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
4.8
The Explosion of Contradictions
Brief summary:
(4.8.1) One of the paradoxes of the strict conditional is: ⊨ (A ∧ ¬A) ⥽ B. By modus ponens we derive: (A∧¬A)⊨B. In other words, contradictions entail everything (any arbitrary formula whatsoever). But this is counter-intuitive, and there are counter-examples that we will consider. (4.8.2) The first counter-example: Bohr knowingly combined inconsistent assumptions in his model of the atom, but on that account the model functioned well. However, explosion does not hold here, because we cannot on the basis of the contradiction infer everything else, like electronic orbits being rectangles. (4.8.3) The second counter-example: we can have inconsistent laws without their contradiction entailing everything. (4.8.4) The third counter-example: there are perceptual illusions that give us inconsistent impressions without giving us all impressions. For example, the waterfall illusion gives us the impression of something moving and not moving, but it does not thereby also give us every other impression whatsoever. The fourth counter-example: there can be fictional situations where contradictions hold but that thereby not all things hold as well.
[The Strict Conditional Involves the Explosion of Contradictions]
[Counter-Example 1: The Bohr Model’s Contradictory Assumptions as Non-Explosive]
[Counter-Example 2: Inconsistent Legislation]
[Counter-Example 3: Perceptual Illusions. Counter-Example 4: Fictional Situations]
Summary
[The Strict Conditional Involves the Explosion of Contradictions]
[One of the paradoxes of the strict conditional is: ⊨ (A ∧ ¬A) ⥽ B. By modus ponens we derive: (A∧¬A)⊨B. In other words, contradictions entail everything (any arbitrary formula whatsoever). But this is counter-intuitive, and there are counter-examples that we will consider.]
[Let us first recall some notions regarding the strict conditional. In section 4.5.2 and section 4.5.3 we learned that the strict conditional is defined as “□(A ⊃ B),” and it is symbolized as A⥽B. In section 4.6.2 and section 4.6.3, we learned that modal systems that can handle conditionality should be systems where modus ponens holds: A, A⥽B ⊨ B. (I did not know why exactly this is necessary, but I guessed it was for the following reason. Suppose modus ponens does not hold. That would mean by affirming the antecedent, we could not obtain the consequent. But were that the case, then we have lost a basic intuition we have about conditionality, namely, that the consequent will follow necessarily from the antecedent.) We learned in section 4.6.2 that for modus ponens to hold in a modal system, it needs the ρ-constraint (reflexivity). (Recall it from section 3.2.3: “ρ (rho), reflexivity: for all w, wRw” p.36.) But we then learned in section 4.6.3 that no matter how many other constraints we add to ρ, we will always obtain the paradoxes of strict implication, with one being: ‘⊨ (A ∧ ¬A) ⥽ B’. Now in our current section, Priest says that by modus ponens, from ⊨ (A ∧ ¬A) ⥽ B we can derive (A∧¬A)⊨B. (I do not know exactly how that works, however. I guess the idea is that if we establish the conditional, and if we have modus ponens, then that means simply from the antecedent being affirmed we can infer the consequent as a semantic consequence. The important philosophical point here is that) the strict conditional in any modal system that can handle conditionality leads us to being able to derive any arbitrary formula whatsoever from a contradiction. As Priest puts it: “Contradictions would entail everything.” But this is counter-intuitive. Priest will now give three counter-examples of situations or theories that are inconsistent but also where we should not be able thereby to infer that everything whatsoever holds.]
The toughest objections to a strict conditional, at least as an account of the indicative conditional, come from the fact that ⊨(A∧¬A)⥽B. If this were the case, then, by modus ponens, we would have (A∧¬A)⊨B. Contradictions would entail everything. Not only is this highly counterintuitive, | there would seem to be definite counter-examples to it. There appear to be a number of situations or theories which are inconsistent, yet in which it is manifestly incorrect to infer that everything holds. Here are three very different examples.
(74-75)
[Counter-Example 1: The Bohr Model’s Contradictory Assumptions as Non-Explosive]
[The first counter-example: Bohr knowingly combined inconsistent assumptions in his model of the atom, but on that account the model functioned well. However, explosion does not hold here, because we cannot on the basis of the contradiction infer everything else, like electronic orbits being rectangles.]
[I do not know much about the first example, so please see the quotation below. The basic idea is that Bohr knowingly combined two inconsistent assumptions in his model of the atom, namely, he assumes “the standard Maxwell electromagnetic equations” but also “that energy could come only in discrete packets (quanta).” Yet, despite its obvious inconsistency, both assumptions were needed for the model to work and “many of its observable predictions were spectacularly verified.” Priest’s philosophical point here is that on the basis of this contradiction, we cannot infer everything else. “Bohr did not infer, for example, that electronic orbits are rectangles” (75).]
The first is a theory in the history of science: Bohr’s theory of the atom (the ‘solar system’ model). This was internally inconsistent. To determine the behaviour of the atom, Bohr assumed the standard Maxwell electromagnetic equations. But he also assumed that energy could come only in discrete packets (quanta). These two things are inconsistent (as Bohr knew); yet both were integrally required for the account to work. The account was therefore essentially inconsistent. Yet many of its observable predictions were spectacularly verified. It is clear though that not everything was taken to follow from the account. Bohr did not infer, for example, that electronic orbits are rectangles.
(75)
[Counter-Example 2: Inconsistent Legislation]
[The second counter-example: we can have inconsistent laws without their contradiction entailing everything.]
[In Priest’s second counter-example, we have two laws that together function together non-problematically in most cases, but in a particular situation they come into contradiction. Priest then says that on the basis of this contradiction, “it would be stupid to infer from this that, for example, the traffic laws are consistent” (75). (I did not quite get how that works. Are we saying that we can consider our two inconsistent laws as presenting a structure like A∧¬A, and “the traffic laws are consistent” is some arbitrary B that we try to derive from it? At any rate, surely at least we might say that from this contradiction we cannot derive any other traffic law we want.)]
Another example: pieces of legislation are often inconsistent. To avoid irrelevant historical details, here is an hypothetical example. Suppose that an (absent-minded) state legislator passes the following traffic laws. At an unmarked junction, the priority regulations are:
(1) Any woman has priority over any man.
(2) Any older person has priority over any younger person.
(We may suppose that clause 2 was meant to resolve the case where two men or two women arrive together, but the legislator forgot to make it subordinate to clause 1.) The legislation will work perfectly happily in three out of four combinations of sex and age. But suppose that Ms X, of age 30, approaches the junction at the same time as Mr Y, of age 40. Ms X has priority (by 1), but has not got priority (by 2 and the meaning of ‘priority’). Hence, the situation is inconsistent. But, again, it would be stupid to infer from this that, for example, the traffic laws are consistent.
(75)
[Counter-Example 3: Perceptual Illusions. Counter-Example 4: Fictional Situations]
[The third counter-example: there are perceptual illusions that give us inconsistent impressions without giving us all impressions. For example, the waterfall illusion gives us the impression of something moving and not moving, but it does not thereby also give us every other impression whatsoever. The fourth counter-example: there can be fictional situations where contradictions hold but that thereby not all things hold as well.]
[The third example is that there are perceptual illusions that can give us inconsistent impressions. For example, the waterfall illusion causes us to see something both in motion and not in motion. But thereby we do not perceive everything else, like for example that everything is red all over. The fourth example is that in fictional situations where there are contradictions, that does not entail that everything holds in that fictional situation. (For some reason the fourth one is placed in a footnote, despite being an excellent and convincing counter-example.)]
Third example: it is possible to have visual illusions where things appear contradictory. For example, in the ‘waterfall effect’, one’s visual system is conditioned by constant motion of a certain kind, say a rotating spiral. If one then looks at a stationary situation, say a white wall, it appears to move in the opposite direction. But, a point in the visual field, | say at the top, does not appear to move, for example, to revolve around to the bottom. Thus, things appear to move without changing place: the perceived situation is inconsistent. But not everything perceivable holds in this situation. For example, it is not the case that the situation is red all over.5
(75-76)
5. A fourth kind of example is provided by certain fictional situations, in which contradictory states of affairs hold. This may well be the case without everything holding in the fictional situation.
(76)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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