# 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 ∧ (QR))’. 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. [oops, see comments!]) 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.

Looking ahead, we define the tautological validity of an inference in an interpreted PL language 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  inferences in a PL language, 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” (heavens, what was it??).  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.)

### 15 thoughts on “Valuations, combinatorial vs ‘realizable’”

1. Hi Peter,

I discuss this distinction in ch.11 of my *Logic: The Laws of Truth* (PUP 2012), in particular in section 11.5, where I distinguish ‘ww-models’ as a subset of all models. Where a ‘ww’ is a ‘way the world could be’ the basic idea is this (from p.258):

“Given a glossary G for some fragment of the logical language, some models of the fragment can be generated from the intensions assigned by that glossary, together with some ww. These are the ww-models (of the fragment under G). Other models cannot be generated in this way: there is no ww that, together with the intensions assigned by G, generates such a model.”

An argument is then NTP (necessarily truth preserving) iff there is no *ww-model* in which the premises are true and the conclusion false, while an argument is valid (as I would say, or tautologically valid, as you would say — i.e. NTP in virtue of form) iff there is no *model* in which the premises are true and the conclusion false.

Best, Nick

PS In this post you are talking about propositional logic, whereas my discussion is framed in terms of predicate logic, but the essential points go through in either setting.

1. Spot on! Though I don’t think I want to buy the “ww-model” terminology (or “ww-valuation for the propositional logic case). For I suspect that “way the world could be” can be misleading for the idea that’s required here, as “way the world could be” could too easily be misread as “way this world could be (could have been)”, and that seems narrower than the all-permissive notion of a possible world.

2. Wilfrid Hodges makes this point in his elementary text ‘Logic’ though he makes it in the context of tableaux rather than truth tables. Tableaux are first introduced (section 10) using interpreted English sentences, then (section 20) with a mixture of English sentences and symbols for truth functional connectives. Only then (section 21) are sentence letters, P, Q, R introduced and they are introduced merely as abbreviations for the interpreted English sentences. Hodges then says (section 21, p. 116 in first edition, p. 92 in second edition):

“If a set of sentences has been completely symbolized, we can test it for consistency by means of sentence tableaux, as in section 20; but there is one difference. Some types of inconsistency become hidden from view when sentences are all symbolized. For example the two sentences:

Schubert died at the age of thirty-one
Schubert died at the age of sixty-eight

are inconsistent. But symbolized, they would look something like this:

P. Q.

There is nothing to indicate that this is inconsistent.”

He then notes that a tableau may have open branches (not containing a formula ‘A’ and its negation ‘~A’) which are nonetheless inconsistent: open branches have to be inspected and the English interpretation referrred to in order to check that the branch is really consistent. He gives an example (pp. 117-18/92-94) of a tableau with some open branches which are not consistent (the inconsistent pair being ‘Only a green colour appears’ and ‘a brown colour appears’) but where there is also an open branch that is consistent.

As far as I can tell, Hodges does not distinguish types of validity. He defines an argument (an argument in an interpreted natural language like English) to be valid if its counter-example set is inconsistent, where ‘inconsistent’ would include the kind of inconsistencies illustrated by the Schubert example. So this corresponds I think to your ‘deductive validity’. He does note (p. 116/92) that if all the branches on a tableau are closed then the original set of sentences is ‘certainly inconsistent’ and there is no need to hunt for ‘hidden’ inconsistencies. But he does not, at this point, introduce a special term (like your ‘tautological validity’) for this kind of inconsistency.

Later in the book (sections 22-25) Hodges introduces a purely formal uninterpreted propositional language, truth tables and what he calls ‘formal’ tableau. But now instead of referring to ‘arguments’ he refers to ‘sequents’ – a set X of formulas of the formal language separted by a turnstile |= from another formula, A) and refers not to the ‘validity’ of such things but to their ‘correctness’: a sequent X |= A is ‘correct’ if every possible assignment of truth values (respecting the truth tables for the connectives) that makes all the formulas in X true, also makes A true (p. 128/102). In that case he says that X ‘semantically entails’ A.

Finally (section 25) ‘formal tableaux’ are introduced as a purely formal syntactic method for proving that sequents are correct. Sequents now have a different turnstile symbol; X |- A is defined to be a ‘correct syntactic seqeuent’ if and only if the formal tableaua for X, ~A is completely closed; in that case X ‘syntactically entails’ A. At this point everything is completely formal, syntactic and uninterpreted. Hodges finishes this part of the book by sketching a soundness and completeness result for ‘syntactic’ and ‘semantic’ entailment and makes some comments on the point of this process of ‘formalization’ as he calls it.

1. Many thanks for this — another reminder how many and varied are the textbook approaches to what is going on in elementary logic! I’ll have to take a look to remind myself how Hodges thinks of ‘formalization’.

2. While it makes sense from a model-theoretic POV to use ‘semantically entails’ in the way Hodges does, it seems odd when looked at from a greater distance, because there’s less semantics involved when using an uninterpreted language than when using an interpreted one which also involves the semantics of English sentences.

So it seems an odd choice of terminology for an elementary, introductory book.

1. I rather agree! (But then, the use by logicians of “semantic” for what is at most in some sense a model of semantics is all too common!!)

3. I know that you did not particularly like my draft textbook http://rocket.csusb.edu/~troy/SLmain.pdf ! But I do treat the issue in sections 5.1 and 9.1. I say that a “glossary” as in your comment is a function resulting in an “intended interpretation” corresponding to any possible world. The intended interpretations are a subset of all the interpretations.

1. Thanks for the reference! Yes, yours is indeed a version of the point I want.

Though I have to say I don’t like the terminology in which you make it. The habit of calling valuations “interpretations” goes along, in many hands, with not thinking of the wffs of PL languages as having interpretations, assignments of Fregean senses, i.e. (as I would say) as not thinking of PL languages as genuine languages in which you can say things. Keeping the interpretations = valuations terminology then deprives one of a natural way of speaking if one also thinks of wffs as having senses which yield to worlds-to-valuations functions (senses which are preserved in translation). A judgement call, yes, but I find it much more natural to distinguish interpretations (assigning meanings that get preserved in translations) from valuations.

4. Where does the idea that “plain deductive validity is a matter of truth-preservation on all realizable valuations” — that it’s about only realizable valuations — appear in IFL1? I’m not sure I’ve seen it in any book ever, but if it’s in IFL1, I must have seen it at some point in the past.

Indeed, IFL1 seems to deny that idea on p 10. When discussing an example about jumping from a 20th floor window, p 9 says “let’s grant that there is no situation that can really obtain in the actual world … in which the premises would be true and the conclusion false” and asks “does that make the inference … deductively valid?” Page 10 answers “No.”

1. Many thanks for the comment. Which brings out that I expressed myself embarrassingly badly by using the phrase “there is no possible way the world might go” in the post. I’ve now re-edited. And I need to check that I didn’t fall into using that very dangerously misleading phrase in the relevant draft chapters. I hope that what I meant is much clearer now. It’s the point of IFL1, §11.4.

1. I think you are (understandably) focusing on part of the blog post after “OK, having set the scene”, while I was wondering primarily about the part right before that: the idea of a “plain deductive validity” and “validity for PL inferences” that isn’t about all combinatorially possible valuations but only about ones possible in some other sense. (That’s what I wasn’t sure I’d seen any book.)

IFL1’s §11.4 is reasonable, but it doesn’t look like it discusses that idea of plain deductive validity. However, it seems the idea is considered in §13.2 which also refers to “our informal idea of logical entailment” in §2.1 (which is where I found the 20th floor window example).

The terminology varies — examples include an “informal idea of logical entailment”, “deductively valid”, “classically valid”, and (in the blog post) “plain deductive validity” and “validity for PL inferences” — and I have to wonder how closely the informal notion matches the more technical ones and the explanations in terms of possible worlds.

Also, what exactly is the relevant notion of possibility? IFL1, §2.1 says “logically possible”, but the examples in the blog post don’t look like straightforward examples of logical possibility / necessity to me.

If I remember Kripke correctly, “water is H2O” was an example of metaphysical, rather then logical necessity; and it’s not clear what status married vs single actually has. Not all that long ago, it may have seemed that “Jill isn’t married” and “Jill is single” would have to the same truth value, but they don’t if Jill is in a civil partnership.

Many things is such areas are in flux, making them rather problematic in examples of possibility and necessity. For example, it used to be that if one partner in a marriage was male, the other would have to be female. In some places, it still is. It used to be if someone was a man, then they couldn’t be pregnant. (That’s now at least contested.)

(a) I shouldn’t have used the “married” vs “single” example in 2018! ( I better check that isn’t in the book!!!) OK replace with “Jill has a sibling” and “Jill is an only child”.

(b) The arm-waving discussion in the early chapters of IFL1 on the informal notion of validity is one of the bits that badly needs revision. And it has got it in IFL2. I was about to put the resulting revised chapters online for comments (and will do after another read through).

But surely the basic thought is a more familiar one that you are allowing. There is an informal notion of necessary preservation of truth — necessarily, if the premisses are true, so is the conclusion (where, yes, the notion of necessity needs explicating). And there there are notions of necessary-preservation-of-truth-in-virtue-of-logical-form, of which e.g. tautological validity is one. Terminology varies — some (me in IFL2) would call the first notion validity, and the second logical validity, and that’s one pretty standard idiom. Others, e.g. Nick Smith in his intro text, reserves “validity” for necessary-preservation-of-truth-in-virtue-of-logical-form. But we agree that there is a more general and a narrower notion here — and an inference can exhibit necessary-preservation-of-truth without exhibiting necessary-preservation-of-truth-in-virtue-of-logical-form. And that will apply whenever we are dealing with interpreted languages, whether natural or artificial — including interpreted PL languages.

1. No, I’m happy with there being an informal notion of necessary preservation of truth and also narrower notions of NPT in virtue of form. I’m even happy with the idea that some combinatorially possible assignments of truth-values might not correspond to possible worlds.

But at some points you seem to be saying there are two forms of validity or inference in propositional logic, one (tautological validity) that considers all lines in a truth table and another (deductive validity) that considers only the lines that correspond to possible worlds (in some sense of “possible” that’s different from combinatorially possible assignments of truth-values). An example is where you say “tautological validity implies validity for PL inferences, but not vice versa”.

Perhaps I’ve forgotten something, but I think my understanding of propositional logic has always been that, when doing propositional logic, you don’t get to look inside bits of English (or other natural language) that are connected by the connectives. Instead, they have to be treated atomically. If you look inside and reason based on what those bits say, you’re not doing propositional logic any more. For example, if you have

(P & Q) -> R

P: All fish can swim.
Q: Nemo is a fish.
R: Nemo can swim.

and you look at that and reason that (P & Q) -> R is necessarily true, you’re doing something like first-order logic, not propositional. But you seem to be saying that would be a “PL inference”.

A related issue arises when you say “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”. If the meaning of a wff includes the natural-language meanings of the bits of English in the glossary, then it’s no longer a formal language: it’s a hybrid.

Similarly, what Hodges is doing with tableaux, where the English interpretation is referred to when checking whether a branch is consistent, isn’t PL. It’s a hybrid of PL trees with other reasoning based on the English meanings.

1. I’d rather say “A valid inference in an interpreted PL language” than baldly a “PL inference” (and I’ve edited for clarity). But that’s not getting to the nub of the matter, the real divergence between us. That turns I think on whether we want elementary logic to be dealing in formalized languages (as in Aristotle or Frege), or in formal languages in the sense, yes, of many a modern textbook. I’ll try to write a piece on this: no prizes for guessing which way I fall.

2. I don’t think our divergence turns on whether elementary logic should deal in interpreted languages. I think it should deal in interpreted languages at least some of the time. However, I agree with what you seem to be saying about PL in §11.4 of IFL1, ‘Possible valuations’:

For our target in developing this language is to give an account of the logic of arguments whose relevant logical materials are just the connectives ‘and’, ‘or’ and ‘not’, i.e. arguments whose validity does not rely on the meaning of their atomic propositional constituents.

If your reasoning uses the meanings of the English sentences that correspond to the atomic propositions, then you’re not doing propositional logic any more. And if the relevant logical materials go beyond the connectives — as they do in the all-fish/Nemo argument — then the argument isn’t suitable for handling using propositional logic.

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