Natural Deduction - Judgments and Propositions

Judgments and Propositions

A judgment is something that is knowable, that is, an object of knowledge. It is evident if one in fact knows it. Thus "it is raining" is a judgment, which is evident for the one who knows that it is actually raining; in this case one may readily find evidence for the judgment by looking outside the window or stepping out of the house. In mathematical logic however, evidence is often not as directly observable, but rather deduced from more basic evident judgments. The process of deduction is what constitutes a proof; in other words, a judgment is evident if one has a proof for it.

The most important judgments in logic are of the form "A is true". The letter A stands for any expression representing a proposition; the truth judgments thus require a more primitive judgment: "A is a proposition". Many other judgments have been studied; for example, "A is false" (see classical logic), "A is true at time t" (see temporal logic), "A is necessarily true" or "A is possibly true" (see modal logic), "the program M has type τ" (see programming languages and type theory), "A is achievable from the available resources" (see linear logic), and many others. To start with, we shall concern ourselves with the simplest two judgments "A is a proposition" and "A is true", abbreviated as "A prop" and "A true" respectively.

The judgment "A prop" defines the structure of valid proofs of A, which in turn defines the structure of propositions. For this reason, the inference rules for this judgment are sometimes known as formation rules. To illustrate, if we have two propositions A and B (that is, the judgments "A prop" and "B prop" are evident), then we form the compound proposition A and B, written symbolically as "". We can write this in the form of an inference rule:

This inference rule is schematic: A and B can be instantiated with any expression. The general form of an inference rule is:

where each is a judgment and the inference rule is named "name". The judgments above the line are known as premises, and those below the line are conclusions. Other common logical propositions are disjunction, negation, implication, and the logical constants truth and falsehood . Their formation rules are below.

frac{Ahbox{ prop} qquad Bhbox{ prop}}{A vee Bhbox{ prop}} vee_F
frac{Ahbox{ prop} qquad Bhbox{ prop}}{A supset Bhbox{ prop}} supset_F
frac{hbox{ }}{tophbox{ prop}} top_F
frac{hbox{ }}{bothbox{ prop}} bot_F

frac{Ahbox{ prop}}{neg Ahbox{ prop}} neg_F

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Famous quotes containing the words propositions and/or judgments:

    Neither moral relations nor the moral law can swing in vacuo. Their only habitat can be a mind which feels them; and no world composed of merely physical facts can possibly be a world to which ethical propositions apply.
    William James (1842–1910)

    The Laws of Nature are just, but terrible. There is no weak mercy in them. Cause and consequence are inseparable and inevitable. The elements have no forbearance. The fire burns, the water drowns, the air consumes, the earth buries. And perhaps it would be well for our race if the punishment of crimes against the Laws of Man were as inevitable as the punishment of crimes against the Laws of Nature—were Man as unerring in his judgments as Nature.
    Henry Wadsworth Longfellow (1807–1882)