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I'm reading Frege's "On Sense and Nominatum" and he uses "Cognitive significance", "Genuine Cognition"...

I kind of infer from the context of each paragraph what he means with those kinds of phrases but I would like to know precisely what he is speaking about.

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You can see Frege's The Thought: A Logical Inquiry (1918-19):

Without wishing to give a definition, I call a thought something for which the question of truth arises. [...] Two things must be distinguished in an indicative sentence: the content, which it has in common with the corresponding sentence-question, and the assertion. The former is the thought, or at least contains the thought.

You can see also:

as well as:

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Just as a rider to the above :

'I' presents me as (roughly) 'the speaker', and this 'mode of presentation' is, of course, very different than that associated with 'he'. The two sentences, 'I am about to be attacked' and 'He is about to be attacked' thus differ in cognitive significance. They can express the same proposition, but, even when they do so, each presents that proposition in a distinctive way, by means of a distinctive cognitive perspective. (Howard Wettstein, 'Cognitive Significance without Cognitive Content', Mind, New Series, Vol. 97, No. 385 (Jan., 1988), p.23, fn.46.)

And to ampllify 'distinctive cognitive perspective' by means of 'associated information' :

We can make the same point - that cognitive significance is not a matter of associated information - with names that ... co-refer. Someone might acquire the names 'Cicero' and 'Tully', associating with them precisely the same information, say 'a famous Roman'. Still the names may differ in 'cognitive value'. It may never strike the speaker that only one person may be in question, and so he may react very differently to sentences that contain one name, than to those that contain the other. (Wettstein, op. cit., 24-5.)

I offer this just as a small addition that may marginally help.

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We call lots of things true or false: pictures, sentences, sounds. But, as he says, we call these things true or false only insofar as they express something. This something that they express is their sense. For Frege a sentence gains "sense" when it makes sense to the person who provides the sentence.

In simple words cognitive significance means that it's my cognitive brain that gives sense to the things I experience. Hence I call them true or false.

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