree there’s a distinction to be made, and agree that what official stories about tautological validity, q-validity (as I’d call it), S5-validity and so give us are accounts of varieties of necessary truth-preservation in virtue of form — special cases of logical validity for me, cases of validity for him. Which is again fine by me.
But now consider this passage from p. 65:
An argument is invalid if there is a possible scenario in which the premises are true and the conclusion false. A truth table tells us whether there is such a possible scenario—but it also does more: if there is, it specifies the scenario for us (and if there is more than one, it specifies them all). For a given argument, we term a scenario in which the premises are true and the conclusion is false a counterexample to the argument. So a truth table does not merely tell us whether an argument is invalid: if it is invalid, we can furthermore read off a counterexample to the argument from the truth table.
Well, suppose we are working with the following glossary (nothing that Smith says, as far as I can see, bans this):
P: Kermit is emerald green
Q: Kermit is green
Or perhaps this glossary:
P: Jill has a twin
Q: Jill has a sibling
Or perhaps this glossary
P: Jill is much taller than Jack
Q: Jack is shorter than Jill.
Then in each case a truth-table tells us that the argument P, so Q is not necessarily-truth-preserving-in-virtue-of-PL-form (where PL form is the aspect of form, i.e. distribution of truth-functional connectives, that propositional logic latches onto). It doesn’t immediately follow from that that the argument is not necessarily-truth-preserving-in-virtue-of-form tout court, but let that pass. For the sake of argument, go with the verdict that the arguments in each case aren’t valid-in-Smith’s-sense. But of course, from the counterexample to tautological validity, meaning the valuation [P] = T, [Q] = F, we can’t in these cases read off a counterexample in Smith’s sense of a possible scenario in which the premiss is true and the conclusion false. In these cases, there simply is no such possible scenario.
Smith has seemingly temporarily forgotten that, once we interpret PL atoms, this allows for relations of necessary connection of truth values that aren’t picked up by truth-tables.
My apologies to my namesake if I am misrepresenting him, and my eye skipped over a qualification or caveat which would make the quoted passage ok after all. (Smith’s book is one of the very best introductions to logic for philosophers, in part because it is long and expansive. However, because it is long and sometimes wordy, with a lot going on in extensive footnotes too, and because the author doesn’t go for boxed definitions/key explanations and/or headline chapter summaries to keep the student on track, it would no doubt be possible to misread important stuff.) But in any case, I am only using this book as an example: some other well-known texts also come more or less near to making the same mistake, or at least are remarkably quiet about the dangers of falling into such a mistake.
So the point needs to be emphasised: Combinatorially possible valuations as listed on lines of a truth-table may, given the interpretations of the atoms, not be valuations corresponding to possible scenarios.
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