Proposition and Judgment

Should we not consider propositions to be judgments, that is to say, claims with truth-values dependent on states-of-affairs ? Then the claim that a proposition is a claim is true. The original meaning of $\vdash P$ is the claim (or judgment) that $P$.  Let the predicate $C$ signify 'being a claim'. Thus $C(\vdash P)$ or rather $\vdash C(\vdash P)$ and hence $C(\vdash C(\vdash P))$.

Truth and judgment are primitive concepts.

There are judgments, for instance the one above.

The judgement that a judgment is a judgment is true.

Hence: there are true judgments and knowledge is possible.

After I wrote this I discovered that Bolzano had already written in the Theory of Science, Theory of Fundamentals §40:

During the time that a person is such a complete sceptic, he cannot form any judgement, no matter what its content. For a judgement is nothing but a proposition which the subject, with more or less confidence, takes to be true. If anybody forms a judgement, he gives us to understand that there is at least one proposition which he takes to be true (with more or less confidence) at the moment when he judges; he believes (with more or less confidence) that there is at least one truth; hence he is not to be called a complete sceptic. Consequently, in order to heal a sceptic, all we have to do is to get him to make a judgement whose truth is so irresistible to him that he tries in vain to doubt it when, a moment later, we call his attention to the fact that in this judgement he has recognised at least one truth.

Husserl, in the Logical Investigations,  distinguishes between judgment and proposition. Propositions correspond to states of affairs.  If we want to deal with propositional attitudes, then there must be a predicate for claiming, for the judgmental aspect discussed above.  Rather than A believes that P or A knows that P we should investigate the more basic A claims that P.  Frege's symbol $\vdash P$ involves an unspoken hidden subject, the subject claiming that P. Of course we can distinguish between explicit manifest claims and implicit claims such as bound-to-hold-that, forced-to-hold-that, etc. Otherwise how can we interpret logical rules of the form: if it is claimed that X then it follows that it is claimed that Y. Thus $\vdash$ should mean something like 'claimable', 'assertible' - the Stoic lekton !

Identity involves synthesis of the manifold via relations, the using of relations in a manifold for identification.  But one may want to distinguish again these elements fused together by identification: fine-grainedness. The act of synthesis presupposes a perception of difference. The elements fused together might themselves be fusions. Thus every category is a quotient category of quotient categories.

There is more than one absolute certainty. Likewise what objection is there that there should be more than one path or proof for arriving at the knowledge that there is absolute certainty ? I.e. both the Leibniz-Bolzano path and the Husserlian path are valid. 

There is much confusion regarding the nature of formal logic and the terms 'formal' and 'formalization'. To us a 'formal logic' is above all a language. A language different from English and other natural languages, but a different language is no other sense.  Formal logic aims at being more syntactically and semantically explicit, unambiguous and detailed than ordinary language. But it is still a language. Our concept of 'formal logic'  might be more properly called a Leibniz-inspired universal formal philosophical language. Here 'formal' thus pertains to its structure rather than its fundamental linguistic nature.

Truth is a relationship between thought and reality and as such both thinkable and a part of reality itself.