In mathematical logic, a judgment (or judgement) or assertion is a statement or enunciation in the metalanguage. 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. In general, a judgment may be any inductively definable assertion in the metatheory.

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, and their conclusion is a judgment as well (thus, hypotheses and conclusions of proofs are judgments). A characteristic feature of the variants of Hilbert-style deduction systems is that the context is not changed in any of their rules of inference, while both natural deduction and sequent calculus contain some context-changing rules. Thus, if we are interested only in the derivability of tautologies, not hypothetical judgments, then we can formalize the Hilbert-style deduction system in such a way that its rules of inference contain only judgments of a rather simple form. The same cannot be done with the other two deductions systems: as context is changed in some of their rules of inferences, they cannot be formalized so that hypothetical judgments could be avoided—not even if we want to use them just for proving derivability of tautologies.

This basic diversity among the various calculi allows such difference, that the same basic thought (e.g. deduction theorem) must be proven as a metatheorem in Hilbert-style deduction system, while it can be declared explicitly as a rule of inference in natural deduction.

In type theory, some analogous notions are used as in mathematical logic (giving rise to connections between the two fields, e.g. Curry-Howard correspondence). The abstraction in the notion of judgment in mathematical logic can be exploited also in foundation of type theory as well.

Logical assertion
In logic, logical assertion is a statement that asserts that a certain premise is true, and is useful for statements in proof. It is equivalent to a sequent with an empty antecedent.

For example, if p = “x is even”, the implication


is thus true. We can also write this using the logical assertion symbol, as


In computer programming and programming language semantics, these are used in the form of assertions; one example is a loop invariant.

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Meanings outside the classical logic
In philosophical logic, the term “judgment” is used instead of the concept “statement”, which is reduced to the logical formal. Correspondingly Aristoteles to Immanuel Kant find divisions of judgments according to categories in a panel of judgment. Kant distinguishes in particular between analytic and synthetic judgments, which relate (a posteriori) to experience or are made prior to all experience (a priori).

Romanticism and German Idealism reject an analytical decomposition into parts as a priority method and give absolute priority to the coherent, unified whole of knowledge, feeling and faith. Friedrich Hölderlin writes in judgment and being that the parts are given their essential purpose by the judgment, but defends itself against the interpretation that the parts such as workpieces could be regarded separately from each other. Novalis notes in his General Brouillon: “One not only wants the sentence or the judgment, but also the acts to do so.”

For the judgment theory of neo-Kantianism, every judgment is affirmative or negative, and consequently implies an opinion on the value of truth, which is why even in the sphere of knowledge one could speak of valuations.

Judgment in the sense of logic can mean something different:

an assertion or statement;
the “final context of a syllogism ” or the “member of a syllogism”;
a conceptual connection or separation or an act of knowledge in the sense of Kant
According to Husserl, the word “judgment” can mean:

the truthfulness;

psychologically language ontological
Judgment (as a mental act) Declarative sentence (sentence)
  • Thought (Frege);
  • Facts (Husserl, formerly Wittgenstein);
  • Proposition (Anglo-Saxon philosophy);
  • Statement (Anglo-Saxon philosophy)
Table according to Tugendhat

If Ernst Tugendhat roughly distinguishes a basic psychological, linguistic and ontological conception of logic, the word has three very different basic meanings (though they are in an analogous context of meaning). What one understands by judgment, therefore, depends on the particular cognitive and conceptual theory.

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