Elementary Propositional Logic

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 Ÿ∙)
  • 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: (PQ)&(QP). The biconditional is also written iff.

Antecedent: the if-clause of a conditional proposition (i.e. P in PQ).

Consequent: the then-clause of a conditional proposition (i.e. Q in PQ).


Example of symbolic reduction:

“If I light the curtains, the house burns down. I light the curtains. Therefore the house burns down.”


If P then Q. P. Therefore Q.


PQ, P, Q


Modus ponens: A valid argument that affirms the antecedent to derive the consequent:

PQ, P, Q


Modus tollens: A valid argument that denies the consequent to derive the negation of the antecedent:

PQ, ~Q, ~P


Fallacy of Affirming the Consequent: an invalid argument that affirms the consequent to mistakenly derive the antecedent:

PQ, Q, P


Fallacy of Denying the Antecedent: an invalid argument that denies the antecedent to mistakenly derive the negation of the consequent:

PQ, ~P, ~Q


©MMXVI Peter Sjöstedt-H