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
4.5
Strict Conditionals
Brief summary:
(4.5.1) We now will examine the conditional in the context of modal logic. (4.5.2) There are contingently true material conditionals that we would not want to say are true on account of their contingency, like, “The sun is shining ⊃ Canberra is the federal capital of Australia.” For, “Things could have been quite otherwise, in which case the material conditional would have been false” (72). To remedy this, we could define the conditional as: “‘if A then B’ as □(A ⊃ B), where □ expresses an appropriate notion of necessity” (72). (4.5.3) This definition of the conditional using modal logic is called the strict conditional, symbolized as A⥽B and defined as □(A⊃B). (4.5.4) The strict conditional does not validate the problematic counter-examples (that we have seen) for the material conditional.
[The Conditional in Modal Logic]
[Problems with the Conditional When It Is Contingently True]
[The Strict Conditional Defined]
[The Strict Conditional as Intuitively Valid, Unlike the Material Conditional]
Summary
[The Conditional in Modal Logic]
[We now will examine the conditional in the context of modal logic.]
Priest says that we have learned the basics of modal logic, and we can now turn to matters regarding the conditional (72).
[Problems with the Conditional When It Is Contingently True]
[There are contingently true material conditionals that we would not want to say are true on account of their contingency, like, “The sun is shining ⊃ Canberra is the federal capital of Australia.” For, “Things could have been quite otherwise, in which case the material conditional would have been false” (72). To remedy this, we could define the conditional as: “‘if A then B’ as □(A ⊃ B), where □ expresses an appropriate notion of necessity” (72).]
[Priest notes that there are ways that a material conditional can be technically true but nonetheless we would not want to say it is true. (The problem here does not seem to be relevance but rather contingency. He gives the example of: The sun is shining ⊃ Canberra is the federal capital of Australia. The problem it seems is that although this can be a true conditional, we might be bothered by the fact that the sun could very well not be shining, and maybe even that some other city could very easily have become the capital. It is not entirely clear to me yet how to understand this problem, but let us look at the solution. It would be to understand structures of the form ‘if A then B’ as □(A ⊃ B). Perhaps the idea is that we can easily think of a world much like ours where “The sun is shining ⊃ Canberra is the federal capital of Australia” is not true ((for, perhaps, in this world on a certain day the sun is shining, but there is some other capital of Australia)). Instead, perhaps, we would want something that would be true even under very strong variations of our world, as for example, “The sun is shining ⊃ it is day” perhaps, or something like that, where it is inconceivable how in any world it could be false. For, that is one sense we give to the conditional, it seems. We might think of it as saying, given any occurrence of the antecedent, under whatever possible circumstance, you will have the consequent too. For otherwise, it may only designate a coincidence.)]
Consider a true material conditional, such as ‘The sun is shining ⊃ Canberra is the federal capital of Australia’. One is inclined to reject this as a true conditional just because the truth of the material conditional is too contingent an affair. Things could have been quite otherwise, in which case the material conditional would have been false. This suggests defining the conditional, ‘if A then B’ as □(A ⊃ B), where □ expresses an appropriate notion of necessity.
(72)
[The Strict Conditional Defined]
[This definition of the conditional using modal logic is called the strict conditional, symbolized as A⥽B and defined as □(A⊃B).]
Priest says that Lewis created modern modal logic out of his dissatisfaction with the material conditional. He favored what is called the strict conditional, symbolized as ⥽ and defined like we saw above in section 4.5.2, and so A⥽B is defined as □(A⊃B).
When Lewis created modern modal logic, he was not, in fact, concerned with modality as such. He was dissatisfied with the material conditional. He defined A⥽B as □(A⊃B), and suggested this as a correct account of the conditional. ⥽ is usually called the strict conditional.
(72)
[The Strict Conditional as Intuitively Valid, Unlike the Material Conditional]
[The strict conditional does not validate the problematic counter-examples (that we have seen) for the material conditional.]
[Priest will now show how this definition of the strict conditional does not validate any of the problematic counter-examples for the material conditional that we saw in section 1.7, section 1.8, and section 1.9. One is
¬(A⥽B) ⊨ A
Priest says that it is a variation on
¬(A ⊃ B) ⊢ A
which we examined in section 1.9. Let us look more closely at it, since we made a possible tableau proof for it, showing it to be valid under the semantics of the material conditional. Here was the possible proof for it:
(This table in not in the text and probably is mistaken. Please trust your own proofs over my attempt below.)
| ¬(A ⊃ B) ⊢ A | ||
| 1. .
2. .
3. .
4. | ¬(A ⊃ B) . ¬A ↓ ¬B ↓ A × | P .
P .
1¬⊃D .
1¬⊃D (4×2) Valid |
(not in Priest’s text, but see section 1.9.1)
We will try to see that under the strict conditional, rendered now as
¬(A⥽B) ⊨ A
this is not valid in Kρστ. To do this, we first translate it as:
¬□(A⊃B) ⊨ A
And we make our tableau using the rules from section 2.4 and 3.3.2. Supposing this to give us open branches, we then use the rules for creating a counter-models from section 2.4.7. So
¬□(A⊃B) ⊨ A
(tested as the following, with counter-modeling, to determine the invalidity of the above:)
¬□(A⊃B) ⊢ A
(This table is not in the text and probably is mistaken. Please trust your own proofs over my attempt below.)
| ¬□(A ⊃ B) ⊢ A | ||
| 1. .
2. .
3. .
4. .
5a. 5b.
. 6. . 7. . 8. .
9. . . | ¬□(A ⊃ B),0 . ¬A,0 ↓ 0r0 ↓ ◊¬(A ⊃ B),0 ↓ 0r1 ¬(A ⊃ B),1 ↓ 1r0 ↓ 1r1 ↓ A,1 ↓ ¬B,1
| P .
P .
1ρrD . 1¬□D .
4◊rD 4◊rD . 5aσrD . 5bρrD . 5b¬⊃D . 5b¬⊃D (open)
|
(not in Priest’s text)
For the counter model, I would guess it is the following, although this is based on the above tableau, which is likely incorrect:
W = {w1, w2}
w1Rw2, w1Rw1, w2Rw2, w2Rw1
vw1(A) = 0, vw1(B) = 0
vw2(A) = 1, vw2(B) = 0
So let us look again at the formulation in question:
¬□(A⊃B) ⊨ A
Recall how we define validity, from section 2.3.11:
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.
(Priest p.23, section 2.3.11)
So in our counter-model, we should have a case where ¬□(A⊃B) is true but A is false. In our counter-model, we have set A as false in world 1. Let us see then if ¬□(A⊃B) is true in world 1. The semantic evaluation for negation is:
vw(¬A) = 1 if vw(A) = 0, and 0 otherwise.
So we next need to know the value of
□(A⊃B)
in world 1. The semantic evaluation for necessity is:
vw(□A) = 1 if, for all w′ ∈ W such that wRw′, vw′(A) = 1; and 0 otherwise.
So □(A⊃B) is true in world 1 if in world 1 and 2 (A⊃B) is true, and it is false if (A⊃B) is false in either world 1 or world 2. Now, as far as I can tell, the semantic evaluation of the conditional in modal logic was not given yet, but somehow I recall it. I just cannot find it at the moment. John Nolt, in his Logics section 11.2.1, defined it as:
v(Φ → Ψ, w) = T iff either v(Φ, w) ≠ T or v(Ψ, w) = T, or both;
v(Φ → Ψ, w) = F iff both v(Φ, w) = T and v(Ψ, w) ≠ T.
(Nolt 315, section 11.2.1)
So let us render that into Priest’s notation:
vw(A ⊃ B) = 1 if vw(A) = 0 or vw(B) = 1, and 0 otherwise.
(not in Priest that I know of or where, yet)
Recall:
vw1(A) = 0, vw1(B) = 0
vw2(A) = 1, vw2(B) = 0
So following the proposed semantics for ⊃ above, we have:
vw1(A ⊃ B) = 1
vw2(A ⊃ B) = 0
Let us now work backwards. Although vw1(A ⊃ B) = 1, still
vw1□(A ⊃ B) = 0
For, vw2(A ⊃ B) = 0, and thus it is not necessarily true in world 1. And since vw1□(A ⊃ B) = 0, that means
vw1¬□(A ⊃ B) = 1
Our original formulation was
¬□(A⊃B) ⊨ A
We see now that it is invalid, because our counter-model gives a premise that is evaluated as true with the conclusion being evaluated as false. Thus:
¬□(A⊃B) ⊭ A
and by translation:
¬(A⥽B) ⊭ A
In all, this shows how the strict conditional gives us a sense of the conditional that seems to correspond with the English conditional.]
It is easy enough to check that all the following are false in Kρστ, and so in all the normal and non-normal logics we have looked at.
B ⊨ A⥽B
¬A ⊨ A⥽B
(A ∧ B)⥽C ⊨ (A⥽C) ∨ (B⥽C)
(A⥽B) ∧ (C⥽D) ⊨ (A⥽D) ∨ (C⥽B)
¬(A⥽B) ⊨ A
But these inferences are the basis of all the objections to the material account of the conditional that we looked at in 1.7–1.9. Hence, the strict conditional is not subject to any of the objections to which the material conditional is.
(72, referencing section 1.7, section 1.8, and section 1.9)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
Or if otherwise noted:
Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.
.
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