# Failure of the deduction theorem in natural language?

It seems as though the deduction theorem can fail in natural language, if we think of a valid inference as one that preserves certainty, rather than truth. What I mean is that if we are certain of (for instance) "If a German wins, it will be Schmitt," we will also be certain of "If the winner isn't Schmitt, it won't be a German"; but in general we're not obliged to be certain of "Either it's not the case that if a German wins, it will be Schmitt, or else if the winner isn't Schmitt, it won't be a German." (That is the material conditional corresponding to the inference.)

My question is whether this has been discussed in the logic literature. I haven't yet found anything in work by Ernest Adams and Vann McGee, but I haven't looked at everything by them, and of course the literature on the logic of conditionals is massive. Wondering if this rings any bells.

I would be happy to discuss the claim of deduction theorem failure on the merits, but my main question is whether it has already been discussed in print.

EDIT to clarify: By the deduction theorem, $\ X \vdash Y$ implies $\; \vdash X \supset Y$, where $\supset$ is the material conditional, so that $\ X \supset\ Y$ can be rewritten as $\; \sim X \; \vee\ Y$. Here, I am supposing that the logical consequence relation $\vdash$ tracks with a semantic relation, $\vDash$, such that $\ X \vDash\ Y$ means that whenever $\ X$ is certain, so is $\ Y$. That is what I mean by defining valid inference in terms of certainty-preservation.

For sentences containing the natural-language conditional connective $\rightarrow$, which is not the material conditional, certainty can be defined numerically (the details are in the writings of Adams and McGee), but the point of my question is that we have enough of an intuitive sense of it to make judgments about it in simple natural-language examples such as the one I gave. In that example, $\ X$ is instantiated as $\ A \rightarrow B$, where $\ A$ is "A German wins" and $\ B$ is "Schmitt wins," and $\ Y$ is instantiated as the contrapositive of $\ X$.

• Your examples are concerned with the law of excluded middle rather than the deduction theorem. The statement that "the deduction theorem fails in natural language" seems to me to be meaningless. – Rob Arthan Jun 17 '18 at 21:43
• You have tagged this with probability, but there is a key difference between the logical contrapositive and the two statements If a German wins, it will probably be Schmitt and If the winner isn't Schmitt, it probably won't be a German – Henry Jun 17 '18 at 22:21
• @Henry: The reason I tagged it with probability is that one leading theory (Adams's) is that the acceptability of a simple indicative conditional sentence is directly related to the corresponding conditional probability. Well, if P(B | A) = 1, then P(~A | ~B) = 1. On the other hand, it's possible for P(B | A) to be high, arbitrarily close to 1 but not equal, and yet for P(~A | ~B) to be quite low. So the inference form called contraposition preserves certainty but not likelihood (using Adams's terminology). That's why I tagged it as a probability topic. (Conditional-probability is not a tag.) – StumpyLeg Jun 17 '18 at 23:36
• @Mauro ALLEGRANZA - Thank you for the question. I edited my original question to clarify. – StumpyLeg Jun 18 '18 at 15:58
• As per your editing above, the issue is the same with $\supset$ and $\lor$ : if we are "certain" about $X$ and about the fact that $X$ implies $Y$ - in a natural language context - we are licensed to assert $Y$. But we simply do not assert $\lnot X \lor Y$... and neither $Y \lor$ whatever. – Mauro ALLEGRANZA Jun 18 '18 at 17:00

## 2 Answers

The deduction theorem says that "for any formulas $\phi$ and $\psi$, if we can prove $\phi$ using $\psi$ as an assumption, then we can prove $\psi \Rightarrow \phi$". This has no useful translation into natural language of the sort you are looking for. The deduction theorem is a statement about proof systems and not just languages.

• OK, let my try to understand your position. Possibly your view is that natural language is a wholly separate domain from formal logic, so that talking about natural-language reasoning as if it were a formal system is misconceived right from the outset. Alternatively, possibly your view is that natural-language reasoning has certain formal properties in common with formal logic, but there's a point where they diverge, and the deduction theorem lies beyond that point. Does either of those describe your position? – StumpyLeg Jun 22 '18 at 4:28
• Probably both of the above $\ddot{\smile}$. The second position seems reasonable if one thinks about mathematical reasoning using natural language: informally, to prove "if $\phi$ then $\psi$", you assume $\phi$ and try to prove $\psi$. However, that allows you to prove that if the moon is made of green cheese then the sky is blue, which would likely be rejected by most people, leading to the first position. – Rob Arthan Jun 22 '18 at 11:48
• Let's hit pause on the moon/sky thing and consider logic vs natural language generically. My thought is (1) there are natural-language inferences (Socrates is a man, all men… etc.). And (2) there are statements we don't need premises for, like "Either there are mortal men or there are not." Finally, (3) natural language can form Boolean compounds using "not" and "or." Okay, so ... (4) for a given natural language inference from A to B, we can ask whether "either not-A or else B" is one of those statements we don't need premises for. Do you disagree with any of this? – StumpyLeg Jun 23 '18 at 6:15

Assuming that the other connectives behave classically, then it follows from the deduction theorem (sometimes called conditional proof in this context) and modus ponens that the conditional is material (see, for example, Hanson (1991) who uses this point to argue that the conditional in English is material).

So most work on non-material conditionals give up the deduction theorem (or rarely, modus ponens). The exception is relevant logics, which keep both (a version of) the deduction theorem and modus ponens, but in which the other connectives behave non-classically too (see Mares (1998) for an introduction).

References

Hanson, William (1991) "The Indicative Conditional is Truth-Functional" Mind 100(1): 53-72.

Mares, Edwin (1998) "Relevance Logic" The Stanford Encyclopedia of Philosophy https://plato.stanford.edu/entries/logic-relevance/

• I upvoted this reply because it gave a reference that sounds useful. However, a couple points: 1. Not having seen the Hanson piece yet, I don’t understand the comment about CP and MP implying that the conditional is material. The strict conditional isn’t material, but it supports CP and MP, right? 2. I’m not talking about failure of CP for a non-material conditional. I’m talking about failure of CP for the material conditional, in the presence of a non-material conditional, when inference is defined in terms of certainty-preservation. – StumpyLeg Dec 20 '18 at 7:17