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How do the symbols for conjunction, disjunction, negation, and implication represent logical operations?
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Modern logic
Representations of the universal affirmative, “All A's are B's” in modern logic.
In formal logic, logical operators are used to form new propositions from existing ones. Their function is truth-functional, meaning the truth value of the resulting compound proposition is determined solely by the truth values of its arguments.1
The standard symbols represent these operations as follows:
- Negation (∼): A monadic operator that reverses the truth value of a proposition. If p is true, ∼p is false; if p is false, ∼p is true.1
- Conjunction (· or ∧): A dyadic operator that joins two propositions. The conjunction p · q is true only when both p and q are true; otherwise, it is false.1
- Disjunction (∨): Represents the assertion that at least one of the two propositions is true. The inclusive disjunction p ∨ q is false only when both p and q are false.12
- Implication (⊃ or →): Represents a conditional relationship ("if p, then q"). Material implication is false only when the antecedent (p) is true and the consequent (q) is false; it is true in all other cases.31