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Notes Dictated to G.E. Moore in Norway: Difference between revisions
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There are ''internal'' relations between one proposition and another; but a proposition cannot have to another ''the'' internal relation which a ''name'' has to the proposition of which it is a constituent, and which ought to be meant by saying it "occurs" in it. In this sense one proposition can't "occur" in another. | There are ''internal'' relations between one proposition and another; but a proposition cannot have to another ''the'' internal relation which a ''name'' has to the proposition of which it is a constituent, and which ought to be meant by saying it "occurs" in it. In this sense one proposition can't "occur" in another. | ||
''Internal'' relations are relations between types, which can't be expressed in propositions, but are all shewn in the symbols themselves, and can be exhibited systematically in tautologies. Why we come to<references /> | ''Internal'' relations are relations between types, which can't be expressed in propositions, but are all shewn in the symbols themselves, and can be exhibited systematically in tautologies. Why we come to call them "relations" is because logical propositions have an analogous relation to them, to that which properly relational propositions have to relations. | ||
Propositions can have many different internal relations to one another. ''The'' one which entitles us to deduce one from another is that if, say, they are φa and φa ⊃ ψa, then φa . φa ⊃ ψa : ⊃ : ψa is a tautology. | |||
The symbol of identity expresses the internal relation between a function and its argument: i.e. φa = (∃x) . φx . x = a. | |||
The proposition (∃x) . φx . x = a : ≡ : φa can be seen to be a tautology, if one expresses the ''conditions'' of the truth of (∃x) . φx . x = a, successively, e.g., by saying: This is true ''if'' so and so; and this again is true ''if'' so and so, etc., for (∃x) . φx . x = a; and then also for φa. To express the matter in this way is itself a cumbrous notation, of which the ab-notation is a neater translation. | |||
What symbolizes in a symbol, is that which is common to all the symbols which could in accordance with the rules of logic = syntactical rules for manipulation of symbols, be substituted for it. [''Cf.'' 3.344.] | |||
The question whether a proposition has sense (''Sinn'') can never depend on the ''truth'' of another proposition about a constituent of the first. E.g., the question whether (x) x = x has meaning (''Sinn'') can't depend on the question whether (∃x) x = x is ''true.'' It doesn't describe reality at all, and deals therefore solely with symbols; and it says that | |||
they must ''symbolize'', but not ''what'' they symbolize. | |||
It's obvious that the dots and brackets are symbols, and obvious that they haven't any ''independent'' meaning. You must, therefore, in order to introduce so-called "logical constants" properly, introduce the general notion of ''all possible'' combinations of them = the general | |||
form of a proposition. You thus introduce both ab-functions, identity, | |||
and universality (the three fundamental constants) simultaneously. | |||
The ''variable proposition'' p ⊃ p is not identical with the ''variable proposition'' ~(p . ~p). The corresponding universals ''would'' be identical. The variable proposition ~(p . ~p) shews that out of ~ (p.q) you get a tautology by substituting ~p for q, whereas the other does not shew this. | |||
It's very important to realize that when you have two different relations (a,b)R, (c,d)S this does ''not'' establish a correlation between a and c, and b and d, or a and d, and b and c: there is no correlation whatsoever thus established. Of course, in the case of two pairs of terms united by the ''same'' relation, there is a correlation. This shews that the theory which held that a relational fact contained the terms and relations united by a ''copula'' () is untrue; for if this were so there would be a correspondence between the terms of different relations. | |||
The question arises how can one proposition (or function) occur in another proposition? The proposition or function itself can't possibly stand in relation to the other symbols. For this reason we must introduce functions as well as names at once in our general form of a<references /> | |||