Symbolic Logic and Logical Form (basic)
Symbolisation
Symbolic logic replaces ordinary language with variables, connectives, predicates, and quantifiers so that form becomes visible. Propositional logic uses sentence letters. Predicate logic adds individual variables, predicates, and quantifiers: ∀x means “for every x”; ∃x means “there exists an x.” Scope determines which quantifier binds which variable.
Categorical forms in symbols
- All S are P: ∀x(Sx → Px).
- No S are P: ∀x(Sx → ¬Px).
- Some S are P: ∃x(Sx ∧ Px).
- Some S are not P: ∃x(Sx ∧ ¬Px).
The A and E forms use a conditional because they say of any thing that if it is S, then it is P or not P. The I and O forms use conjunction because they assert that something exists and has both features.
Worked example
“All students are curious” becomes ∀x(Sx → Cx). It does not assert that students exist. “Some students are curious” becomes ∃x(Sx ∧ Cx); this does assert existence. The difference explains why the first can be true even if there are no students.
Traps that cost marks
- Symbolising “All S are P” as ∀x(Sx ∧ Px), which wrongly asserts everything is S.
- Symbolising “Some S are P” as ∃x(Sx → Px), which is too weak.
- Confusing universal with existential quantifiers.
- Ignoring quantifier scope and variable binding.
- Treating ordinary-language “some” as “not all.”