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6.11
The propositions of logic therefore say nothing. (They are the analytical
propositions.) |
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6.111
Theories which make a proposition of logic appear substantial are always
false. Once could e.g. believe that the words "true" and
"false" signify two properties among other properties, and then
it woud appear as a remarkable fact that every proposition possesses one
of these properties. This now by no means appears self-evident, no more so
than the proposition "All roses are either yellow or red" would
seem even if it were true. Indeed our proposition now gets quite the
character of a proposition of natural science and this is a certain
symptom of its being falsely understood. |
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6.112
The correct explanation of logical propositions must givem them a peculiar
position among all propositions |
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6.113 It
is the characteristic mark of logical propositions that one can perceive
in the symbol alone that they are true; and this fact contains in itself
the whole philosophy of logic. And so also it is one of the most important
facts that the truth or falsehood of non-logical propositions can not
be recognized from the propositions alone. |
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6.12
The fact that the
propositions of logic are tautologies shows the formal -- logical
-- properties of language, of the world.
That its
constituent parts connected together in this way give a tautology
characterizes the logic of its constituent parts. |
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6.1201
That e.g. the propositions "p" and "~p"
in the connexion "~p . ~p" give a
tautology shows that they contradict one another. That the propositions. |
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6.1202 It
is clear that we could have used for this purpose contradictions instead
of tautologies |
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6.1203
In order to
recognize a tautology as such, we can, in cases in which no sign of
generality occurs in the tautology, make use of the following intuitive
method: I write instead of "p", "q",
"r, etc., "TpF", "TqF",
"TrF", etc. The truth-combinations I express by brackets,
e.g.:
and the co-ordination of the truth or falsity of the whole proposition
with the truth-combinations of the truth-arguments by lines in the
following way:
This
sign, for example, would therefore present the proposition p q now
I will proceed to inquire. The form "~ " is written in our
notation the form " " thus: - -
Hence
the proposition ~(p . ~q) runs thus :--
whether
such a proposition as ~(p . ~p) (The Law of
Contradiction) is a tautology
If
here we put "p" instead of "q" and
examine the combination of the outermost T and F with the innermost, it is
seen that the truth of the whole proposition is co-ordinated with all
the truth-combinations of its argument, its falsity with none of the
truth-combinaation. |
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6.121 (end)
The propositions of
logic demonstrate the logical properties of propositions, by combining
them into propositions which say nothing.
This
method could be called a zero-method. In a logical proposition
propositions are brought into equilibrium with one another, and the state
of equilibrium then shows how these propositions must be logically
constructed |
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In
order that propositions connected together in a definite way may give a
tautology they must have definite properties of structure. That they give
a tautology when so connected shows therefore that they possess
these properties of structure. |
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6.122
Whence it follows that we can get on without logical propositions, for we
can recognize in an adequate notation the formal properties of the
propositions by mere inspection.
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6.123
It is clear that
the laws of logic cannot themselves obey further logical laws.
(There is not, as
Russell supposed, for every "type" a special law of
contradiction; but one is sufficient, since it is not applied to itself.)
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6.124 (end)
The logical
propositions describe the scaffolding of the world, or rather they present
it. They "treat" of nothing. They presuppose that names have
meaning, and that elementary propositions have sense. And this is their
connexion with the world. It is clear that it must show something about
the world that certain combinations of symbols -- which essentially have a
definite character -- are tautologies. Herein lies the decisive point. We
said that in the symbols which we use something is arbitrary, something
not. In logic only this expresses: but this means that in logic it is not we
who express, by means of signs, what we want, but in logic the nature of
the essentially necessary signs itself asserts. That is to say, if we know
the logical syntax of any sign language, then all the propositions of
logic are already given.
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6.125 It
is possible, also with the old conception of logic, to give at the outset
a description of all "true" logical propositions |
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6.126
Whether a
proposition belongs to logic can be calculated by calculating the logical
properties of the symbol.
And this we do when
we prove a logical proposition. For without troubling ourselves about a
sense and a meaning, we form the logical propositions out of others by
mere symbolic rules.
We prove a logical
proposition by creating it out of other logical propositions by applying
in succession certain operations, which again generate tautologies out of
the first. (And from a tautology only tautologies follow.)
Naturally
this way of showing that its propositions are tautologies is quite
unessential to logic. Because the propositions, from which the proof
starts, must show without proof that they are tautologies. |
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6.127 All
propositions of logic are of equal rank; there are not some which are
essentially primitive and others deduced from there.
Naturally this way of showing that its propositions are tautologies
is quite unessential to logic. Because the propositions, from which the
proof starts, must show without proof that they are tautologies.
Every
tautology itself shows that it is a tautology. |
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6.13 (end)
Logic
is not a theory but a reflexion of the world.
Logic is trascendental. |
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