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On Sophistical Refutations   

the expression' means in this doctrine 'not to be directed against the

thought'. For if not all are directed against either expression or

thought, there will be certain other arguments directed neither

against the expression nor against the thought, whereas they say

that all must be one or the other, and divide them all as directed

either against the expression or against the thought, while others

(they say) there are none. But in point of fact those that depend on

mere expression are only a branch of those syllogisms that depend on a

multiplicity of meanings. For the absurd statement has actually been

made that the description 'dependent on mere expression' describes all

the arguments that depend on language: whereas some of these are

fallacies not because the answerer adopts a particular attitude

towards them, but because the argument itself involves the asking of a

question such as bears more than one meaning.

It is, too, altogether absurd to discuss Refutation without first

discussing Proof: for a refutation is a proof, so that one ought to

discuss proof as well before describing false refutation: for a

refutation of that kind is a merely apparent proof of the

contradictory of a thesis. Accordingly, the reason of the falsity will

be either in the proof or in the contradiction (for mention of the

'contradiction' must be added), while sometimes it is in both, if

the refutation be merely apparent. In the argument that speaking of

the silent is possible it lies in the contradiction, not in the proof;

in the argument that one can give what one does not possess, it lies

in both; in the proof that Homer's poem is a figure through its

being a cycle it lies in the proof. An argument that does not fail

in either respect is a true proof.

But, to return to the point whence our argument digressed, are

mathematical reasonings directed against the thought, or not? And if

any one thinks 'triangle' to be a word with many meanings, and granted

it in some different sense from the figure which was proved to contain

two right angles, has the questioner here directed his argument

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