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by Corry Shores
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[Priest, Introduction to Non-Classical Logic, entry directory]
[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]
Summary of
Graham Priest
An Introduction to Non-Classical Logic: From If to Is
Part I:
Propositional Logic
4.
Non-Normal Modal Logics; Strict Conditionals
4.4
The Properties of Non-Normal Logics
Brief summary:
(4.4.1) K interpretations are special cases of N interpretations, where all the worlds are normal (W = N). This means that K is an extension of N, because “if truth is preserved at all worlds of all N-interpretations, it is preserved at all worlds of all K-interpretations” (67). (4.4.2) The extensions of K are also extensions of their respective N extensions, such that for example “Kστ is an extension of Nστ and so on” (67). (4.4.3) Each K-logic is a proper extension of its corresponding N-logic. (4.4.4) Kρστ is the strongest logic we have seen so far. It is a normal logic, and every other normal logic we have seen is contained in it. Moreover, every non-normal system we have seen is contained in its corresponding normal system (with none yet being stronger than Kρστ and thus every non-normal system is weaker than Kρστ). In fact, “N is the weakest system we have met. It is contained in every non-normal system, and also in K, and so in every normal system” (68). (4.4.5) If we now instead define logical validity as truth preservation over all worlds, including non-normal ones, then certain formulas like □(A ∨ ¬A) will no longer be valid, because no necessary formulations are true in non-normal worlds. (4.4.6) The Rule of Necessitation is: for any normal system, ℒ, if ⊨ℒ A then ⊨ℒ □A. (4.4.7) The Rule of Necessitation fails in non-normal systems, because it will not work when applied doubly on the same formula. (4.4.8) On account of the failure of the Rule of Necessitation in non-normal systems, “Non-normal worlds are, thus, worlds where ‘logic is not guaranteed to hold’” (69).
[K Logics as Extensions of N Logics]
[K Extensions as Extensions of N Extensions]
[K-Logics as Proper Extensions of Their Corresponding N-Logics]
[Kρστ as the Strongest Logic Thus Far. N as the Weakest.]
[An Alternate Definition of Validity: Truth-Preservation Over All Worlds]
[The Rule of Necessitation]
[The Failure of the Rule of Necessitation in Non-Normal Systems]
[Logic as Not Guaranteed to Hold in Non-Normal Systems]
Summary
[K Logics as Extensions of N Logics]
[K interpretations are special cases of N interpretations, where all the worlds are normal (W = N). This means that K is an extension of N, because “if truth is preserved at all worlds of all N-interpretations, it is preserved at all worlds of all K-interpretations” (67).]
[Recall from section 2.1.2 that we call the most basic modal logic K. In section 3.2.2 we learned that K is the most basic “normal” modal logic. Then in section 4.2 Priest explained the semantics for non-normal modal logics N. In section 4.2.2 Priest defines non-normal modal logic interpretations in the following way:
A non-normal interpretation of a modal propositional language is a structure, ⟨W, N, R, v⟩, where W, R and v are as in previous chapters, and N ⊆ W. Worlds in N are called normal. Worlds in W−N (the worlds that are not normal) are called non-normal.
(p.64, section 4.2.2)
In section 4.3 we learned how to make tableaux for non-normal modal logics. Priest now says that K interpretations are special cases of N interpretations, namely, ones where all the worlds in question are normal. I still struggle with this notion of extensions. But for some reason, it seems (but check the quotation as I am not sure), all inferences and formulas that are valid in N interpretations are also valid in K interpretations. Thus K logic is an extension of N logic. (I am sorry that I am not sure how to understand why this is so. As far as I can tell, the reason might have something to do with the fact that even when there are non-normal worlds, validity is defined in terms solely with regard to truth preservation in normal worlds.
Logical validity is defined in terms of truth preservation at normal worlds, thus:
∑ ⊨ A iff for all interpretations ⟨W, N, R, v⟩ and all w ∈ N: if vw(B) = 1 for all B ∈ ∑ then vw(A) = 1.
⊨ A iff φ ⊨ A, i.e., iff for all ⟨W, N, R, v⟩ and all w ∈ N, vw(A) = 1.
(p.65, section 4.2.5)
So maybe that is why when there are non-normal worlds that will not add any new valid inferences. But I am guessing. Also see the discussion in section 4.1.1 on strong and weak and extensions.)]
A K-interpretation is simply a special case of an N-interpretation, namely, one where W = N. Hence, if truth is preserved at all worlds of all N-interpretations, it is preserved at all worlds of all K-interpretations. Hence, the logic K is an extension of N. (Another way of seeing this is to note that any tableau that closes under the rules for N must also close under the rules for K.)
(67)
[K Extensions as Extensions of N Extensions]
[The extensions of K are also extensions of their respective N extensions, such that for example “Kστ is an extension of Nστ and so on” (67).]
[In section 4.2.6 we discussed extensions of K like Kστ and their parallel N extensions like Nστ. Priest now expands the prior point and says that the extension relation between K and N holds also for each extension within K and N, such that Kστ is an extension of Nστ and so on.]
The same is true for the corresponding extensions of K and N: Kρ and Nρ, Kρτ and Nρτ, etc.
(67)
[K-Logics as Proper Extensions of Their Corresponding N-Logics]
[Each K-logic is a proper extension of its corresponding N-logic.]
[I will not summarize the following properly, so it is best simply to skip to the quotation below. I will for now make the following guess-comments. We are dealing with extensions of logics, which as I noted above is something I have little grasp of. But we discuss them in the comments to section 4.2.6, where in bracketed and disclaimed commentary I write: “The idea as I understand it so far (and I do not quite grasp it fully yet) is that to be an extension means that a logic has all the same valid inferences as the one it is extending, and probably more in addition to that. See the discussion in section 3.2.8 and the examples from sections 3.3.3, 3.3.4, and 3.3.5”. I am now guessing quite, quite wildly that a logic that is a proper extension of another is one that has all the same valid inferences and certainly more in addition to those. Priest says that each such K-logic (I think the extension mentioned above in section 4.4.2) are proper extensions of their corresponding N-logics. He says we can see that with a particular example formula from section 4.3.4 that we proved to be invalid in N. Priest says that it is valid however in K. Thus K is a proper extension of N, because it has all of N’s valid inferences, and at least this one over and beyond that set. I understand very little at all of the final point. The main idea is no restrictions on the R relation, the intuitive sense of which are notated in the tableaux as r, will make the formula valid. The reasoning here, which I have not figured out, is that no additional such r rule can make world 1 □-inhabited. (Recall from section 4.3.1 that “If world i occurs on a branch of a tableau, call it □-inhabited if there is some node of the form □B,i on the branch.”) Thus on that account the tableau must remain open and for that reason, I am guessing, a counter-example is always possible.]
But each K-logic is a proper extension of the corresponding N-logic. It is easy enough to check that ⊢K □(p ⊃ □(q ⊃ q)) (and a fortiori any of K’s extensions), but as the tableau of 4.3.4 shows, it is not valid in N. Moreover, adding any of the rules for r to this tableau does not close it, either. None of the rules makes world 1 □-inhabited; hence, it remains open. Hence, this inference is not valid in any of the non-normal extensions of N either.
(67)
[Kρστ as the Strongest Logic Thus Far. N as the Weakest.]
[Kρστ is the strongest logic we have seen so far. It is a normal logic, and every other normal logic we have seen is contained in it. Moreover, every non-normal system we have seen is contained in its corresponding normal system (with none yet being stronger than Kρστ and thus every non-normal system is weaker than Kρστ). In fact, “N is the weakest system we have met. It is contained in every non-normal system, and also in K, and so in every normal system” (68).]
Priest now notes a number of facts that I think need no summarization. [I would only note that we discussed the notion of a logic being stronger or weaker in section 4.1.1. There I made the following bracketed and disclaimed guess-comment: “To be a weaker system means that it has fewer valid inferences. Non-normal modal logics are weaker than normal logics, thus they would have fewer valid inferences.” Again, I am not in the least sure about these things, but that might be consistent with what was said above in section 4.4.3 where Priest explains that every K-logic is a proper extension of its corresponding N-logic, with a demonstration of that being a formula which is valid in K but not in N. Priest writes here in our current section: “N is the weakest system we have met. It is contained in every non-normal system, and also in K, and so in every normal system.”]
Note that Kρστ(Kυ) is the strongest of all the logics we have looked at: every normal system that we looked at is contained in Kρστ (3.2.9), and every non-normal system that we looked at is contained in the corresponding normal system (4.4.1, 4.4.2). N is the weakest system we have met. It is contained in every non-normal system, and also in K, and so in every normal system.
(68)
[An Alternate Definition of Validity: Truth-Preservation Over All Worlds]
[If we now instead define logical validity as truth preservation over all worlds, including non-normal ones, then certain formulas like □(A ∨ ¬A) will no longer be valid, because no necessary formulations are true in non-normal worlds.]
[Recall again from section 4.2.5 that we defined semantic validity for non-normal modal logics in the following way:
Logical validity is defined in terms of truth preservation at normal worlds, thus:
∑ ⊨ A iff for all interpretations ⟨W, N, R, v⟩ and all w ∈ N: if vw(B) = 1 for all B ∈ ∑ then vw(A) = 1.
⊨ A iff φ ⊨ A, i.e., iff for all ⟨W, N, R, v⟩ and all w ∈ N, vw(A) = 1.
(p.65, section 4.2.5, boldface mine)
I again will missummarize the following, but I will note the main ideas as best I can. We wonder now, suppose we have a logic with a non-normal semantics, but we instead change the definition of validity to truth preservation at all worlds, including non-normal ones. He then notes that for all normal worlds of any interpretation, ⊨ □(A ∨ ¬A). Thus it is valid under the normal rule of validity. But □(A ∨ ¬A) cannot be valid in non-normal worlds (by definition, see section 4.2.3). So if we change the definition of validity such that it is required that an inference preserve truth in all non-normal worlds too, that means □(A ∨ ¬A) would not be a valid logical truth.]
It might be wondered what happens if we define a logic with non-normal semantics, and validity defined in terms of truth preservation at all worlds (normal and non-normal). This gives a sub-logic of the corresponding non-normal logic. (If truth is preserved at all worlds of an interpretation, it is preserved at all normal worlds.) In fact, it is a proper sub-logic. In any non-normal modal logic, for example, ⊨ □(A ∨ ¬A). But since □(A ∨ ¬A) is not true at non-normal worlds, □(A ∨ ¬A) is not valid if logical truth is defined with reference to all worlds. Hence, this definition can be used to create logics weaker than N.2
(68)
2.The logics which are the same as S2 and S3, except that validity is defined in terms of truth preservation at all worlds, are sometimes called E2 and E3.
(68)
[The Rule of Necessitation]
[The Rule of Necessitation is: for any normal system, ℒ, if ⊨ℒ A then ⊨ℒ □A.]
[The next idea seems to be the following, but I am guessing. So it is best to skip to the quotation. It seems we are returning to the normal definition of validity as truth preservation over all normal-worlds, putting aside non-normal ones. We then note that if a formula is valid in some normal system, that means, it is true for all worlds of all interpretations of that system (all worlds in a normal system of course are normal worlds). Now recall from section 2.3.5 that:
vw(□A) = 1 if, for all w′ ∈ W such that wRw′, vw′(A) = 1; and 0 otherwise.
(p.22, section 2.3.5)
So if a formula is valid in a normal system and is thus true for all worlds of all interpretations of that system, that means it must be true for all worlds associated with some world. That fulfills the definition for necessity, and so the necessity formulation of that formula must be valid in that normal system. This is called the Rule of Necessitation, which is formulated as: “For any normal system, ℒ, if ⊨ℒ A then ⊨ℒ □A.” Let me quote, as I probably have that wrong:]
Let us finally, now, return to the question of the meaning of non-normal worlds. For any normal system, ℒ, if ⊨ℒ A then ⊨ℒ □A. (This is sometimes called the Rule of Necessitation.) For if ⊨ℒ A then A is true at all worlds of all ℒ-interpretations. Hence, if w is any such world, A is true at all worlds accessible from w. Hence, □A is true at w. Thus, ⊨ℒ □A.
(68)
[The Failure of the Rule of Necessitation in Non-Normal Systems]
[The Rule of Necessitation fails in non-normal systems, because it will not work when applied doubly on the same formula.]
[Again it is best to skip to the quotation, but I will give my best guess commentary. We will now see why the Rule of Necessitation fails in every non-normal logic. The point we seem to arrive at is that certain formulas can hold at all worlds including non-normal ones, thus these formulas with the necessity operator will hold for all normal worlds and therefore their necessity formulation is valid. But since necessity formulas do not hold for non-normal worlds, we cannot say that the formula now with two necessity operators holds in normal worlds that access the non-normal ones. This would be problematic, because the rule of necessitation says the following: if ⊨ℒ A then ⊨ℒ □A. We have already said that the formula with a singular necessity operator in the normal world is valid. The rule says that therefore the formula now with two necessity operators would be valid too. But that cannot be, because it assumes that the formula with just one necessity operator holds in all worlds accessible to the normal one in question, which is not so in this interpretation. I quote:]
The Rule of Necessitation fails in every non-normal logic, ℒ, however. Consider, for example, A∨¬A. This holds at all worlds, normal or non-normal. Hence, □(A∨¬A) holds at all normal worlds, i.e., ⊨ℒ □(A∨¬A). But at any non-normal world, □(A∨¬A) is false. Now consider an interpretation where there is a normal world that accesses such a world. Then □□(A∨¬A) is false at that world. So, ⊭ℒ □□(A∨¬A).3
(68)
3.Similarly, the principle that if A ⊨ B then □A ⊨ □B, which holds in all normal logics, as we saw in 3.6.10, also fails in non-normal logics. If □A is true at a normal world of an interpretation, it follows that A is true at all worlds in it; but it does not follow from this and A ⊨ B that B is true at all the worlds in it – only that it is true at all normal worlds in it.
(68)
[Logic as Not Guaranteed to Hold in Non-Normal Systems]
[On account of the failure of the Rule of Necessitation in non-normal systems, “Non-normal worlds are, thus, worlds where ‘logic is not guaranteed to hold’” (69).]
[Because certain logical truths can fail to hold at non-normal worlds (as for example □(A∨¬A) holding in an interpretation’s normal worlds but not in its non-normal ones, as we saw in section 4.4.7 above), that means the Rule of Necessitation fails in non-normal systems. Priest says that this perhaps is “the most distinctive feature of non-normal systems.” (I am not sure why, but it seems to have to do with the philosophical import of what it implies, namely, that) this means that “Non-normal worlds are, thus, worlds where ‘logic is not guaranteed to hold’” (69). We return to this point later. (I am not sure exactly what it means for logic itself not hold in non-normal worlds, so we should reserve speculation until later. But I wonder if it means that the normal laws of logic may not hold, maybe in some cases none of them. We will see later.)]
The failure of the Rule of Necessitation is, perhaps, the most distinctive feature of non-normal systems. And it fails, as we have just seen, because | logical truths may fail to hold at non-normal worlds. Non-normal worlds are, thus, worlds where ‘logic is not guaranteed to hold’. We come back to this insight in a later chapter.
(68-69, boldface mine)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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