Suppose that some contradictions are true – for example, that as I walk through the door, I’m inside and I’m not inside. Then we argue 'if I'm walking through the door, I'm inside; I'm not inside; therefore, I'm not walking through the door' is an invalid instance of modus tollens

Blumson, Ben

Helke, Theresa

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DIALETHEISM AND MODUS TOLLENSBEN BLUMSON AND THERESA HELKEAccording to dialetheists, some contradictions are true. In other words,according to dialetheists there is a sentence A such that it and its negation¬A are both true. Dialetheists have argued for many examples of truecontradictions. But for the sake of illustration, we will employ a purportedexample of a true contradiction arising from motion, given by Graham Priest(2006: In Contradiction, Oxford University Press, 161), who writes: “I amin a room. As I walk through the door, am I in the room or out of (not in)it?” Priest argues that the answer to this question is a true contradiction –as I walk through the door, I’m in the room, but also not in the room.Modus tollendo tollens, or modus tollens for short, is the argument formA → B,¬B ∴ ¬A, in which the major premise is a conditional, the minorpremise the negation of its consequent, and the conclusion the negation ofits antecedent. For example, the argument ‘if it’s raining, then it’s cloudy;it’s not cloudy; therefore, it’s not raining’ is an instance of modus tollenswhere the major premise is the conditional ‘if it’s raining, then it’s cloudy’,the minor premise is ‘it’s not cloudy’, the negation of the consequent, andthe conclusion is ‘it’s not raining’, the negation of the antecedent.Date: May 4, 2022.12 BEN BLUMSON AND THERESA HELKEWe argue that if dialetheism is true, then modus tollens is invalid. To seewhy, suppose that both ‘I’m in the room’ and its negation ‘I’m not in theroom’ are true, but that ‘I’m walking through the door’ is true while ‘I’mnot walking through the door’ is not. Then consider the following instanceof modus tollens:(1) If I’m walking through the door, I am in the room.(2) I am not in the room.(3) ∴ I’m not walking through the door.By hypothesis, the second premise is true while the conclusion is not. In thescenario described, the first premise is intuitively true as well. But if so, wehave an instance of modus tollens with true premises and a false conclusion,so modus tollens is invalid.Whereas we think the instance of modus tollens described above has truepremises and a false conclusion, we don’t have similar intuitions about anycorresponding instance of modus ponens (the argument form A → B,A ∴B). Consider, for example, the following instance of modus ponens:(1) If I am in the room, I’m in bed.(2) I am in the room.(3) ∴ I’m in bed.If ‘I’m in the room’ is both true and false, but ‘I’m in bed’ is just false,then ‘if I am in the room, I’m in bed’ is just false. Unlike the correspondinginstance of modus tollens, this instance of modus ponens is intuitively valid.DIALETHEISM AND MODUS TOLLENS 3The issue is notable for at least two reasons. First, many putatively in-valid instances of modus tollens have been given even in the classical context.Most of these putative counterexamples involve an embedded conditional ormodal in the consequent of the major premise, which is putatively negatedin the minor premise (for an overview, see Theresa Helke (2018: On Con-ditionals, PhD Thesis, National University of Singapore)). But the logicalcomplexity of the embedded conditional or modal makes it arguable whetherthese examples are truly instances of modus tollens. In contrast, the examplewe gave above is a perfectly straightforward instance of modus tollens.Secondly, it’s often observed that “one person’s modus ponens is another’smodus tollens”. In the context of classical logic, the validity of modus po-nens and modus tollens stand and fall together (as recently emphasised inthis journal by Lina Lissia (2020: ‘On some analogies between the counterex-amples to modus ponens (and modus tollens)’, The Reasoner, 35-7)). Butwe have just seen that in the context of dialetheism, the validity of modusponens and modus tollens can come apart – some instances of modus tollens,but not of modus ponens, are intuitively invalid.11We thank Jay Garfield for discussion of this paper.

Dialetheism and Modus Tollens

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Dialetheism and Modus Tollens

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