by Corry Shores
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[Gottlob Frege, Entry Directory]
[Frege, Begriffsschrift, Chapter 1, Entry Directory]
[The following is summary. Bracketed commentary is my own. Please forgive my typos, as proofreading is incomplete.]
In the expression for a judgment, the complex symbol to the right of ⊢ may always be regarded as a function of one of the symbols that occur in it. Let us replace this argument with a Gothic letter, and insert a concavity in the content-stroke, and make this same Gothic letter stand over the concavity: e.g.:This signifies the judgment that the function is a fact whatever we take its argument to be. A letter used as a functional symbol, like Φ in Φ(A), may itself be regarded as the argument of a function; accordingly, it may be replaced by a Gothic letter, used in the sense I have just specified. The only restrictions imposed on the meaning of a Gothic letter are the obvious ones: (i) that the complex of symbols following a content-stroke must still remain a possible content of judgment (§2); (ii) that if the Gothic letter occurs as a functional symbol, account must be taken of this circumstance. All further conditions imposed upon the allowable substitutions for a Gothic letter must be made part of the judgment. From such a judgment, therefore, we can always deduce any number we like of judgments with less general content, by substituting something different each time for the Gothic letter; when this is done, the concavity in the content-stroke vanishes again. The horizontal stroke that occurs to the left of the concavity inis the content-stroke for [the proposition] that Φ(ɑ) holds good whatever is substituted for ɑ; the stroke occurring to the right of p. 2o] the concavity is the content-stroke of Φ(ɑ) – we must here imagine something definite substituted for ɑ.(Frege 16)
By what was said before about the meaning of the judgment-stroke, it is easy to see what an expression like| means.(16-17)
This expression may occur as part of a judgment, as in(17)
It is obvious that from these judgments we cannot infer less general judgments by substituting something definite for ɑ, as we can from[seems to end the sentence here.]serves to deny that Χ(ɑ) is always a fact whatever we substitute for ɑ. But this does not in any way deny the possibility of giving ɑ some meaning Δ such that X(Δ) is a fact.means the case in whichis affirmed and A denied does not occur. But this does not in any way deny the occurrence of the case in which X(Δ) is affirmed and A denied; for, as we have just seen, X(Δ) may be affirmed and neverthelessdenied. Thus here likewise, we cannot make an arbitrary substitution for a without prejudice to the truth of the judgment.(17)
This explains why we need the concavity with the Gothic letter written on it; it delimits the scope of the generality signified by the letter. A Gothic letter retains a fixed meaning only within its scope; the same Gothic letter may occur within various scopes in the same judgment, and the meaning we may ascribe to it in one scope does not extend to any other scope. The scope of one Gothic letter may include that of another, as is shown in p. 21]In this case different letters must be chosen; we could not replace e by a. It is naturally legitimate to replace a Gothic letter everywhere in its scope by some other definite letter, provided that there are still different letters standing where different letters | stood before. This has no effect on the content. Other substitutions are permissible only if the concavity directly follows the judgment stroke, so that the scope of the Gothic letter is constituted by the content of the whole judgment. Since this is a specially important case, I shall introduce the following abbreviation: an italic letter is always to have as its scope the content of the whole judgment, and thus scope is not marked out by a concavity in the content stroke. If an italic letter occurs in an expression not preceded by a judgment stroke, the expression is senseless.(Frege 18)
An italic letter may always be replaced by a Gothic letter that does not yet occur in the judgment; in this case the concavity must be inserted immediately after the judgment-stroke. E.g. forwe may putsince a occurs only in the argument-position within X(a).Likewise it is obvious that fromwe may deduceif A is an expression in which a does not occur, and a occupies only argument-positions in Φ(a). Ifis denied, we must be able to specify a meaning for a such that Φ(a) is denied. Thus if
were denied and A affirmed, we should have to be able to specify a meaning for a such that A was affirmed and Φ(a) denied.
(18)
But since we havewe cannot do so; for this formula means that whatever a may be the case in which Φ(a) would be denied and A affirmed does not | occur. Hence we likewise cannot both deny
and affirm A: i.e.... Similarly when we have several conditional strokes.(Frege 18-19)
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