Assertion and Information in Frege's Logic

João Schmidt

Frege famously held, in the Grundlagen der Arithmetik (FREGE, 1953), that logic can be informationally augmentative. Precisely, he argued how analytic judgements can be ampliative. Given that Frege endorsed the existence of ampliative analytical judgements, he thus endorsed the non-triviality of his logical system, viz., the concept-script (FREGE, 1967 and 2013). This was an important point for logicism as he envisage it: if arithmetic is a more developed logic, and if arithmetic is a proper science as he defended, then logic must embrace discoveries as well. This give rises to the problem on how can logic be informative if such information plays no significant role in deduction. Hintikka (1970) called this as the “scandal of deduction”: the fact that, although being essentially tautological, no satisfactory account has been given for the highly informational capabilities of logical truths.

Another important aspect of fregean logicism is it's realistic approach towards logical truth and propositional content. He envisaged a contentful logical language suitable for deriving and describing the contentful science of arithmetic. But his logic has an important non-descriptive, i.e. normative, sign: the judgement-stroke . Such sign does not denote a truth-value, or any object whatsoever, but indicate an act: the act of asserting, or judging, something as true. This raised the objection that Frege’s judgement stroke is only psychologically relevant[1]. But in fact, in Frege’s eyes, this shows the proper essence of logic: judging. It is noteworthy that Frege does not have, or presuppose, a metatheory in his logical system, as Van Heijenoort (1967) famously noted. In this scenario, there is no room to consider the truth-predicate inside the language without some circularity, and for that reason, Frege takes that for a proposition (or fregean Thought) to be true, it must be asserted as such. Thus, the assertoric force associated with such an act is what properly answer to the essence of logic[2].

In this talk, I aim to link these two important aspects of Frege’s logicism: the informativity of logic and it’s normative character. The main idea is the following: Frege’s logic, being a second-order system, is undecidable by nature, meaning that no decidable/computable procedure can be given to offer the set of its validities. It is this condition that marks logic as informative[3]. Since logic is both an human activity and an objective science, it must be voided of psychological conditions. The judgement-stroke, therefore, helps labeling the validity of a deductive chain in every point, avoiding one to wrongly take something to be true without a corresponding proof. The chain is a valid proof-procedure for the given proposition, and one that no mechanical procedure could assuredly find. With this in mind, I argue that Frege’s judgement-stroke plays a significant role in marking logic as informative.

[1] This objection is well restated, and equally answered, by Pedriali (2017).

[2] This is better explained by Greimann (2014).

[3] This remark was, to my knowledge, first made by Landini (2012).