Peter Smith's Blog, page 74

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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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 approach 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, 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, which for PL wffs means truth-preservation 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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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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[Security, etc.]

[Just a temporary notice to say that over the next week I plan to install an SLL certificate so that this becomes a “secure” site served via https, which will stop some browsers telling you this isn’t a secure site. I’m also moving domain registrar and other stuff behind the scenes. There well may be unintended hiccups, outages, etc.!]

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The Epistemic Lightness of Truth

Using my hefty discount at the CUP shop, I bought a copy of The Epistemic Lightness of Truth: Deflationism and its Logic by Cezary Cieśliński when it came out at the very end of last year. I mentioned it briefly here, saying that first impressions were very good. I then read some more;  but, life being as it is, I got distracted, and I never returned to say more about what struck me as an excellent book — a must-read if you are tempted by/interested in a broadly deflationist approach to truth.

In fact, I’ve not been keeping up quite closely enough with the literature here to give a fully informed judgement of Cieśliński’s achievement without more homework than I have had time for. However, Leon Horsten is in as good a position as anyone to assess the state of play. And he has now written an extensive and detailed review for Notre Dame Philosophical Reviews. His summary judgement? “I cannot praise this book too highly. I predict that it will constitute indispensable reading for any researcher in the field (professional or postgraduate) for years to come.” So read the very helpful review. Order the book for your library. And let’s hope that CUP issue a more modestly priced paperback sooner rather than later.

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Singing on the river



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At Kettle’s Yard

Edge III, 2012, Antony Gormley

To cheer up after a depressing work day yesterday — one of those days you lose faith in the book you are writing — a cheering outing to the newly opened exhibition at Kettle’s Yard, with five works by Antony Gormley, an exhibition which makes wonderful use of the spaces in the new galleries there. Very striking and thought provoking.

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As when the dove



To distract you from Trump, Brexit, and other woes, the wonderful Lucy Crowe sings “As when the dove” from Handel’s Acis and Galatea. The whole new recording from the Early Opera Company under Christian Curnyn on Chandos is terrific.

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Thin Objects

Øystein Linnebo’s book Thin Objects: An Abstractionist Account is out from OUP. If you’ve been following his contributions to debates on neo-Fregean philosophy of mathematics and related issues over some fifteen years, you won’t be surprised by the general line; but you will be pleased to have the strands of thought brought together in a shortish and (at least relative to the topic) accessible book. If you are new to Linnebo’s brand of neo-neo-neo-Fregeanism, this is your chance to catch up!

I confess that, having read Frege at an impressionable age, I still rather want something broadly Fregean to be right. I want there to be mathematical objects OK, but for them to be “thin”, to use Linnebo’s word — which he cashes out (perhaps not entirely happily, I’d say) as “not making a substantial demand on the world”. Linnebo gets his thin objects, Frege-style — his Platonism on the cheap — by conjuring the objects into being by abstraction principles. But unlike Frege and the Hale/Wright neo-Fregeans, Linnebo insists that the defensible and harmless principles need to be predicative. And it is well known that predicative abstraction principles by themselves are too weak to be useful for the foundations of much mathematics. Linnebo’s distinctive response is to allow indefinite iteration of abstraction principles (what he calls “dynamic” abstraction). But allowing completed infinite iterations would seem to entangle us in some pretty robust commitments again; so Linnebo wants to regiment his ideas about dynamic abstraction by going modal.

Does this work? Is getting thinness at the cost of going modal a good trade? I’ll let you know (given world enough and time).

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IFL2 Chapters 18 to 20: Propositional truth trees

Back then to the excitements of the second edition of An Introduction to Formal Logic. I have been revising the revisions of the chapters on propositional trees. I’ve streamlined the presentation, and some material is relegated to an online Appendix (yet to come). So there are now three short chapters, just 26 pages plus 3 pages of Exercises (also yet to come), as opposed to four chapters and 39 pages in the first edition. I do hope the result is still a very clear introduction to the truth-tree method.

All comments and/or corrections (either here or to the email address in the watermarked header) are as always most welcome.

[Added: I should say that tree-rules for biconditionals and examples with biconditionals are a topic for the planned end-of-chapter Exercises.]

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