by Corry Shores

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[Logic and Semantics, entry directory]

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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

9.

Logics with Gaps, Gluts and Worlds

9.7

Impossible Worlds and Relevant Logic

Brief summary:

(9.7.1) We will now discuss philosophical matters regarding *K*_{4} , *N*_{4} , *K*_{∗} , and *N*_{∗}*. *(9.7.2) We will now call non-normal worlds “logically impossible worlds,” because they are worlds where the laws of logic are different*. *(9.7.3) Just as there is no problem in conceiving physically impossible worlds, there should likewise be no problem in conceiving logically impossible worlds*. *(9.7.4) We already seem to suppose such logically impossible worlds when we note how certain laws of logic fail in particular non-classical logics, as for example when we say: “if intuitionist logic were correct, the law of double negation would fail*.” *(9.7.5) Objections to logically impossible worlds do not work*. *For, we cannot simply require that the laws of logic admit of no variation, when in fact that is what we are successfully and fruitfully modelling. (9.7.6) In a logically impossible world, it could still be that no normal laws of logic be broken, just like how in a physically impossible world, normally-impossible physical events can take place, but for contingent reasons happen not to*. *(9.7.7) Logically impossible worlds can also in fact be ones where laws of logic indeed are broken*. *(9.7.8) Relevant propositional logics are ones where whenever “*A *→* B* is logically valid, *A *and* B* have a propositional parameter in common” (172)*. *(9.7.9) But *N*_{4} is a relevant logic, on account of how conditionals are evaluated in normal worlds (they depend on the values in non-normal worlds) in combination with the arbitrarity of their value assignments in non-normal worlds*. *(9.7.10) In a similar way, *N*_{∗} is also a relevant logic*. *(9.7.11) Relevant logics tend to our intuitions that there should be relevance between antecedent and consequent of conditionals, and this can be done by requiring them to share parameters*. *(9.7.12) There is another sort of relevant logic that is of a whole different class, called filter logics, in which “a conditional is taken to be valid iff it is classically valid *and* satisfies some extra constraint, for example that antecedent and consequent share a parameter” (173)*. *(9.7.13) Relevance in our systems here however is not conditions added on top of classical validity*. *(9.7.14) If we wanted to keep this system but reserve a real world where truth operates in a more conventional way, then we can designate an @ actual world that has certain constraints. For example, we could add exhaustion and exclusion constraints to eliminate truth gaps and gluts in the actual real world @*.*

[Moving to a Discussion on *K*_{4} , *N*_{4} , *K*_{∗} , and *N*_{∗}]

[Logically Impossible Worlds]

[Logically Impossible Worlds as Admissible]

[Our Seeming Assumption of Logically Impossible Worlds]

[Failure of Objections to Impossible Worlds]

[The Non-Necessity for Logical Laws To Be Broken in Logically Impossible Worlds]

[Logically Impossible Worlds Where Normal Laws of Logic Are In Indeed Broken]

[Relevant Logics and the Conditional]

[*N*_{4} as Relevant]

[*N*_{∗} as Relevant]

[Relevant Logics Meet Our Intuitions About Conditionals and Relevance]

[Filter Logics]

[More Than Classical Relevance]

[Preserving Conventional Truth in This System]

Summary

[Moving to a Discussion on *K*_{4} , *N*_{4} , *K*_{∗} , and *N*_{∗}]

[We will now discuss philosophical matters regarding *K*_{4} , *N*_{4} , *K*_{∗} , and *N*_{∗}*.*]

[Recall the semantics and tableau constructions from previous sections: *K*_{4} (9.2 and 9.3); *N*_{4} (9.4 and 9.5) and *K*_{∗} and *N*_{∗} (9.6). Now Priest will discuss philosophical matters regarding these constructions.]

We are now in a position to make some comments on the import of the previous constructions.

(171)

[Logically Impossible Worlds]

[We will now call non-normal worlds “logically impossible worlds,” because they are worlds where the laws of logic are different*.*]

[Recall from section 4.2.3 that for modal logics, non-normal worlds are ones where nothing is necessary and all is possible; for, at non-normal worlds, all necessary propositions (those starting with □) are always false, and all possible propositions (those starting with ◊) are always true. In section 9.2, we discussed a possible worlds *First Degree Entailment *(and thus four value-situationed) system called *K*_{4}. In section 9.4, we noted that *K*_{4} has the following problematic valid formula: ⊨ *p *→ (*q* → *q*). This was problematic, because we want to be able to say things about what would follow if certain laws of logic were suspended. So the following would be valid, even though we would want it to be invalid: “if every instance of the law of identity failed, then, if cows were black, cows would be black. If every instance of the law failed, then it would precisely not be the case that if cows were black, they would be black” (p.167, section 9.4.3 ). But “we need to countenance worlds where the laws of logic are different, and so where laws of logic, like the law of identity, may fail. This is exactly what non-normal worlds are” (p.167). We thus incorporated non-normal worlds into *K*_{4} in order to get *N*_{4}. But we note here that in non-normal worlds, the normal laws of logic are different, and we will call such non-normal worlds “logically impossible worlds.”]

As we saw (9.4.4 – 9.4.6), non-normal worlds of the kind we have employed in this chapter are worlds where the laws of logic are different. Let us call these ‘logically impossible worlds’.

(171)

[Logically Impossible Worlds as Admissible]

[Just as there is no problem in conceiving physically impossible worlds, there should likewise be no problem in conceiving logically impossible worlds*.*]

[In section 3.6.5 (not yet summarized), Priest discussed physically impossible worlds: “Something is physically necessary if it is determined by the laws of nature, and physically possible if it is compatible with the laws of nature. Thus, it is physically impossible for me to jump thirty metres into the air (though this is not a logical impossibility)” (46). Priest says now that there is no reason why there cannot likewise be logically impossible worlds.]

There seems to be no reason why there should not be logically impossible worlds, in whatever sense there are possible worlds. Physically impossible worlds, where the laws of physics are different, are entirely routine (see 3.6.5). And just as there are worlds where the laws of physics are different, there must be worlds where the laws of logic are different.

(171)

[Our Seeming Assumption of Logically Impossible Worlds]

[We already seem to suppose such logically impossible worlds when we note how certain laws of logic fail in particular non-classical logics, as for example when we say: “if intuitionist logic were correct, the law of double negation would fail*.”*]

[In fact, when we discuss non-classical logics, we seem to suppose such “logically impossible” worlds (even though in fact the real world might be one where the laws of classical logic do indeed fail). It is implied for example when we say that “if intuitionist logic were correct, the law of double negation would fail.”]

After all, we seem to envisage just such worlds when we evaluate conditionals such as ‘if intuitionist logic were correct, the law of double negation would fail’ (true), ‘if intuitionist logic were correct, the law of | identity would fail’ (false). Even if one is a modal realist (2.6), why should there not be such worlds?

(171-172)

[Failure of Objections to Impossible Worlds]

[Objections to logically impossible worlds do not work*. *For, we cannot simply require that the laws of logic admit of no variation, when in fact that is what we are successfully and fruitfully modelling.]

[Priest next deals with some objections to the idea that there can be logically impossible worlds. {1} Objection: Someone might say that logical laws should always hold at possible worlds, by definition. Reply: we are not dealing with possible worlds but rather impossible ones. {2} Objection: One might say that some proposed logically law for a possible world that breaks one of our normal logical laws cannot be the case, simply because it is breaking a normal logical law. For example, suppose someone claims that there is a world where it is a logical law that *A *→ (*B* ∧ ¬*B*) holds and so does *A*. Then, by *modus ponens* we can infer that *B* ∧ ¬*B*. The objector can say that this is a contradiction and it cannot be the case. Reply: {2a} Some might have philosophical reasons to say that the normal laws can be broken. For example, a dialetheist would say that the law of non-contradiction is breakable. {2b} The objection assumes that *modus ponens* holds in this world. But as an impossible world, it may not.]

One might suggest that there can be no worlds at which logical laws fail: by definition, logical laws hold at all possible worlds. Maybe so. But it is precisely

impossibleworlds that we are dealing with here. Or one might say: take a world in which it is a logical law thatA→ (B∧ ¬B) and in whichAis also true. It would follow thatB∧ ¬Bis true at that world, which cannot be the case. This argument is hardly likely to persuade someone who accepts the possibility of truth-value gluts. But in any case, it is fallacious. For who says thatmodus ponensholds at that world? In the semantics we have looked at, it is entirely possible to have bothAandA→Cholding at a non-normal world, withoutCholding there.(172)

[The Non-Necessity for Logical Laws To Be Broken in Logically Impossible Worlds]

[In a logically impossible world, it could still be that no normal laws of logic be broken, just like how in a physically impossible world, normally-impossible physical events can take place, but for contingent reasons happen not to*.*]

[Priest next notes that logically impossible worlds do not necessarily have cases of broken laws of logic, just like how physically impossible worlds may allow for certain alternate physical situations without them ever obtaining. However, one could specifically define logically impossible worlds as ones where the laws of logic are in fact broken.]

Note that one might take ‘logically impossible world’ to mean something other than ‘world where the laws of logic are different’. One might equally take it to mean ‘world where the logically impossible happens’. This need not be the same thing. If this is not clear, just consider physically impossible worlds. The fact that the laws of physics are different does not necessarily mean that physically impossible things happen there (though the converse is true). For example, even if the laws of physics were to permit things to accelerate past the speed of light, it does not follow that anything actually would. Things at that world might be accelerating very slowly, and the world might not last long enough for any of them to reach super-luminal speeds.

(172)

[Logically Impossible Worlds Where Normal Laws of Logic Are In Indeed Broken]

[Logically impossible worlds can also in fact be ones where laws of logic indeed are broken*.*]

[I might be mistaken about this next point. It might be that we know there are logically impossible worlds where the laws of logic are broken, because we have already seen that there is a world where *A* and *A *→* C *are true, but* C *is not. In the footnote Priest mentions some inferences that do not hold in any impossible world. However, these are instances without conditionals, and it is conditionals that express the laws of logic (but I do not myself know why that is.)]

But logically impossible worlds, in the sense that these occur in the semantics we have been looking at, may be logically impossible in the second sense as well. For example, there are, as has just been noted, worlds where

AandA→Care true, butCis not.^{6}(172)

6. There are no worlds at which

A∧Bis true, butAis not, or at which ¬¬Ais true, butAis not. But it is conditionals that express the laws of logic, not conjunctions or negations. That is why it is their behaviour (and only theirs) that changes at non-normal worlds.(172)

[Relevant Logics and the Conditional]

[Relevant propositional logics are ones where whenever “*A *→* B* is logically valid, *A *and* B* have a propositional parameter in common” (172)*. *]

[Priest now defines relevant logic: “A propositional logic is *relevant* iff whenever *A *→* B* is logically valid, *A *and* B* have a propositional parameter in common” (172). This may seem odd, because *A* and *B* would seem to be propositional parameters, and surely we are not saying that conditions need to be of the form *A *→* A* to be relevant. So recall from section 1.2.3: “I use capital Roman letters, *A*, *B*, *C*, ..., to represent arbitrary formulas of the object language. Lower-case Roman letters, *p*, *q*, *r*, ..., represent arbitrary, | but distinct, propositional parameters” (4-5). So maybe we would need to look at the lower-case sorts of formulations, when looking for relevance. We will see some examples. Or maybe *A *→* B *is shorthand for more complex formulations, like we will see below, as with ⊨ *A *⥽ (*B* ∨ ¬*B*)*. *He says that conditionals that suffer from the paradoxes of implication, including the strict conditional, are not relevant. Let us look at the paradoxes of strict implication. In section 4.6.3 we saw that the following are valid for the strict conditional.

⊨

A⥽ (B∨ ¬B)⊨ (

A∧ ¬A) ⥽B

We might fill them out with propositional parameters I am going to guess in the following way.

⊨

p⥽ (q∨ ¬q)⊨ (

p∧ ¬p) ⥽q

But I am not sure about much here yet. Yet we can see that the strict conditional is not part of a relevant logic. And these same forms are not valid in *K*_{4}.

⊭

_{K4 }p→q∨ ¬q⊭

_{K4}(_{ }p∧ ¬p) →q

Nonetheless, Priest says that neither *K*_{4} and *K*_{∗} are relevant. To see why, we first recall from section 9.4.2 that in *K*_{4}, ⊨ *p *→ (*q* → *q*) is valid, and in section 9.6.6 we saw that it if valid in *K*_{∗} too.]

A propositional logic is

relevantiff wheneverA→Bis logically valid,AandBhave a propositional parameter in common. Obviously, any conditional that suffers from paradoxes of implication (material implication, | strict implication, the intuitionist conditional) is not relevant. Neither areK_{4}andK_{∗}relevant, as we have seen (9.4.2 and 9.6.6).(172-173)

[*N*_{4} as Relevant]

[But *N*_{4} is a relevant logic, on account of how conditionals are evaluated in normal worlds (they depend on the values in non-normal worlds) in combination with the arbitrarity of their value assignments in non-normal worlds*.*]

[Priest will now show that *N*_{4} is a relevant logic. It gets very technical, and I am the wrong person to summarize this, so please skip to the quotation below. I will try to say some things still, but they probably will not help you. Recall that in *N*_{4}, there are non-normal worlds. And recall from section 4.2.5 and 9.4.9 that inferences are valid only if they preserve truth in all interpretations at all normal worlds. I might have this wrong, but I think that means it cannot be that the premises are at least true and the conclusion not at least true. If I am following even a little here (and probably not), Priest is going to do the following. Let me first note that I am not certain if we are dealing with an inference, like in his cited problem, or a simple conditional, like mentioned in the last line of this paragraph. I am also not sure if it makes a difference. Let us for now say that we are dealing with the conditional *A *→* B*, and we want to know if it would be valid/true in *N*_{4} whenever there is no relevance of the antecedent *A* to the consequent *B*. Priest will make a model where it is false, even though in *K*_{4} presumably it would be true. It seems that the way this will work will have to do with the fact that in normal worlds we evaluate conditionals on the basis of all other worlds, whether normal or not. But as we saw in section 9.4.6, we do not evaluate the conditionals in non-normal worlds compositionally in terms of the component terms’ values but rather we assign their values arbitrarily however we please. Priest will exploit those two features of *N*_{4} in order to make a non-relevant *A *→* B *be false/invalid in a normal world. So our model will have two worlds, 0 and 1, and world 1 is the non-normal one. It is still unclear to me if we are dealing with an inference from *A* to *B *or a conditional, but I am guessing wildly that Priest is covering both options. (Sorry, please read the text). For non-relevance, we suppose that we have antecedent *A* and consequent *B *(or premise(s) *A* and conclusion *B*), but *A *and *B* share no propositional parameters in common. We also consider a propositional parameter (or conditional) called *D*, which can be included either in antecedent *A* or consequent *B* (or in premise(s) *A* or conclusion *B*); but it cannot be in both, because as we said, *A* and *B *share no parameters in common, thus if *D* is in one, it cannot be in the other. We will assign our values for the conditional in non-normal world 1 arbitrarily, as that is how it works in non-normal worlds (see section 9.4.6). So we say, if *D* is among the antecedent (or premises), then we assign it both as true and also as false. Or, if instead it is in the consequent (or conclusion), then we assign it neither true nor false. Now, let us stick with conditionals for a second. Recall from section 9.2.4 that this is how we evaluate conditionals in *N*_{4}:

A→Bρ1 iff for all_{w}w′ ∈Wsuch thatAρ_{w}_{′}1,Bρ_{w}_{′}1

A→Bρ0 iff for some_{w}w′ ∈W,Aρ_{w}_{′}1 andBρ_{w}_{′}0(p.164, section 9.2.4)

We ask, *A *→* B**ρ _{w}*

_{0}??? In other words, we want to determine the value of

*A*→

*B*in world 0, the normal world. Priest’s way of proving this uses the induction method, which I have not learned yet (see section 0.2). Were we to perform it, we would find that somehow, regardless of whether

*D*is in the antecedent (or premises) or in the consequent (or conclusion),

*A*will be both true and false (and thus at least true) and

*B*will be neither true nor false (and thus not at least true). And hence the formula will be false/invalid in world 0, the normal world. Please read the quotation, as I am not grasping this one very well at all.]

But

N_{4}is a relevant logic. This can be seen by modifying the argument of 8.10, problem 5. Suppose thatAandBshare no propositional parameters, and consider an interpretation ⟨W,N, ρ⟩, whereW= {w_{0},w_{1}};N={w_{0}}; ifDis a propositional parameter or a conditional inA,Dρ_{w}_{1}1 andDρ_{w}_{1}0; ifDis a propositional parameter or a conditional inB, neitherDρ_{w}_{1}1 norDρ_{w}_{1}0. (Dcannot occur in both, sinceAandBhave no parameters in common.) It is easy to check thatAρ_{w}_{1}1 andAρ_{w}_{1}0, but neitherBρ_{w}_{1}1 nor =Bρ_{w}_{1}0.^{7}In particular,Ais true atw_{1}andBis not. HenceA→Bis not true atw_{0}.(173)

7.

Proof: For the first, what we show is that every formula made up from the propositional parameters occurring inA– and so, in particular,A– the result holds. Similarly forB. This is proved by induction on the construction of sentences, but an induction slightly different from the normal kind. Note that every formula can be built up from conditionals and parameters using the extensional connectives. Hence, the result may be proved by induction, with parameters and conditionals as the basis case, and induction cases for the extensional connectives. The basis case is true by definition. The induction cases are as in the notes to 8.4.6 and 8.4.9.(173)

[*N*_{∗} as Relevant]

[In a similar way, *N*_{∗} is also a relevant logic*.*]

[Priest next shows how in a similar way *N*_{∗} is a relevant logic. Please consult the text for the details.]

A similar argument shows that

N_{∗ }is a relevant logic. Take a ∗ interpretation ⟨W,N, ∗,v⟩, whereW= {w_{0},w_{1},w_{2}};N= {w_{0}},w^{*}_{o}=w_{0},w^{*}_{1}=w_{2},w^{*}_{2}=w_{1}; for every propositional parameter or conditional,D, inA,v_{w1}(D) = 1 andv_{w2}(D) = 0; for every propositional parameter or conditional,D, inB,v_{w1}(D) = 0 andv_{w2}(D) = 1. One can check thatv_{w1}(A) = 1, andv_{w1}(B) = 0. Hencev_{w0}(A→B) = 0. Details are left as an exercise.(172)

[Relevant Logics Meet Our Intuitions About Conditionals and Relevance]

[Relevant logics tend to our intuitions that there should be relevance between antecedent and consequent of conditionals, and this can be done by requiring them to share parameters*.*]

[We have the intuition already that “for a conditional to be true there must be some connection between its antecedent and consequent” (172). But it is not always obvious how to do that in a formalized way. Yet we saw in section 9.7.8 that one way is to require shared parameters.]

It is a natural thought that for a conditional to be true there must be some connection between its antecedent and consequent. It was precisely this idea that led to the development of relevant logic. A sensible notion of connection is not so easy to spell out, however (as we saw, in effect, in 4.9.2). The parameter-sharing condition of 9.7.8 gives some content to the idea.

(172)

[Filter Logics]

[There is another sort of relevant logic that is of a whole different class, called filter logics, in which “a conditional is taken to be valid iff it is classically valid *and* satisfies some extra constraint, for example that antecedent and consequent share a parameter” (173)*.*]

[Priest then notes another sort of relevant logic called filter logics. Here “a conditional is taken to be valid iff it is classically valid *and* satisfies some extra constraint, for example that antecedent and consequent share a parameter.” But it is a different sort of logic than the ones of this book. Priest notes that often times filter logics break the principle of transitivity.]

There are some approaches to relevant logic where a conditional is taken to be valid iff it is classically valid

andsatisfies some extra constraint, for example that antecedent and consequent share a parameter. (These are | sometimes calledfilter logics, since the extra constraint filters out ‘undesirables’.) Characteristically, such approaches give rise to relevant logics of a kind different from those considered in this book. For example, if the parameter-sharing filter is used, (p∧ (¬p∨q)) →qis valid, which it is not in the relevant logics of this, and subsequent, chapters. Typically (though not invariably), a feature of filter logics is the failure of the principle of transitivity: ifA⊨BandB⊨CthenA⊨C(thus breaking the argument of 4.9.2).(173-174)

[More Than Classical Relevance]

[Relevance in our systems here however is not conditions added on top of classical validity*.*]

[The way that we are dealing with relevance here “is not some extra condition imposed *on top* of classical validity” (174). Rather, it is something else (but I am not sure I understand what it is and what the distinction is. So see the quote below.)]

In the present approach, relevance is not some extra condition imposed

on topof classical validity. Rather, relevance, in the form of parameter sharing, falls out of something more fundamental, namely the taking into account of a suitably wide range of situations.(174)

[Preserving Conventional Truth in This System]

[If we wanted to keep this system but reserve a real world where truth operates in a more conventional way, then we can designate an @ actual world that has certain constraints. For example, we could add exhaustion and exclusion constraints to eliminate truth gaps and gluts in the actual real world @*.*]

[Priest then notes some additional concerns. We might want within this four value-situationed non-normal worlds logic to reserve certain conventional properties of truth for the “real” or actual world. Priest then explains how this would work. We symbolize @ as the actual world, and we say truth in this restricted sense is truth at @, and validity is truth preservation at @ for all interpretations. Then, we can add constraints in @ to model the properties we think truth should have in the actual real world. For example, we could add exhaustion and exclusion constraints to eliminate truth gaps and gluts (see section 8.4.6 and section 8.4.9).]

One final comment: one might hold that truth – real truth, not just truth in some world – has some special properties; that unlike truth in an arbitrary world, truth itself can have no gaps or gluts. To accommodate this view, one could take an interpretation to include a distinguished normal world, @ (for actuality), such that truth (

simpliciter) is truth at @. Validity would then be defined as truth preservation at @ in all interpretations.^{8}The special properties of truth would be reflected in semantic constraints on @. Thus, if it be held that there are no truth value gluts in @, one would impose the constraint that ρ@ satisfy the conditionExclusionof 8.4.6. If it be held that there are no truth-value gaps in @, then one would impose the constraint that ρ@ satisfy the conditionExhaustionof 8.4.9.^{9}Or in a ∗ interpretation, one might require that @ = @∗, which rules out gaps and gluts. But from the present | perspectives, these conditions would require justification by some novel considerations.(174-175)

8 One could, in fact, set up all the possible-world semantics that we have had till now in this way. But since these semantics contain nothing to distinguish @ from any other normal world, this would have had no effect on validity.

9 Strictly speaking, these conditions are not sufficient. To rule out truth-value gluts and gaps with formulas containing →

s, we need to make another change as well. Specifically, to rule out truth-value gaps, the falsity conditions forA→Bat @ have to read:

A→Bρ_{@}0 iff (for somew′,Aρ_{w}_{′}1 andBρ_{w}_{′}0) or (it is not the case thatA→Bρ_{w}_{@}1) and to rule out truth-value gluts, they have to read:

A→Bρ_{@}0 iff (for somew′,Aρ_{w}_{′}1 andBρ_{w}_{′}0) and (it is not the case thatA→Bρ_{w}_{@}1).(174)

From:

Priest, Graham. 2008 [2001]. *An Introduction to Non-Classical Logic: From If to Is*, 2nd edn. Cambridge: Cambridge University.

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