Notes Dictated to G.E. Moore in Norway: Difference between revisions

E.g., take ''φ''a, ''φ''a ⊃ ''ψ''a, ''ψ''a. By merely looking at these three, I can see that 3 follows from 1 and 2; i.e. I can see what is called the truth of a logical proposition, namely, of [the] proposition ''φ''a . ''φ''a ⊃ ''ψ''a : ⊃ : ''ψ''a. But this is ''not'' a proposition; but by seeing that it is a tautology I can see what I already saw by looking at the three propositions: the difference is that I ''now'' see {{small caps|that}} it is a tautology. <!--[''Cf''. 6.1221.]-->

(1) Take ''φ''x. We want to explain the meaning of 'In "''φ''x" a ''thing'' symbolizes'. The analysis is:—

:(∃y) . y symbolizes . y = "x" . "''φ''x"

["x" is the name of y: "''φ''x" = '"''φ''" is at [the] left of "x"' and ''says'' ''φ''x.]

(N.B. In [the] expression (∃y) . ''φ''y, one ''is'' apt to say this means "There is a ''thing'' such that...". But in fact we should say "There is a y, such that..."; the fact that the y symbolizes expressing what we mean.)

(2) The point can here be brought out as follows. Take ''φ''a and ''φ''A: and ask what is meant by saying, "There is a thing in ''φ''a, and a complex in ''φ''A"?

:(1) means: (∃x) . ''φ''x . x = a

:(2) means: (∃x, ''ψξ'') . ''φ''A = ''ψ''x . ''φ''x.<!--<ref>''ξ'' is Frege's mark of an ''Argumentstelle'', to show that ''ψ'' is a ''Funktionsbuchstabe''. [''Edd''.]</ref>-->

The reason why ~x is meaningless, is simply that we have given no meaning to the symbol ~''ξ''. I.e. whereas ''φ''x and ''φ''p look as if they were of the same type, they are not so because in order to give a meaning to ~x you would have to have some ''property'' ~''ξ''. What symbolizes in ''φξ'' is ''that'' ''φ'' stands to the left of ''a'' proper name and obviously this is not so in ~p. What is common to all propositions in which the name of a property (to speak loosely) occurs is that this name stands to the left of a ''name-form''.

''φ'' cannot possibly stand to the left of (or in any other relation to) the symbol of a property. For the symbol of a property, e.g., ''ψ''x is ''that'' ''ψ'' stands to the left of a name form, and another symbol ''φ'' cannot possibly stand to the left of such a ''fact'': if it could, we should have an illogical language, which is impossible.

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 ''φ''a and ''φ''a ⊃ ''ψ''a, then ''φ''a . ''φ''a ⊃ ''ψ''a : ⊃ : ''ψ''a is a tautology.

The symbol of identity expresses the internal relation between a function and its argument: i.e. ''φ''a = (∃x) . ''φ''x . x = a.

The proposition (∃x) . ''φ''x . x = a : ≡ : ''φ''a can be seen to be a tau­tology, if one expresses the ''conditions'' of the truth of (∃x) . ''φ''x . 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) . ''φ''x . x = a; and then also for ''φ''a. To express the matter in this way is itself a cumbrous notation, of which the ab-notation is a neater translation.

Logical functions all presuppose one another. Just as we can see ~p has no sense, if p has none; so we can also say p has none if ~p has none. The case is quite different with ''φ''a, and a; since here a has a meaning independently of ''φ''a, though ''φ''a presupposes it.