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In his critique of Boole and the other algebraic logicians, Frege made a strong use of what is known as "the Priority Principle".

The Priority Principle expresses Frege's belief that our investigation of thoughts, should start with considering judgments (thoughts) and not concepts.

What I got is that Frege thinks that the Booleans just developed concepts upon Aristotele's logic (giving a mathematical foundation to it), without investigating if the judgements (categorical and hypothetical sentences?) really represent our thoughts. From that came his attempt to revisit what sentences really mean in our language and the consequent introduction of quantifiers...Did I get it right?

It may possibly be all wrong because as stated in the question I didn't find a general definition of what judgment and concept mean for Frege. So I tried to extrapolate them but I think I still need clarification.

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See: Kevin Klement, Frege and the Logic of Sense and Reference (2002), page 84:

Let us take a closer look at the passage from Frege’s “Notes for Ludwig Darmstädter” quoted above[*]. There, Frege was attempting to distinguish his approach to logic from the approach taken in classical logic. The logical approach of Leibniz and Boole, continuous with the Aristotelian tradition, analyzes arguments in terms of judgments, and judgments into one fixed form: subject concept – copula – predicate concept.

Here, the concepts are taken as basic and complete, and it is the copula that provides the unity of the judgment. In Frege’s approach, however, concepts are incomplete, and this displaces the need for a copula in his approach. It is really the incompleteness of concepts in particular Frege is stressing in this passage. On the reading of the Sinne of function expressions (including, of course, concept expressions) given above, the Sinn of a concept expression would be nothing other than a set of description information describing a truth-value—i.e., a Gedanke—from which a component is removed or certain information remains to be supplied. The Sinn of a concept can thus be seen as a Gedanke from which a component has been removed. It is in this sense that Gedanken are prior to the Sinne of concepts; the Sinne of concepts are Gedanken with parts removed.

Note:

[*] What is distinctive about my conception of logic is that I begin by giving pride of place to the content of the word “true”, and then immediately go on to introduce a Gedanke as that to which the question “Is it true?” is in principle applicable. So I do not begin with concepts and put them together to form a Gedanke or judgment; I come by the parts of a Gedanke by analysing the Gedanke. This marks off my Begriffsschrift from the similar inventions of Leibniz and his successors.

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