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by Corry Shores
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[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
4.2
Non-Normal Worlds
Brief summary:
(4.2.1) We will first examine the technical elements of non-normality. (4.2.2) Our interpretations of non-normal modal logics take the structure ⟨W, N, R, v⟩. W is the set of worlds. R is the accessibility relation. v is the valuation function. And N is the set of normal worlds, with all the remaining worlds in W being non-normal ones. (4.2.3) The semantics are the same for non-normal worlds, except that at non-normal worlds, all necessary propositions (those starting with □) are always false, and all possible propositions (those starting with ◊) are always true. For, in non-normal worlds, nothing is necessary and all is possible. (4.2.4) At every world, including non-normal ones, ¬□A and ◊¬A have the same truth value. ¬◊A and □¬A do too. (4.2.5) Inferences are valid only if they preserve truth in all interpretations at all normal worlds. (4.2.6) Non-normal modal logics with the structure ⟨W, N, R, v⟩ in which R is a binary relation on W are called N, with such R constraints as ρ, σ, τ etc. creating extensions of N like Nρ, etc. (So here we have N for non-normal modal logics where we previously had K and its extensions for normal modal logics.) And, “As for normal logics, Nρτ is an extension of Nρ, which is an extension of N, etc.” (4.2.7) Nρ = S2; Nρτ = S3; and Nρστ = S3.5, with the first two S’s being “Lewis systems” and the last one being a “non-Lewis system”. (4.2.8) Although non-normal worlds originally were fashioned solely for technical reasons, in fact they have a philosophical meaning too.
[Preview: Technicalities]
[The Components of Non-Normal Interpretations]
[The Universal Possibilities and the Absence of Necessities in Non-Normal Worlds]
[Equivalences Holding in Non-Normal Worlds Too]
[Validity]
[N Non-Normal Modal Logic and Its Extensions]
[N Logics in Terms of Lewis Systems]
[Origins and Philosophical Potential of Non-Normal Logics]
Summary
[Preview: Technicalities]
[We will first examine the technical elements of non-normality.]
We will now first look “at the technicalities concerning non-normality,” and we later gradually will “discuss what they mean” (64).
[The Components of Non-Normal Interpretations]
[Our interpretations of non-normal modal logics take the structure ⟨W, N, R, v⟩. W is the set of worlds. R is the accessibility relation. v is the valuation function. And N is the set of normal worlds, with all the remaining worlds in W being non-normal ones.]
[Recall the semantics for modal logic that we saw in section 2.3. The following comes from our brief summary there:
In our modal semantics, we add to our propositional language two modal operators, □ for ‘necessarily the case that’ and ◊ for ‘possibly the case that’. An interpretation in our modal semantics takes the form ⟨W, R, v⟩, with W as the set of worlds, R as the accessibility relation, and v as the valuation function. ‘uRv’ can be understood as either, “world v is accessible from u,” “in relation to u, situation v is possible,” or “world u access world v.” Negation, conjunction, and disjunction are evaluated (assigned 0 or 1) just as in classical propositional logic, except here we must specify in which world the valuation holds.
vw(¬A) = 1 if vw(A) = 0, and 0 otherwise.
vw(A ∧ B) = 1 if vw(A) = vw(B) = 1, and 0 otherwise.
vw(A ∨ B) = 1 if vw(A) = 1 or vw(B) = 1, and 0 otherwise.
(p.21, section 2.3)
A formula is possibly true in one world if it is also true in another world that is possible in relation to the first. A formula is necessarily true in a world if it is also true in all worlds that are possible in relation to it.
For any world w ∈ W:
vw(◊A) = 1 if, for some w′ ∈ W such that wRw′, vw′(A) = 1; and 0 otherwise.
vw(□A) = 1 if, for all w′ ∈ W such that wRw′, vw′(A) = 1; and 0 otherwise.
(p.22, section 2.3)
In non-normal logics, most of this stays the same, with the following exceptions. We designate a subset of worlds that are normal, meaning that all the above semantics holds for them, and all the rest of the worlds are non-normal, meaning that for them all necessity formulations are false and all possibility formulations are true. In other words, in non-normal worlds, nothing is necessarily the case, and all things are possible. (For the set notation towards the end of the quote below, see sections 0.1.6 and 0.1.8.)]
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.
(64)
[The Universal Possibilities and the Absence of Necessities in Non-Normal Worlds]
[The semantics are the same for non-normal worlds, except that at non-normal worlds, all necessary propositions (those starting with □) are always false, and all possible propositions (those starting with ◊) are always true. For, in non-normal worlds, nothing is necessary and all is possible.]
[In the above section 4.2.2, we mentioned the semantics for the truth functions. Priest says they are the same I think for both normal and non-normal worlds. The truth conditions for necessity and possibility, also given above in section 4.2.2, hold still for normal worlds, but we have different rules for non-normal worlds. If something is necessary, then it is assigned the value false, and if something is possible, it is assigned the value true. This carries the sense that in non-normal worlds, nothing is necessary, and everything is possible.]
The truth conditions for the truth functions, ∧,∨,¬, etc. are the same as before (2.3.4). The truth conditions for □ and ◊ at normal worlds are also as before (2.3.5). But if w is non-normal:
vw(□A) = 0
vw(◊A) = 1
In a sense, at non-normal worlds, everything is possible, and nothing is necessary.
(64, boldface mine)
[Equivalences Holding in Non-Normal Worlds Too]
[At every world, including non-normal ones, ¬□A and ◊¬A have the same truth value. ¬◊A and □¬A do too.]
[In section 2.3.9 Priest gives a proof for why ¬◊A at any (normal) world is equivalent to □¬A. I was not able to summarize the reasoning in that proof adequately. It seemed to me at the time that the idea was the following. If it is not that something is necessary, then that means that at least in one related world it is not true. In that world where it is not true, its negated form would be true (under classical assumptions about negation). So for the first world that we started with, in a related world the negation is true, thus the negation is possible in that first world. I am just guessing, sorry. Now his point is that this holds also for non-normal worlds. I am not entirely sure how and why. My best guess is that by assigning all necessities as 0 and all possibilities as 1 does not thereby change the rules governing the way the worlds are related. Thus we will find the same conditions that would cause these equivalence to remain. I quote:]
Note that at every world, w,¬□A and ◊¬A still have the same truth value, as do ¬◊A and □¬A. We saw this to be the case for normal worlds in 2.3.9 and 2.3.10. It is easy to see that this is also true if w is non-normal.
(65)
[Validity]
[Inferences are valid only if they preserve truth in all interpretations at all normal worlds.]
[Now we discuss validity. Recall from section 2.3.11 that an inference is valid (as a semantic consequence) if it is truth-preserving in all worlds of all interpretations, (that is, if in all worlds in all interpretations, whenever the premises are true, so too is the conclusion). And, a logical truth (or tautology) is a formula that is true in all worlds of all interpretations. Here is a quotation from that section, with the formal definition.
An inference is valid if it is truth-preserving at all worlds of all interpretations. Thus, if Σ is a set of formulas and A is a formula, then semantic consequence and logical truth are defined as follows:
Σ ⊨ A iff for all interpretations ⟨W, R, v⟩ and all w ∈ W: if vw(B) = 1 for all B ∈ Σ, then vw(A) = 1.⊨ A iff φ ⊨ A, i.e., for all interpretations ⟨W, R, v⟩ and all w ∈ W, vw(A) = 1.
(p.23, section 2.3.11)
In our non-normal modal logics, we have the added issue of non-normal worlds. Priest says that an inference or formula is valid so long as it preserves truth at all normal worlds (leaving open the possibility that it is not preserved in non-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.
(65)
[N Non-Normal Modal Logic and Its Extensions]
[Non-normal modal logics with the structure ⟨W, N, R, v⟩ in which R is a binary relation on W are called N, with such R constraints as ρ, σ, τ etc. creating extensions of N like Nρ, etc. (So here we have N for non-normal modal logics where we previously had K and its extensions for normal modal logics.) And, “As for normal logics, Nρτ is an extension of Nρ, which is an extension of N, etc.”]
[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, and in section 3.2.3 Priest explains how we may fashion extensions of K by placing constraints on the R relation, like reflexivity, etc.:
Other normal modal logics are obtained by defining validity in terms of truth preservation in some special class of interpretations. Typically, the special class of interpretations is one containing all and only those interpretations whose accessibility relation, R, satisfies some constraint or other. Some important constraints are as follows:
ρ (rho), reflexivity: for all w, wRw.
σ (sigma), symmetry: for all w1, w2, if w1Rw2, then w2Rw1.
τ (tau), transitivity: for all w1, w2, w3, if w1Rw2 and w2Rw3, then w1Rw3.
η (eta), extendability: for all w1, there is a w2 such that w1Rw2.
(p.36, section 3.2.3)
A normal modal logic K under the symmetry and transitivity constraints, for example, will be called Kστ. Priest is saying now that instead of K, when R is a binary relation on W, then our logic is now called N, with the R constraints producing such extensions of N as Nστ, for example. My guess is that for the construction to be an N, the R relation and its constraints would need to apply to all worlds, normal and non-normal. The last notion we should note is this idea of being an extension, such that “Nρτ is an extension of Nρ, which is an extension of N”. 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.]
If the accessibility relation, R, may be any binary relation on W, the logic this construction gives will be called N.1 As with normal modal logics, additional logics can be formed by placing constraints on R, such as reflexivity, transitivity, symmetry, etc. (as in 3.2). In fact, of course, how R behaves at non-normal worlds is irrelevant, since this plays no role in determining truth values. We use Nρ to refer to the non-normal logic determined by the class of all interpretations where R is reflexive; Nστ, to refer to the non-normal logic determined by the class of all interpretations where R is symmetric and transitive, and so on. As for normal logics, Nρτ is an extension of Nρ, which is an extension of N, etc.
(65)
1. The name is not standard, but is sensible enough. Note that N is also used for the normal worlds in an interpretation. Context, however, will disambiguate. (65)
[N Logics in Terms of Lewis Systems]
[Nρ = S2; Nρτ = S3; and Nρστ = S3.5, with the first two S’s being “Lewis systems” and the last one being a “non-Lewis system”.]
[In section 3.2.5, Priest noted that Kρτ = S4 and Kρστ = S5. Now Priest says that Nρ = S2 and Nρτ = S3, with both S’s being “Lewis systems”, and Nρστ = S3.5, with that being a “non-Lewis system.” I am not familiar with the idea of a “Lewis system”, so I will have to quote for now.]
Historically, Nρ and Nρτ are the Lewis systems S2 and S3 respectively. Nρστ is the non-Lewis system S3.5.
(65)
[Origins and Philosophical Potential of Non-Normal Logics]
[Although non-normal worlds originally were fashioned solely for technical reasons, in fact they have a philosophical meaning too.]
[Priest now gives some interesting and philosophically important historical information about non-normal logics. For these ideas we need the notions of stronger and weaker, which I have not quite grasped yet. We discussed these ideas in section 4.1.1, where my guess was that the extensions are stronger because they have more valid formulas. But I really do not know if that is so. I will quote now, as I cannot summarize any better than what is written:]
Non-normal worlds were originally invented purely as a technical device to give a possible-world semantics for the Lewis systems weaker than S4. As we shall see in due course, though, they have a perfectly good philosophical meaning. For the record, Lewis thought that the correct system of modal logic for logical necessity was S2.
(65)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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