# Frege on “if”

I’ve been looking at the passage early in the Begriffsschrift where Frege introduces the material conditional — not, of course, using that label, and not of course with our notation. He notes that $$A \supset B$$ can be affirmed when A is denied or when B is affirmed, and in those cases “there need not exist a causal connection between the two contents” (the content of A and the content of B). One can also

make the judgment $$A \supset B$$ without knowing whether A and B are to be affirmed or denied. For example, let denote the circumstance that the Moon is in quadrature [with the sun] and B the circumstance that it appears as a semicircle. In this case $$A \supset B$$ can be translated with the aid of the connective ‘if’: ‘If the moon is in quadrature, then it appears as a semicircle’. The causal link implicit in the word ‘if’, however, is not expressed by our symbols, although a judgement of this kind can be made only on the basis of such a link.

That’s Michael Beaney’s translation, with notation changed: but other translations don’t differ in relevant ways. In particular, they all use the word ‘causal’ in rendering Frege’s remarks. And this is what caught my eye.

For Frege seems to be intending to make general claims here. To judge  $$A \supset B$$ we need not suppose that there is a causal connection or link between A and B, it suffices (of course) to be in a position to deny A or assert B. By contrast, however, a judgement if A then B can only be made on the basis of a causal link. And doesn’t that strike us as an odd line for him to take, given that Frege’s first interest is in the language of arithmetic and the  language of analysis, where causation doesn’t come into it? True arithmetical ‘if’s aren’t causal ‘if’s — or so many of us English-speaking analytic philosophers would be inclined to say (not least because we have read our Frege!).

We might wonder, then about the shared translation here. But the relevant German is “ursächlicher Zusammenhang” and “ursächliche Verknüpfung”; and according to the dictionary ‘ursächlich’ means ‘causal’. So it seems that the translations are right.

Though this sets me musing. In English (or at least, in my corrupted-by-philosophy English) there is something of a disconnect between ‘cause’ and ‘because’. If we have A true and this fact causes B to be true, then I am happy to say B, because A. But this doesn’t reverse: in particular, in mathematical cases where  I am happy to say something of the form B, because A, I’d usually balk at talking about causation. For example, I’m quite happy to say of a particular function that it is computable because it is primitive recursive, but would balk (wouldn’t you?) at saying that its being primitive recursive causes it to be computable.

Now I suppose English could have had the notion of becausal link, i.e. some connection or other that holds when B, because A is true (not necessarily causal in the narrow sense). And then we could imagine the view that “a becausal link is implicit in the word ‘if'” (however exactly we are to spell out ‘implicit’ here).

So that raises a question: when Frege talks about ‘if’s and causal connections, does he in fact mean anything stronger than becausal connections (assuming that a ‘because’ need not be causal ‘because’). How are things in philosophical German? Does “ursächliche Verknüpfung” definitely connote a causal as opposed to, more generally, becausal link?

### 6 thoughts on “Frege on “if””

1. I’m not at liberty to go look it up right now but I vaguely remember there was always something tricky about the meaning of vera causa, the Latin translation of a Greek term in Aristotle and others that could mean something more like true reason than our modern concept of physical cause.

1. Yes, it’s this sort of thought that was at the back of my mind, that “cause” talk has been used to cover more in the past, and so wondering how widely Frege may have meant it here.

2. Native speaker here, well Austrian, but I guess that is good enough. Concerning your question: Nowadays “ursächlicher Zusammenhang” means “causal connection”, that is pretty clear. (Actually I had more doubts about whether “Zusammehang” should be translated as “connection”, because the German term seems slightly vaguer to me.)
Here is a further note, that might be relevant: German does not have two (non-technical) terms for “if” and “when” as English does. Usually both are just “wenn”. If you think that “when” has more of a causal ring to it, this might explain why Frege discusses causality.

1. A remark about this: The word Ursache is often used, today as in the 19th century, as the English word reason, say when one explains one’s actions. Whenever causality in nature is referred to, the standard German word would be kausal, as in die kausalen Zusammenhänge der Natur (causal connections in nature). Frege’s implications come from reasoning, for the *admission* (Bejahung) of an implication is a *judgment* (Urteil), a mental act expressed by the use of the turnstile.

3. Without knowing how others had translated, I translated this “ursachliche” as “substantial connection.” Paolo Mancosu then told me that others used causal, but that he agrees about substantial as an alternative. My reason is that NO German mathematician of the 19th century would have thought that there is anything causal about connections in mathematics, and none think that way even today.

The discussion is continued with what Peano called formal implication, as with the induction clause A(x) -> A(x+1). The variable connects the antecedent and consequent, so that’s the substantial connection, certainly not causal in any reasonable sense.

The difficulty with conditionals comes directly from the logic being classical. There is no genuine implication if A -> B is just -A V B. Add to this the usual mistake of implicitly applying the disjunction property to the latter, and you make the Russellian fallacy by which A -> B doesn’t state anything because either A is false and A -> B true but useless or B is true and the addition of A as a condition is superfluous.

My translation will be “official” in a book on the development of logic and foundations that comes out in June, google The Great Formal Machinery Works.

4. My favorite passage for illustrating the difference between logical antecedents and causal or temporal antecedents comes from Warren S. McCulloch.

In 1923 I gave up the attempt to write a logic of transitive verbs and began to see what I could do with the logic of propositions. My object, as a psychologist, was to invent a kind of least psychic event, or “psychon”, that would have the following properties: First, it was to be so simple an event that it either happened or else it did not happen. Second, it was to happen only if its bound cause had happened — shades of Duns Scotus! — that is, it was to imply its temporal antecedent. Third, it was to propose this to subsequent psychons. Fourth, these were to be compounded to produce the equivalents of more complicated propositions concerning their antecedents.

Warren S. McCulloch, “What Is a Number, that a Man May Know It, and a Man, that He May Know a Number?”, Ninth Alfred Korzybski Memorial Lecture, General Semantics Bulletin, Numbers 26 and 27, Institute of General Semantics, Lakeville, CT, 1961, pp. 7–18. Reprinted in Embodiments of Mind, MIT Press, Cambridge, MA, 1965, pp. 1–18.

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