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
4.6
The Paradoxes of Strict Implication
Brief summary:
(4.6.1) We wonder if the definition of the strict conditional – A⥽B is defined as □(A⊃B) – is adequate. But first we need to address the matter of its variance under different systems of modal logic. (4.6.2) To model conditionality in general and the strict conditional in particular, we need modus ponens to hold, as it is a basic inferential principle that should hold when the conditional has its normal semantics. But in systems without the ρ-constraint (reflexivity), modus ponens will fail. Thus our system at least needs the ρ-constraint . (4.6.3) We need not narrow our systems down any further than systems with the ρ-constraint, because no matter what, they will all lead to the paradoxes of strict implication: ‘□B ⊨ A ⥽ B’, ‘¬◊A ⊨ A ⥽ B’; and also ‘⊨ A ⥽ (B ∨ ¬B) ’, ‘⊨ (A ∧ ¬A) ⥽ B’.
[The Question of the Adequacy of the Strict Conditional ⥽]
[The Need for the ρ-Constraint]
[The Paradoxes of Strict Implication]
Summary
[The Question of the Adequacy of the Strict Conditional ⥽]
[We wonder if the definition of the strict conditional – A⥽B is defined as □(A⊃B) – is adequate. But first we need to address the matter of its variance under different systems of modal logic.]
[Recall from section 4.5 the notion of the strict conditional. In section 4.5.3 we learn that the strict conditional, symbolized as ⥽, is defined in the following way: A⥽B is defined as □(A⊃B). (p.72, section 4.5.3). We now ask if this definition of the conditional is adequate. But as the properties of the strict conditional will vary according to the modal logic system at hand, we need to say more on this matter.]
Does it provide an adequate account of the conditional? Each system of modal logic gives ⥽ different properties. Hence, before we can answer that question, we need to address the question of which system of modal logic it is that is at issue. Let me make two comments on this.
(72)
[The Need for the ρ-Constraint]
[To model conditionality in general and the strict conditional in particular, we need modus ponens to hold, as it is a basic inferential principle that should hold when the conditional has its normal semantics. But in systems without the ρ-constraint (reflexivity), modus ponens will fail. Thus our system at least needs the ρ-constraint .]
[Priest first notes that modus ponens fails in systems without the ρ-constraint (the reflexivity constraint; see section 3.2.3). It seems that for conditionality we would want modus ponens to hold: A, A⥽B ⊨ B. I do not know the exact reason why, but it would seem that conditionality should allow us to infer the consequent from an affirmation of the antecedent. For otherwise, what is the sense of the conditional without that also holding? But, Priest says, in systems without the reflexivity constraint, modus ponens will not hold. Thus we at least need the reflexivity constraint.]
First, it is natural to suppose that any notion of necessity that is to be employed in defining a notion of conditionality must be at least as strong as Kρ (or Lρ if one is countenancing non-normal systems). This is because, without ρ, modus ponens fails: A, A⥽B ⊭ B. With it, it holds, as simple tableau tests verify.
(73)
[The Paradoxes of Strict Implication]
[We need not narrow our systems down any further than systems with the ρ-constraint, because no matter what, they will all lead to the paradoxes of strict implication: ‘□B ⊨ A ⥽ B’, ‘¬◊A ⊨ A ⥽ B’; and also ‘⊨ A ⥽ (B ∨ ¬B) ’, ‘⊨ (A ∧ ¬A) ⥽ B’.]
[Priest’s next point I might not summarize properly, but I am guessing it is the following. So far we specified that for the strict conditional we need systems with the reflexivity constraint. But we learn now that we need not narrow our systems down any further, because no matter how constrained we make them, all systems will lead to certain paradoxes. And it is these paradoxes that lead us to question the notion that the strict conditional models the English conditional.]
Second, a further determination of this question is not very important for what follows. This is because the major objections to the claim that English conditionals are strict hinge on a feature that the strict conditional possesses in all systems of modal logic. In all systems of modal logic the following hold:
□B ⊨ A ⥽ B
¬◊A ⊨ A ⥽ B
These facts are sometimes called the ‘paradoxes of strict implication’. A tableau test verifies that these hold in L, and so in all the normal and non-normal systems that we have looked at. Since, in all systems, we also have ⊨□(B∨¬B) and ⊨¬◊(A∧¬A), this gives us as special cases:
⊨ A ⥽ (B ∨ ¬B)
⊨ (A ∧ ¬A) ⥽ B
(73)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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