Frege was the father of modern mathematical logic. In 1879 he published Concept-notation, a work that included a formal language which was able to express generality through the quantifier-variable notation. This work also set forth a version of second order quantificational logic whcih he used to develop a logical definition for the ancestral of a relation. He was incredibly influential on the later works of Wittgenstein, Russell, George Boole, and Ernst Schroeder.

TRUTH "When entering upon the study of a science, we need to have some idea, if only a provisional one, of its nature. We want to have in sight a goal to strive towards; we want some point to aim at that will guide our steps in the right direction. The word `true' can be used to indicate such a goal for logic, just as can `good' for ethics and `beautiful' for aesthetics. Of course all the sciences have truth as their goal, but logic is concerned with the predicate `true' in a quite special way, namely in a way analogous to that in which physics has to do with the predicates `heavy' and `warm' or chemistry with the predicates `acid' and `alkaline'. There is, however, the difference that these sciences have to take into account other properties besides these we have mentioned, and that there is no one property by which their nature is so completely characterized as logic is by the word `true'. (...)

Now it would be futile to employ a definition in order to make it clearer what is to be understood by `true'. If, for example, we wished to say `an idea is true if it agrees with reality' nothing would have been achieved, since in order to apply this definition we should have to decide whether some idea or other did agree with reality. Thus we should have to presuppose the very thing that is being defined. The same would hold of any definition of the form `A is true if and only if it has such-and-such properties or stands in such-and-such a relation to such-and-such a thing'. In each case in hand it would always come back to the question whether it is true that A has suchand-such properties, or stands in such-and-such a relation to such-and-such a thing. Truth is obviously something so primitive and simple that it is not possible to reduce it to anything still simpler. Consequently we have no alternative but to bring out the peculiarity of our predicate by comparing it with others. What, in the first place, distinguishes it from all other predicates is that predicating it is always included in predicating anything whatever."

From: Gottlob Frege - Logic (1897) - in: Posthumous Writings - Edited by Hans Hermes, Friedrich Kambartel, Friedrich Kaulbach - Chicago, The University of Chicago Press

1979 pp. 128-129.

7. What true is, I hold to be indefinable.

8. The expression in language for a thought is a sentence. We also speak in an extended sense of the truth of a sentence.

12. Logic only becomes possible with the conviction that there is a difference between truth and untruth.

13. We justify a judgement either by going back to truths that have been recognized already or without having recourse to other judgements. Only the first case, inference,

is the concern of Logic.

14. The theory of concepts and of judgement is only preparatory to the theory of inference.

15. The task of logic is to set up laws according to which a judgement is justified by others, irrespective of whether these are themselves true.

16. Following the laws of logic can guarantee the truth of a judgement only insofar as our original grounds for making it, reside in judgements that are true.

17. No psychological investigation can justify the laws of logic."

From: Gottlob Frege - [17 Key sentences on Logic] (1906 or earlier) - in: Posthumous Writings - Edited by Hans Hermes, Friedrich Kambartel, Friedrich Kaulbach - Chicago, The University of Chicago Press 1979 pp. 174-17