Symbolic Logic and Logical Connectives
Why symbolise?
Symbolic logic replaces ordinary language with compact notation so that form becomes visible and proofs can be tested mechanically. Statement letters (P, Q, R) stand for whole simple propositions; the connectives ~, •, ∨, ⊃, ≡ correspond to not, and, or, if–then, and if and only if.
Translation rules
- Not both P and Q: ~(P • Q), not ~P • ~Q.
- Neither P nor Q: ~(P ∨ Q), not ~P ∨ ~Q.
- P unless Q: ~Q ⊃ P (equivalently P ∨ Q).
- P only if Q: P ⊃ Q. P if Q: Q ⊃ P.
- P if and only if Q: P ≡ Q.
Well-formed formulas and scope
Brackets fix the scope of each operator; the main operator has the widest scope and determines the form of the whole formula. ~(P ⊃ Q) differs from ~P ⊃ Q: in the first, negation governs the conditional; in the second, the conditional is the main operator.
Worked example
“You will pass only if you work hard” → P ⊃ W. The direction is fixed by “only if”: working hard is necessary, not sufficient. If the sentence were “You will pass if you work hard,” it would be W ⊃ P.
Traps that cost marks
Reversing “only if”; translating “neither…nor” as ~P ∨ ~Q; dropping brackets in P ⊃ Q ∨ R, which changes the main operator. Statement letters stand for simple propositions, never for terms or classes.