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
9.
Logics with Gaps, Gluts and Worlds
9.4
Non-normal Worlds Again
Brief summary:
(9.4.1) On account of the potential for truth-value gaps and gluts, the conditional in our possible worlds First Degree Entailment system K4 does not suffer from the following paradoxes of the strict conditional: ⊨ p → (q ∨ ¬q), ⊨ (p ∧¬p) → q. (9.4.2) In K4, if ⊨ A then ⊨ B → A. That means ⊨ p → (q → q) is valid, because ⊨ q → q is valid. (9.4.3) Even though ⊨ p → (q → q) , which contains the law of identity, is valid, we can think of a paradoxical instance of this that shows how the law of identity can fail: “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” (167). (9.4.4) As we noted, the conditional should be able to express things that go against the laws of logic, like the law of identity. We should be able to formulate sentences in which the antecedent supposes some law of logic not to hold, and then the consequent would express what sorts of things would follow from that. Non-normal worlds are ones where the normal laws of logic may fail; so we should implement non-normal worlds: “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” (167). (9.4.5) We thus need to consider non-normal worlds where the laws of logic fail and, given how the conditionals express those laws, where the conditional takes on values different than it would in normal worlds (K4). (9.4.6) At a non-normal world, A → B might be able to take on any sort of value, because the laws of logic may change in that world. (9.4.7) We “take an interpretation to be a structure ⟨W, N, ρ⟩, where W is a set of worlds, N ⊆ W is the set of normal worlds (so that W − N is the set of non-normal worlds), and ρ does two things. For every w, ρw is a relation between propositional parameters and the truth values 1 and 0, in the usual way. But also, for every non-normal world, w, ρw is a relation between formulas of the form A → B and truth values” (167). (9.4.8) The truth conditions for connectives in our non-normal worlds K4 are the same as for K4, except in non-normal worlds, the conditional is assigned its value not recursively but in advance by the ρ relation. Here are the truth conditions for normal worlds:
A ∧ Bρw1 iff Aρw1 and Bρw1
A ∧ Bρw0 iff Aρw0 or Bρw0
(p.164, section 9.2.3)
A ∨ Bρw1 iff Aρw1 or Bρw1
A ∨ Bρw0 iff Aρw0 and Bρw0
¬Aρw1 iff Aρw0
¬Aρw0 iff Aρw1
(not in the text)
A → Bρw1 iff for all w′ ∈ W such that Aρw′1, Bρw′1
A → Bρw0 iff for some w′ ∈ W, Aρw′1 and Bρw′0
(p.164, section 9.2.4)
(9.4.9) Our non-normal worlds FDE system will be called N4, and it will define validity in the same way as for K4, namely, as truth preservation at all normal worlds of all interpretations.
[K4 as Immune to Certain Paradoxes of the Strict Conditional]
[The Validity of ⊨ p → (q → q)]
[An Instance Where the Law of Identity Fails]
[The Need for Non-Normal Worlds for the Conditional]
[The Conditional in Non-Normal Worlds]
[Conditionals Taking on Different Truth-Values]
[A Formalization of Non-Normal Worlds K4]
[Truth Conditions in Non-Normal Worlds K4]
[Non-Normal Worlds K4 as N4. Validity in N4.]
Summary
[K4 as Immune to Certain Paradoxes of the Strict Conditional]
[On account of the potential for truth-value gaps and gluts, the conditional in our possible worlds First Degree Entailment system K4 does not suffer from the following paradoxes of the strict conditional: ⊨ p → (q ∨ ¬q), ⊨ (p ∧¬p) → q.]
[Recall from section 9.1.1 and section 9.2.1 that K4 combines {1} the four value-situation semantics of First Degree Entailment (FDE), meaning that it uses the ρ relation to relate a formula to either just 1 (true), just 0 (false), both 1 and 0, or neither 1 nor 0 (see section 8.2) with {2} the possible worlds logics of normal modal logic Kυ (he says in section 9.2.6 that “A natural name for this logic would be Kυ4. We will call it, more simply, K4” (164).) This means that when we use the ρ relation to assign values, we also specify the world where the formula takes that value. For example, if formula A is true in world 1, then we would write: Aρw11. Priest here says, “As is to be expected, and is not difficult to check, the following do not hold in K4: ⊨ p → (q ∨ ¬q), ⊨ (p ∧¬p) → q.” Regarding whether or not it is expected, let us first consider how we might expect that ⊭ p → (q ∨ ¬q) in K4. Recall from section 8.6.1 and section 8.4.11 that in FDE:
p ⊬FDE q ∨ ¬q
p ⊭FDE q ∨ ¬q
Recall from section 8.2.8 that semantic validity is defined in FDE as:
Σ ⊨ A iff for every interpretation, ρ, if Bρ1 for all B ∈ Σ then Aρ1
(p.144, section 8.2.8)
And recall from section 8.2.6 that the evaluation rule in FDE for disjunction is:
A ∨ Bρ1 iff Aρ1 or Bρ1
A ∨ Bρ0 iff Aρ0 and Bρ0
(p.143, section 8.2.6)
That means that if both p and q have neither value, they do not fulfill the conditions to be either true or false (or to be both), and so they are neither. So we see why:
p ⊭FDE q ∨ ¬q
Here, suppose p is true and q is neither value. That means the premises are true and the conclusion is not at least true, and so it is invalid. But I am not sure how we go from that to
⊨ p → (q ∨ ¬q)
I would note from section 9.2.4 that the rule for semantically evaluating the condition is:
A → Bρw1 iff for all w′ ∈ W such that Aρw′1, Bρw′1
A → Bρw0 iff for some w′ ∈ W, Aρw′1 and Bρw′0
(p.164, section 9.2.4)
and from section 9.2.5 that for our possible worlds FDE
semantic consequence is defined in terms of truth preservation at all worlds of all interpretations:
Σ ⊨ A iff for every interpretation, ⟨W, ρ⟩, and all w ∈ W: if Bρw1 for all B ∈ Σ, Aρw1
(p.164, section 9.2.5)
So they seem to have similar definitions. But in the end, we might just use the tableau rules of K4 (section 9.3) to show the invalidity.
| ⊬K4 p → (q ∨ ¬q) | ||
| 1. .
2. .
3. .
4. .
5. | p → (q ∨ ¬q),–0 ↓ p,+1 ↓ q ∨ ¬q,–1 ↓ q,–1 ↓ ¬q,–1 | P .
1→– .
1→– .
3∨– . 3∨– Open Invalid |
(not in Priest’s text and likely mistaken)
Hence, we would make a model with one world, and in it, p is true, and neither q nor ¬q have a value. So we might expect that ⊭K4 p → q ∨ ¬q, because truth value gaps can make the antecedent true and the consequent not at least true. Now why would we expect that ⊭FDE (p ∧¬p) → q ? Similarly we can see from section 8.4.8 and 8.6.1 that
p ⊬FDE (p ∧ ¬p) → q
p ⊭FDE (p ∧ ¬p) → q
And we would note that we evaluate conjunctions in FDE like:
A ∧ Bρ1 iff Aρ1 and Bρ1
A ∧ Bρ0 iff Aρ0 or Bρ0
And we note that in FDE, there can be truth-value gluts. In this case, we set p to 1 and also to 0, and q we set to just 0, all in the same world. Let us try at least to see if we get something similar with the tableau.
| ⊬K4 (p ∧ ¬p) → q | ||
| 1. .
2. .
3. .
4. .
5.
| (p ∧ ¬p) → q,–0 ↓ p ∧ ¬p,+1 ↓ q,–1 ↓ p,+1 ↓ ¬p,+1
| P .
1→– .
1→– .
2∧+ . 2∧+ Open |
For our counter-model, we have that q is false but p is true and p is false. As such, ¬p is false and ¬p is also true. Thus, (p ∧ ¬p) is at least true (perhaps it is also false, I am not sure. See section 8.2.7. But that is inconsequential here). That means for
(p ∧ ¬p) → q
we have the antecedent as true and the consequent as false. And so it is invalid, and hence both:
⊭K4 p → q ∨ ¬q
⊭K4 (p ∧ ¬p) → q
Lastly recall from section 4.6.3 that these are two of the paradoxes of strict implication. Since they are invalid in FDE, that means FDE is free of them.]
As is to be expected, and is not difficult to check, the following do not hold in K4: ⊨ p → (q ∨ ¬q), ⊨ (p ∧¬p) → q. The conditional of K4 does not, therefore, suffer from these paradoxes of the strict conditional.
(166)
[The Validity of ⊨ p → (q → q)]
[In K4, if ⊨ A then ⊨ B → A. That means ⊨ p → (q → q) is valid, because ⊨ q → q is valid.]
[Priest next notes that even in K4, if ⊨ A then ⊨ B → A. (But I am not sure how to show that. Priest gives the informal account “If A is true at all worlds of all interpretations, it is true at all worlds of all | interpretations where B is true” (166-167).) Now, since ⊨ q → q, that means we can have ⊨ p → (q → q).
But, as is also easy to see, it is still the case that if ⊨ A then ⊨ B → A . (If A is true at all worlds of all interpretations, it is true at all worlds of all | interpretations where B is true).2 In particular, for example, since ⊨ q → q, ⊨ p → (q → q).
(166-167)
2 The dual (if ⊨ ¬A then ⊨ A → B) does not hold. For example, even though ⊨ ¬¬(p → p), ⊭ ¬(p → p) → q, as may be checked.
(167)
[An Instance Where the Law of Identity Fails]
[Even though ⊨ p → (q → q) , which contains the law of identity, is valid, we can think of a paradoxical instance of this that shows how the law of identity can fail: “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” (167).]
[So we saw in section 9.4.2 above that ⊨ p → (q → q). Priest now sets up a paradoxical instance of this. He first notes that q → q is an instance of the law of identity. And he finds as an example, “if cows were black, cows would be black.” But then he makes the p term mean that “if every instance of the law of identity failed”. But the consequent to that is an instance of the law of identity. So while formally the problem might not be evident, it seems we can show the problem by exemplifying the terms in this problematic way.]
This may well be felt to be unsatisfactory. q → q is an instance of the law of identity. Yet the following conditional would hardly seem to be true: 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.
(167)
[The Need for Non-Normal Worlds for the Conditional]
[As we noted, the conditional should be able to express things that go against the laws of logic, like the law of identity. We should be able to formulate sentences in which the antecedent supposes some law of logic not to hold, and then the consequent would express what sorts of things would follow from that. Non-normal worlds are ones where the normal laws of logic may fail; so we should implement non-normal worlds.]
[(I cannot say I follow all this too very well. But above in section 9.4.3 we saw an instance of the valid formula ⊨ p → (q → q) which contains a formulation for the law of identity (q → q), but which says verbally that if the law itself fails generally, then in a specific case it will hold. And this we thought should be false. I am really not sure, but I am thinking maybe the idea here is the following, but I am guessing. We can formulate this statement, namely, “if every instance of the law of identity failed, then, if cows were black, cows would be black.” And although it structurally is valid, we would wish it to instead say that “if every instance of the law of identity failed, then, if cows were black, cows would not be black.” So we want a way to make such a formulation, even though structurally, under our current provisions, it cannot be made. And we would want to make such a formulation, because we would want to be able to consider what would happen if the law of identity failed. Under normal circumstances, we maybe lack the formulational machinery to express it. Now at this point we should recall some things about non-normal worlds, which we studied in section 4. Here is the brief summary for section 4.2:
(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.) [...]
(From our brief summary of section 4.2)
In section 4.4, we discussed the properties of non-normal worlds. Recall that according to the rule of necessitation, formulas that are valid in normal worlds are necessary (meaning they are true in all normal worlds). But the rule of necessitation fails for non-normal worlds, meaning that the laws of logic may not hold in them.
[...] (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).
(From our brief summary of section 4.4)
We saw in section 4.4a.14 a case of the failure of the rule of necessitation is in the non-normal modal logic L. There Priest writes:
The rule of Necessitation fails in L for essentially the same reason that it fails in N and its extensions (4.4.7). Indeed, that ‘logic need not hold’ at non-normal worlds in L is patent: if A is a logical truth, □A can behave any old way at such a world.
(71)
In our current situation, we want to be able to model situations where the law of identity may not hold. So to accommodate such propositions, we will turn to non-normal worlds.]
Clearly, if we are thinking in terms of worlds, to do justice to this conditional, 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, as we saw in 4.4.8 and 4.4a.14. Hence, it is natural to augment the semantic machinery with appropriate non-normal worlds.
(167)
[The Conditional in Non-Normal Worlds]
[We thus need to consider non-normal worlds where the laws of logic fail and, given how the conditionals express those laws, where the conditional takes on values different than it would in normal worlds (K4).]
[Priest then says that conditionals are what express the laws of logic. (But I do not know how that works yet. We have only seen how q → q is an instance of the law of identity.) And conditionals also guarantee truth preservation from the antecedent to the consequent in all worlds. (But, since we are dealing with instances where the laws of logic fail, that means we need to consider non-normal worlds where the conditional takes on different values than it does in normal worlds.)]
Now, it is exactly conditionals – which guarantee truth preservation from antecedent to consequent at all worlds – that express laws of logic. (A conditional such as ‘If it does not rain, we will go to the cricket’ does not express a law of logic, of course. But, as we noted in 5.2.4, such a conditional is not, arguably and strictly speaking, true.) Hence, we need to consider worlds where formulas of the form A → B may take values different from the values they may take in K4.
(167)
[Conditionals Taking on Different Truth-Values]
[At a non-normal world, A → B might be able to take on any sort of value, because the laws of logic may change in that world.]
[Recall the modal logic system L from section 4.4a:
(4.4a.1) L is a type of non-normal modal logic. Modal formulas are “sentences of the form □A and ◊A.” And in L, “modal formulas are assigned arbitrary truth values at non-normal worlds” (69). (4.4a.2) In L, the evaluation function v “assigns each modal formula a truth value at every non-normal world” (69). (4.4a.3) The tableau rules for L are the same as for N, except “there are no rules applying to modal formulas or their negations at worlds other than 0. That is, the rules of 2.4.4 apply at world 0 and world 0 only” (69). [...]
(from our brief summary of section 4.4a.)
As we see, in the system L, in non-normal worlds the modal formulas in non-normal worlds are assigned arbitrary truth values. (Since we are here going to assign different truth values for conditionals in our system here, we see the parallels to L.)]
How different? If logical laws may change, then there would seem to be no a priori bound on how this may happen. Hence, at a non-normal world A → B might be able to take on any sort of value. It therefore behaves in exactly the same way as do modal formulas in the logic L of 4.4a.
(167)
[A Formalization of Non-Normal Worlds K4]
[We “take an interpretation to be a structure ⟨W, N, ρ⟩, where W is a set of worlds, N ⊆ W is the set of normal worlds (so that W − N is the set of non-normal worlds), and ρ does two things. For every w, ρw is a relation between propositional parameters and the truth values 1 and 0, in the usual way. But also, for every non-normal world, w, ρw is a relation between formulas of the form A → B and truth values” (167).]
[Recall first from section 2.3 some basic notational conventions for giving the formal structure of modal logics:
[...] 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. [...]
(from the brief summary of section 2.3)
Now recall from section 4.2.2 the formalization of non-normal modal semantics.
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)
Now Priest gives a formalization of the structure of our non-normal worlds K4 system. As we will see, there is no R relation here. My best guess for why is the following. Recall from section 9.2.6 that K4 is another name for Kυ4. And recall from section 3.5.1 that the υ here means that the system has the universal constraint: “let an υ-interpretation – ‘υ’ (upsilon) for universal – be an interpretation in which R satisfies the following condition: for all w1 and w2, w1Rw2 – everything relates to everything” (p.45, section 3.5.1). I am guessing, but maybe our current system can put aside the R relation, because it is going to hold universally anyway, at least for normal worlds. The other change in the formalization is that instead of the truth-valuing function v, we now have the truth-valuing relation ρ. Priest says however ρ performs two functions. On the one hand, in normal worlds it relates propositional parameters to the two truth values 1 and 0 in the usual way (allowing of course for the four value situations of FDE). But in non-normal worlds, it relates conditional formulas to truth values (perhaps in an arbitrary or stipulative way.)
A way of making these ideas precise is to take an interpretation to be a structure ⟨W, N, ρ⟩, where W is a set of worlds, N ⊆ W is the set of normal worlds (so that W − N is the set of non-normal worlds), and ρ does two things. For every w, ρw is a relation between propositional parameters and the truth values 1 and 0, in the usual way. But also, for every non-normal world, w, ρw is a relation between formulas of the form A → B and truth values.
(167)
[Truth Conditions in Non-Normal Worlds K4]
[The truth conditions for connectives in our non-normal worlds K4 are the same as for K4, except in non-normal worlds, the conditional is assigned its value not recursively but in advance by the ρ relation.]
[Now recall from section 9.2.3 and section 9.2.4 the truth-value conditions for K4.
A ∧ Bρw1 iff Aρw1 and Bρw1
A ∧ Bρw0 iff Aρw0 or Bρw0
(p.164, section 9.2.3)
A ∨ Bρw1 iff Aρw1 or Bρw1
A ∨ Bρw0 iff Aρw0 and Bρw0
¬Aρw1 iff Aρw0
¬Aρw0 iff Aρw1
(not in the text)
A → Bρw1 iff for all w′ ∈ W such that Aρw′1, Bρw′1
A → Bρw0 iff for some w′ ∈ W, Aρw′1 and Bρw′0
(p.164, section 9.2.4)
Priest says that the truth conditions for all connectives in our non-normal system will be the same as in K4, only now we do not determine the values of → recursively using these rules, but rather they are already assigned their values by the ρ value-assigning relation.]
The truth conditions for all the connectives are exactly as in K4 (9.2.4), except that at non-normal worlds, the truth values of → formulas are not determined recursively: they are already determined by ρ.
(168)
[Non-Normal Worlds K4 as N4. Validity in N4.]
[Our non-normal worlds FDE system will be called N4, and it will define validity in the same way as for K4, namely, as truth preservation at all normal worlds of all interpretations.]
[Recall from section 4.2.5 that in K4
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)
Our non-normal worlds FDE system will be called N4, and it will define validity in the same way.]
Validity is defined in terms of truth preservation at all normal worlds of all interpretations, as in 4.2.5. (After all, we are interested in what follows from what in the worlds where logic is not different.) Call this logic N4.3
(168)
3. Since the logic is conceptually much closer to the non-normal modal logic L than N, ‘L4’ would be a more appropriate name. (Similarly for N∗ in 9.6.) However, ‘N4’ was the name used in the first edition of this book, and it would seem to cause less confusion to stick with this.
(168)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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