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3-1-2 Proposition

A proposition is a sentence which can be uniquely determined as either True (i.e., correct) or False (i.e., incorrect).

“I am Japanese.” ··· This is a proposition. (The statement can be determined as either true or false.) “I am pretty.” ··· This is not a proposition. (The definition of “pretty” is unclear, so this statement cannot be clearly determined as true or false.)

Propositional logic is a concept which analyzes relationships among multiple propositions and deduces new propositions (i.e., compound proposition) by other propositions with the use of join operations ( ¬: negation, →: if, then). Propositional logic is explained with the use of the following two propositions (P and Q) as examples. · 1 question

Proposition P: It is raining. Proposition Q: I am putting up an umbrella.

The following four relationships can be derived from the two propositions P and Q.

Relationship	Expression	Meaning
Implication	P→Q	If P, then Q. (If it is raining, then I am putting up an umbrella.)
Converse	Q→P	If Q, then P.

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		(If I am putting up an umbrella, then it is raining.)
Reverse	¬P→¬Q	If not P, then not Q. (If it is not raining, then I am not putting up an umbrella.)
Contraposition	¬Q→¬P	If not Q, then not P. (If I am not putting up an umbrella, then it is not raining.)

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In this example, the statement of the contraposition is the only one where the fact of the true or false matches the implication (P→Q) regardless of the true or false values of the individual propositions P and Q.

In propositional logic, the propositions themselves are not considered for analysis. For a concept of analysis of propositions as subjects and predicates, a predicate logic is used. Inference is used in both propositional logic and predicate logic. Inference is the process of extracting a new principle from an established event (i.e., fact), and has the following methods:

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The deduction panel pairs X→Y with Tama is a cat, and Y→Z with Cat says meow. Its arrow leads to X→Z and Tama says meow. The induction panel pairs X1→Y with The cat in my house says meow, and X2→Y with The neighbor's cat says meow. Its arrow leads to X→Y and Cat says meow.

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• Deduction inference If X→Y and Y→Z, then it is inferred that X→Z. This is a common concept that is known as syllogism. X→Y Tama is a cat. => X→Z Tama says meow. Y→Z Cat says meow. • Inductive Inference If X1→Y and X2→Y, then it is inferred that X {X1, X2, ···}→Y. This is a concept of generalizing an individual event. The cat in my house X1→Y says meow. => X→Y Cat says meow. The neighbor’s cat says X2→Y meow.

A relational model is a model that is devised from set theory and predicate logic. Propositional logic and predicate logic are also used in relational databases which use relational models.