Two propositions are opposed
to one another if there is no proposition with a sense, that affirms them
both. Every proposition that contradicts another negate it.
5.13 When the truth of one proposition follows from the truth of others, we
can see this from the structure of the proposition.
5.131 If the truth of one proposition follows from the truth of others,
this finds expression in relations in which the forms of the propositions
stand to one another: nor is it necessary for us to set up these relations
between them, by combining them with one another in a single proposition;
on the contrary, the relations are internal, and their existence is an
immediate result of the existence of the propositions.
5.1311 When we infer q from p C q and Pp, the relation between the
propositional forms of 'p C q' and 'Pp' is masked, in this case, by our
mode of signifying. But if instead of 'p C q' we write, for example, 'p|q .
| . p|q', and instead of 'Pp', 'p|p' (p|q = neither p nor q), then the
inner connexion becomes obvious. (The possibility of inference from (x) .
fx to fa shows that the symbol (x) . fx itself has generality in it.)
5.132 If p follows from q, I can make an inference from q to p, deduce p
from q. The nature of the inference can be gathered only from the two
propositions. They themselves are the only possible justification of the
inference.
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