Judgments

https://en.wikipedia.org/wiki/Judgment_(mathematical_logic)

A judgment or assertion is a statement in the metalanguage. In general, a judgment may be any inductively definable assertion in the metatheory.

For example, typical judgments in first-order logic would be that a string is a well-formed formula, or that a proposition is true. Similarly, a judgment may assert the occurrence of a free variable in an expression of the object language, or the provability of a proposition.

Judgments are used in formalizing deduction systems:

• a logical axiom expresses a judgment

• premises of a rule of inference are formed as a sequence of judgments

• their conclusion is a judgment

• thus, hypotheses and conclusions of proofs are judgments as well

A judgment is an evident object of knowledge, if one has a proof of it. In mathematical logic however, evidence is often not directly observable, but rather deduced from more basic judgments - this process of deduction is what constitutes a proof.

The most important judgments in logic are of the form "A is true", where A stands for any proposition; therefore, the truth of a judgment depends on a more primitive judgment, that which asserts that "A is a proposition".

Of course, if A is not a proposition, the logic is outta window, so we'll assume that we're dealing with freaking propositions, that are syntactically valid and obtained by translating declarative statements with a truth value from some natural language.

Besides the everlasting judgement "A is true", many other judgments have been studied, including:

• "A is false" in classical logic

• "A is true at time t" in temporal logic

• "A is necessarily true" or "A is possibly true" in modal logic

• "A is achievable from the available resources" in linear logic

• "program M has type τ" in type theory

References

https://en.wikipedia.org/wiki/Judgment_(mathematical_logic) https://en.wikipedia.org/wiki/Natural_deduction https://en.wikipedia.org/wiki/Rule_of_inference https://en.wikipedia.org/wiki/Sequent_calculus