Proposition: a sentence that makes an assertion, a claim.
Premise: a proposition that conditions a conclusion.
Syllogism: an argument with two premises leading to a conclusion.
Valid Argument: one in which the conclusion follows from the premises.
Sound Argument: one which is valid with true premises.
- Note that all sound arguments are valid, though not all valid arguments are sound.
Invalid Argument: one in which the conclusion does not follow from the premises – a fallacy.
Propositional Variable: a clause within a proposition, or the proposition alone (symbolized by a capital letter usually starting from P).
Logical Operators: the logical relations amongst propositional variables:
- The Conditional (If P then Q): →
- The Conjunct (P and Q): & (or ∧, or ∙)
- The Disjunct (P or Q): ∨ (not mutually exclusive)
- The Negation (not-P): ~
- The Biconditional: (If and only if P, then Q): ↔
- Note the Biconditional expresses a sufficient (if) and necessary (only if) condition: (P→Q)&(Q→P). The biconditional is also written iff.
Antecedent: the if-clause of a conditional proposition (i.e. P in P→Q).
Consequent: the then-clause of a conditional proposition (i.e. Q in P→Q).
Example of symbolic reduction:
“If I light the curtains, the house burns down. I light the curtains. Therefore the house burns down.”
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If P then Q. P. Therefore Q.
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P→Q, P, ∴Q
Modus ponens: A valid argument that affirms the antecedent to derive the consequent:
P→Q, P, ∴Q
Modus tollens: A valid argument that denies the consequent to derive the negation of the antecedent:
P→Q, ~Q, ∴~P
Fallacy of Affirming the Consequent: an invalid argument that affirms the consequent to mistakenly derive the antecedent:
P→Q, Q, ∴P
Fallacy of Denying the Antecedent: an invalid argument that denies the antecedent to mistakenly derive the negation of the consequent:
P→Q, ~P, ∴~Q
©MMXVI Peter Sjöstedt-H