by Corry Shores
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[The following is summary. All boldface, underlying and bracketed commentary are my own.]
Begriffsschrift, Chapter 1
When we have a conditional, if we affirm that the antecedent is true, we can thereby conclude that the consequent is true. All inferences can – and in this text will – be formulated using this conditional structure.
[Recall what Frege said in the prior section about conditional statements. He made this diagram to notate their structure.
This represents “B implies A”. Thus if we assert B, we may infer A.] Suppose we have the following two judgments:
We see that we can infer from these judgments the new judgment.
[Frege then has us recall the four possible cases for the conditional in question (quoting from §5):
If A and B stand for possible contents of judgment (§ 2), we have the four following possibilities:
(i) A affirmed, B affirmed;
(ii) A affirmed, B denied;
(iii) A denied, B affirmed;
(iv) A denied, B denied.
Now I will quote from the current text, and I will discuss it after:]
Of the four cases enumerated above, the third is excluded by
and the second and fourth by:
so that only the first remains....
[The third is, “A denied, B affirmed”. In other words, the antecedent is affirmed but the consequent is denied. This of course is invalid. The second and fourth are: “A affirmed, B denied,” and “A denied, B denied”. As you can see, in both cases, B (the antecedent) is denied. Hence we see why Frege claims above that these cases are excluded by ⊢B; for, ⊢B affirms B rather than denies it as in those cases. This means that if we begin with “B implies A” and “B”, then we are left with the first option, where the affirmation of the antecedent B is coupled with the affirmation of the consequent A.]
[I am not certain about the following point. (By the way, the ellipses in this translation are for parts that have been excluded, so this is just the next point in this translation, not in the original). It seems that Frege is saying that although there are many sorts of inferences that do not use the conditional form, they all can be reduced to the conditional. You will have to read and judge the text for yourself, which I will quote. He has us think of two propositions that serve as antecedents to some consequent. So perhaps the idea is that we regard all inferences to have the basic structure of the conditional, where the premises are the antecedents and the conclusion is the consequent. Let me quote:]
Following Aristotle, logicians enumerate a whole series of kinds of inference; I use just this one – at any rate in all cases where a new judgment is derived from more than one single judgment. For the truth implicit in another form of inference can be expressed in a judgment of the form: if M holds and N holds then Λ holds also; symbolically,
From this judgment, and ⊢N, and ⊢M, ⊢Λ will then follow as above. An inference, of whatever kind, may he reduced to our case in this way. Accordingly it is possible to get along with a single form of inference; and therefore perspicuity demands that we should do so. Moreover, if we did not there would be no reason to confine ourselves to the Aristotelian forms of inference; we could go on adding new forms indefinitely.... This restriction to a single form of inference is however in no way meant to express a psychological proposition; we are just settling a question of formulation, with a view to the greatest convenience for our purpose....
(7, turnstile symbols have much longer horizontal stems in the original)
Frege, Gottlob. “Begriffsschrift (Chapter 1)”. Transl. P.T. Geach. In Translations from the Philosophical Writings of Gottlob Frege. Eds. P.T. Geach and Max Black. Oxford: Basil Blackwell, 1960, second edition (1952 first edition).