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Elementary Logic Truth-table tests

I. Truth-tables and propositions:

A. If, for a given proposition, it’s impossible to construct a row of a truth-table in which that proposition is true, then the proposition is a contradiction.

- Thus if you want to determine whether a proposition is a (self-) contradiction (in propositional logic), try to construct a row of a truth-table in which that proposition is true. If this proves impossible, the proposition is a contradiction.

If, for a given proposition, it’s impossible to construct a row of a truth-table in which that proposition is false, then the proposition is a tautology.

- Thus if you want to determine whether a proposition is a tautology (in propositional logic), try to construct a row of a truth-table in which that proposition is false. If this proves impossible, then the proposition is a tautology.

II. Using truth-tables to test arguments for validity:

Construct a conditional statement whose antecedent is the conjunction of all the premises, and whose consequent is the conclusion.

If the whole conditional statement is a tautology (i.e., it’s impossible for it to be false), then the argument is valid.

Thus a good strategy to use is to try to construct a single row of a truth-table in which the whole conditional statement is false – i.e., in which the antecedent (= all the premises) is true but the consequent (= the conclusion) is false:

- If it’s impossible to construct such a row, then the argument is valid (in propositional logic).

- If it’s possible to construct such a row, then the argument is invalid (in propositional logic).

III. Using truth-tables to test a set of statements for consistency:

Construct a sentence that is the conjunction of all the statements in question.

If the whole conjunction is a contradiction (i.e., it’s impossible for all the conjuncts to be true together), then the statements are inconsistent.

Thus a good strategy to use is to try to construct a single row of a truth-table in which all the conjuncts are true.

- If it’s impossible to construct such a row, then the statements are inconsistent.

- If it’s possible to construct such a row, then the statements are consistent in propositional logic.

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