Contradiction and contrariety

It seems so simple: The opposite of yes is no; of doing is not doing. However, I came to realize that matters are more complicated when I recently reread Elisabeth Anscombe’s Intention and read John Schwenkler’s guide to this book. Anscombe’s Intention is one of the most influential books in analytical philosophy. I have bought it already many years ago. It’s a little book with less than hundred pages but it is full of ideas. Anscombe was a student and friend of Wittgenstein and so it is not surprising that Intention has been written in a Wittgensteinian style. Anscombe was also one of the executors of Wittgenstein’s literary legacy. Enough reason to reread Anscombe’s book. It’s a pity that in my blogs I can discuss only some aspects. Here I want to write about contradiction and contrariety.
Take this example, also used by Anscombe: A man is operating a pump in order to replenish the water supply of a house. What is the opposite of this action? The first answer that probably comes to your mind is not pumping: The man does nothing. However, suppose that there is a hole in the pipe leading from the pump to the house. The man doesn’t know and keeps pumping (but so doing he isn’t replenishing the water supply). Can we say then that this is the same as doing nothing (not pumping) and that pumping without the intended effect is opposite to pumping with the intended effect? I think that this cannot be true.
Take now this example used by Anscombe, which she had read in a newspaper: “A certain soldier was court-martialled (or something of the sort) for insubordinate behaviour. He had, it seems, been ‘abusive’ at his medical examination. The examining doctor had told him to clench his teeth; whereupon he took them out, handed them to the doctor and said ‘You clench them’.” (§ 31). The soldier was court-martialled because he had refused an order: Not executing the order is considered here the same as refusing the order. But what if it is impossible to do what you are ordered to do? Apparently the soldier had thought (and in a sense he was right): “My teeth are false, so I cannot clench my teeth, for I don’t have them anymore.” Looking that way at the order to clench your teeth, not doing so is opposite to the order in a different way than refusing the order is.
Discussing these examples by Anscombe Schwenkler – following Aristotle – distinguished two kinds of opposition between statements: contradictories and contraries. “In Aristotelian logic”, so Schwenkler, “a pair of statements are contradictories just in case the truth of each entails the falsehood of the other and vice versa; and contraries just in case the truth of each entails the falsehood of the other, but the falsehood of neither entails the other’s truth.” ... A “contradictory pair says precisely that the other member is false. But Aristotelian contraries are opposed as well, as each member of a pair of contraries is such that its truth would entail the falsity of the other …” (pp. 109-110; italics Schwenkler), but I want to add: “though not the other way round”.
With the help of the distinction contradictory-contrary we can understand what went wrong in calling both not pumping and pumping without the intended effect as opposite to pumping, or, taking the other example, why court-martialling the soldier because he should have refused an order resulted from a misunderstanding. If the pumping man says “I am replenishing the house water supply” and you say “You aren’t, but I am going to repair the hole in the pipe”, we have a contradiction. However, if you say, “You aren’t for there is a leak in the pipe”, we have a contrariety. And the same so for the case of the soldier. The contradiction of “Clench your teeth” is “Do not clench your teeth”. That’s what the doctor thought and so he thought that the soldier simply refused to clench his teeth, and that he refused an order. However, the soldier thought: “I cannot clench them. I have no teeth. My teeth are false”, and not clenching his teeth for this reason is not contradictory but contrary to the doctor’s order. You cannot refuse what you cannot do. See here the reason of the misunderstanding between the doctor and the soldier that made that the latter was court-martialled.
There are many cases where the distinction contradiction-contrariety can be relevant. One such a case where the distinction can be meaningfully applied is telling the truth versus lying versus being silent. Another case is doing versus allowing versus doing nothing. 

Sources
- G.E.M. Anscombe, Intention. Oxford: Basil Blackwell, 1976 (my edition).
- Schwenkler, John, Anscombe’s Intention. A Guide. Oxford: Oxford University Press, 2019.