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
10.
Relevant Logics
10.2
The Logic B
Brief summary:
(10.2.1) We can strengthen relevant logics like N4 and N∗ to accommodate certain intuitively correct principles regarding the conditional by incorporating non-normal worlds and a ternary accessibility relation on worlds, Rxyz. (10.2.2) The intuitive sense of the ternary relation Rxyz is: for all A and B, if A → B is true at x, and A is true at y, then B is true at z. (10.2.3) We will focus on the ternary relation ∗ semantics, as they have been the ones studied historically speaking. (10.2.4) The ternary ∗ interpretation is a structure, ⟨W, N, R, ∗, v⟩, 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)”, “for all w ∈ W, w∗∗ = w; v assigns a truth value to every parameter at every world, and to every formula of the form A → B at every non-normal world,” and R is any ternary relation on worlds. (So, technically, R ⊆ W × W × W.)” (167; 170; 189). (10.2.5) Priest next gives the truth conditions for connectives.
vw(A ∧ B) = 1 if vw(A) = vw (B) = 1, otherwise it is 0.
vw(A ∨ B) = 1 if vw(A) = 1 or vw (B) = 1, otherwise it is 0.
vw(¬A) = 1 if vw*(A) = 0, otherwise it is 0.
(p.151, section 8.5.3; p.169. section 9.6.6, see 9.6.2)
at normal worlds, the truth conditions for → are:
vw(A → B) = 1 iff for all x ∈ W such that vx(A) = 1, vx(B) = 1
The exception is that if w is a non-normal world:
vw(A → B) = 1 iff for all x, y ∈ W such that Rwxy, if vx(A) = 1, then vy(B) = 1
(189)
(10.2.6) Validity is truth preservation over all normal worlds. (10.2.7) This logic is named B, and it is a sub-logic of K∗, while N∗ is a sub-logic of B. (10.2.8) The normality condition is Rwxy iff x = y. By implementing it in the conditional rule, we can simply it so that it works for all worlds: vw(A → B) = 1 iff for all x ∈ W such that vx(A) = 1, then vx(B) = 1. (10.2.9) Finally Priest notes that “the normality condition falls apart into two halves. From left to right: if Rwxy then x = y and from right to left, since x = x: Rwxx” (190).
[Strengthening Relevant Logics with Non-Normal Worlds and a Ternary Accessibility Relation on Worlds, Rxyz]
[The Intuitive Sense of Rxyz]
[Focusing on the Ternary Relation ∗ Semantics]
[The Structure of Ternary ∗ Interpretations]
[Truth Conditions for Connectives]
[Validity]
[B]
[Simplifying the Conditional’s Truth Conditions Using the Normality Condition]
[The Two Halves of the Normality Condition]
Summary
[Strengthening Relevant Logics with Non-Normal Worlds and a Ternary Accessibility Relation on Worlds, Rxyz]
[We can strengthen relevant logics like N4 and N∗ to accommodate certain intuitively correct principles regarding the conditional by incorporating non-normal worlds and a ternary accessibility relation on worlds, Rxyz.]
[Recall from section 9.7.8 that “A propositional logic is relevant iff whenever A → B is logically valid, A and B have a propositional parameter in common” (p.172, section 9.7.8). Next 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. And recall from section 9.4 we can add to it impossible worlds to get N4. In section 9.7.9 and 9.7.10 we saw that N4 and N∗ are relevant logics. Priest now notes that N4 and N∗ “are too weak, on the ground that there are intuitively correct principles concerning the conditional that they do not validate” (188). Priest now explains that we can incorporate these principles using non-normal worlds and a ternary accessibility relation on worlds, Rxyz.]
N4 and N∗ are relevant logics, but, as relevant logics go, they are relatively weak. Many proponents of relevant logic have thought that the relevant logics of the last chapter are too weak, on the ground that there are intuitively correct principles concerning the conditional that they do not validate. A way to accommodate such principles within a possible-world semantics is to use a relation on worlds to give the truth conditions of conditionals at non-normal worlds. Unlike the binary relation of modal logic, xRy, though, this relation is a ternary, that is, three-place, relation, Rxyz.1
(188)
1. Using a binary relation would produce irrelevance, since p → p would be true at all worlds, and hence, q → (p → p) would be logically valid.
(188)
[The Intuitive Sense of Rxyz]
[The intuitive sense of the ternary relation Rxyz is: for all A and B, if A → B is true at x, and A is true at y, then B is true at z. ]
[Priest next gives the intuitive sense of the ternary relation Rxyz: for all A and B, if A → B is true at x, and A is true at y, then B is true at z. (In other words, Rxyz is a particular sort of relation between worlds. We suppose that A → B is true in world x, and A is true in world y. On those conditions, when Rxyz holds, then also B must be true in world z. Priest will come back to the issue of understanding what this all means.]
Intuitively, the ternary relation Rxyz means something like: for all A and B, if A → B is true at x, and A is true at y, then B is true at z. What philosophical sense to make of this, we will come back to later.
(188)
[Focusing on the Ternary Relation ∗ Semantics]
[We will focus on the ternary relation ∗ semantics, as they have been the ones studied historically speaking.]
[We can apply this technique (maybe, the incorporation of non-normal worlds and a ternary accessibility relation on worlds, Rxyz) to the relational semantics (maybe K4 and N4) and the ∗ semantics (maybe K∗ and N∗). Recall from section 9.6.9 that K4 and N4 are not equivalent to K∗ and N∗. For example, K∗ and N∗ validate contraposition (p → q ⊨ ¬q → ¬p), but K4 and N4 do not, and from section 9.6.10 that K4 and N4 verify p ∧ ¬q ⊨ ¬(p → q), but K∗ and N∗ do not. At any rate, it happens that the ternary relation ∗ semantics are what have been more studied, so we will focus on them.]
The technique can be applied to both the relational semantics and the ∗ semantics. As we noted in 9.6.9 and 9.6.10, these semantics diverge once we add → to the language. Though the ternary relation relational semantics are perfectly good, it is, as a matter of historical fact, the logics with the ternary relation ∗ semantics that occur in the literature. Hence, we look only at those.
(189)
[The Structure of Ternary ∗ Interpretations]
[The ternary ∗ interpretation is a structure, ⟨W, N, R, ∗, v⟩, 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)”, “for all w ∈ W, w∗∗ = w; v assigns a truth value to every parameter at every world, and to every formula of the form A → B at every non-normal world,” and R is any ternary relation on worlds. (So, technically, R ⊆ W × W × W.)” (167; 170; 189).]
[Priest now defines the ternary ∗ interpretation. It has a structure, ⟨W, N, R, ∗, v⟩. Here “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)” (p.167, section 9.4.7), “for all w ∈ W, w∗∗ = w; v assigns a truth value to every parameter at every world, and to every formula of the form A → B at every non-normal world” (p.170, section 9.6.6), and R is any ternary relation on worlds. (So, technically, R ⊆ W × W × W.)” (189).]
A ternary (∗) interpretation is a structure ⟨W, N, R, ∗, v⟩, where W, N, ∗ and v are as in the semantics for N∗ (9.6.6), and R is any ternary relation on worlds. (So, technically, R ⊆ W × W × W.)
(189)
[Truth Conditions for Connectives]
[Priest next gives the truth conditions for connectives.]
[Recall from section 9.6.6 the truth conditions for connectives in N∗. To these we will exclude the conditional rule for N∗ and replace it with one specific for our new system:
vw(A ∧ B) = 1 if vw(A) = vw (B) = 1, otherwise it is 0.
vw(A ∨ B) = 1 if vw(A) = 1 or vw (B) = 1, otherwise it is 0.
vw(¬A) = 1 if vw*(A) = 0, otherwise it is 0.
(p.151, section 8.5.3; p.169. section 9.6.6, see 9.6.2)
at normal worlds, the truth conditions for → are:
vw(A → B) = 1 iff for all x ∈ W such that vx(A) = 1, vx(B) = 1
The exception is that if w is a non-normal world:
vw(A → B) = 1 iff for all x, y ∈ W such that Rwxy, if vx(A) = 1, then vy(B) = 1
(189)
]
With one exception, the truth conditions for all connectives are as for N∗. In particular, at normal worlds, the truth conditions for → are:
vw(A → B) = 1 iff for all x ∈ W such that vx(A) = 1, vx(B) = 1
The exception is that if w is a non-normal world:
vw(A → B) = 1 iff for all x, y ∈ W such that Rwxy, if vx(A) = 1, then vy(B) = 1
(189)
[Validity]
[Validity is truth preservation over all normal worlds.]
[Recall from section 9.6.6 that in N∗, “Validity is defined in terms of truth preservation at normal worlds” (p.170, section 9.6.6). It is the same here too.]
Validity is defined as truth preservation over all normal worlds, as in N∗.
(189)
[B]
[This logic is named B, and it is a sub-logic of K∗, while N∗ is a sub-logic of B.]
[We call this logic B, which stands for ‘basic’. Priest notes that B is a sub-logic of K∗, and N∗ is a sub-logic of B (see the quotation for details why.)]
The logic generated in this way is usually called B (for basic).2 Clearly, B is a sub-logic of K∗ (since any K∗ interpretation is a B interpretation, with W − N = φ). Moreover, any B interpretation, , is equivalent to an N∗ interpretation. We just take that N∗ interpretation which is the same as , except that it assigns to each conditional at each non-normal world, w, whatever value it has at w in . Hence, N∗ is a sub-logic of B.
(189)
2. We continue to use B as a letter for formulas, too. Context will disambiguate.
(189)
[Simplifying the Conditional’s Truth Conditions Using the Normality Condition]
[The normality condition is Rwxy iff x = y. By implementing it in the conditional rule, we can simply it so that it works for all worlds: vw(A → B) = 1 iff for all x ∈ W such that vx(A) = 1, then vx(B) = 1.]
[Recall from section 10.2.2 that the intuitive sense of the ternary relation Rxyz is: for all A and B, if A → B is true at x, and A is true at y, then B is true at z. And recall from section 10.2.5 above that
at normal worlds, the truth conditions for → are:
vw(A → B) = 1 iff for all x ∈ W such that vx(A) = 1, vx(B) = 1
The exception is that if w is a non-normal world:
vw(A → B) = 1 iff for all x, y ∈ W such that Rwxy, if vx(A) = 1, then vy(B) = 1
(189, 10.2.5)
Let us take a closer look at the conditional rules. Suppose we are dealing with a normal world. A conditional is true only if for all other worlds whenever the antecedent is true in that other world, so too is the consequent true in that other world. I am not sure however if non-normal worlds are included in those other worlds. Suppose instead we are dealing with a non-normal world. Then a conditional is true in the non-normal world only if for any other pair of worlds (normal or not), where the ternary relation holds (that is to say, whenever the conditional is true in the non-normal world in question, and the antecedent is true also in the first other world, then the consequent is true in the second other world), then the consequent indeed is true in the second other world. I am not following this at all really, so you will need to simply read the quotation below. And I cannot really summarize the rest of this section, so you will need to read it for yourself. The next point is that we can simplify the bipartite truth conditions above by defining R at normal worlds using the normality condition: Rwxy iff x = y. I am guessing that this is added to the conditions already given for the intuitive sense of the ternary relation. The final idea is that by means of this, we can reduce our truth conditions for the conditional to hold for all worlds and to be formulated simply as:
vw(A → B) = 1 iff for all x ∈ W such that vx(A) = 1, then vx(B) = 1
]
The bipartite truth conditions of → can be simplified if one thinks of R as defined at normal worlds. Specifically, if w is normal, we specify R by the following condition:
Rwxy iff x = y
Call this the normality condition. If we define R at normal worlds in this way, we may take the ternary truth conditions to govern conditionals at all worlds. For, given this condition, the ternary truth conditions:
for all x, y ∈ W such that Rwxy, if vx (A) = 1, then vy(B) = 1
|
become:
for all x, y ∈ W such that x = y, if vx(A) = 1, then vy(B) = 1
And given the standard properties of =, this is logically equivalent to:
for all x ∈ W such that vx(A) = 1, vx(B) = 1
which gives the standard truth conditions of → at normal worlds. We adopt this simplification in what follows.
(189-190)
[The Two Halves of the Normality Condition]
[Finally Priest notes that “the normality condition falls apart into two halves. From left to right: if Rwxy then x = y and from right to left, since x = x: Rwxx” (190).]
[Recall from section 10.2.8 above that the normality condition is: Rwxy iff x = y. Priest then notes the following, but I do not know what he means:]
Notice that the normality condition falls apart into two halves. From left to right: if Rwxy then x = y and from right to left, since x = x: Rwxx.
(190)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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