[The original version of this post misread the quoted passage from Nick Smith’s book — see his comment! I’ve edited so that his comment still fits, but so that someone browsing through and not delving into comments doesn’t go away with the wrong impression!]
As a footnote to my last post, I want to consider a passage in Nick Smith’s Logic: The Laws of Truth.
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 just 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 just 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 possible situation in which which the premiss is true and the conclusion false. In these cases, there simply is no such possible situation.
So has Smith has 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? It turns out not so, contrary to my first impression — see his comment below. For when he talks about a counterexample as a “scenario” in which the premises are true and the conclusion is false, he is using “scenario” in a special sense which is equivalent to (combinatorial) valuation and not — as the incautious reader whose eye skipped (me!) might assume — to mean, erm, scenario, i.e. situation in a possible world.
My original post then misrepresented my namesake: we agree that 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 in a natural sense, though they are ”scenarios” in his official (I’d say, misleadingly labelled!) sense. Good! Still, what I wanted to stress in my previous posting here is that the point we do agree on — however expressed — needs to be made loud and clear. Apologies to Nick for explicitly suggesting he’d fumbled the point as opposed, perhaps, to (like others I could mention!) not flagging it in the most perspicuous way.
3 thoughts on “Valuations again”
Thanks for this. I hadn’t forgotten that crucial distinction! :) I am using the term ‘possible scenario’ in a special sense here — I am not using it to mean ‘possible world’. I say in n.1 attached to p.63:
Throughout this book I use “scenario” in a special sense: as an abbreviation for “way of making propositions true or false.”
and then on p.63 I say:
In the PL framework, an assignment of truth values to basic propositions determines the truth values of all compound propositions as well, via the truth tables for the connectives. So we can think of a possible scenario — a possible way of making propositions true or false — as an assignment of truth values to basic propositions.
So when you say in your post:
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.
I don’t agree, given my meaning of ‘possible scenarios’ here (on which a possible scenario just is a valuation) — but I agree entirely, if one reads ‘possible scenarios’ as ‘possible worlds’ (cf. my comment on your previous post).
Many thanks for this. I’ve edited my original post appropriately (I hope!!). Cheers!
It looks like footnote 25 on page 479 is an entry point to Smith’s discussion of the relevant issues; it points to a discussion in Chapter 11:
Re “the PL framework”, see page 63 and this on page 120: “Now part of the assumption of the PL framework is that all combinations of assignments to truth values to basic propositions are possible.”
I find the discussion in Chapter 11 too complicated to digest and summarise at this hour.