Introduction to TFL
Capitals P, Q, R, S, …, Z : These represent atomic sentences (i.e. sentences not containing any connectives). (P … Z are called ‘propositional variables’. We will use A, B, C … as ‘meta-variables’.)
Symbols ¬, ∧, ∨, →, biconditional
These represent the five truth-functional connectives: ¬ negation (‘It is not the case that…’) ∧ conjunction (‘Both… and …’) ∨ disjunction (‘Either… or …’) → conditional (‘If … then …’) biconditional (‘… if and only if …’)
Parentheses ( , )
We use ( , ) for grouping.
Truth Table connectives: Negation
The characteristic truth-table of negation, 𝑨 ¬𝑨
T F
F T
Truth Table connectives: Conjunction:
The characteristic truth-table of conjunction,
𝑨 𝑩 𝑨 ∧ 𝑩
T T T
T F F
F T F
F F F
Truth Table connectives: Disjunction:
The characteristic truth-table of (inclusive) disjunction,
𝑨 𝑩 𝑨 ∨ 𝑩
T T T
T F T
F T T
F F F
Truth Table connectives: Conditional:
The characteristic truth-table of the material conditional,
𝑨 𝑩 𝑨 → 𝑩
T T T
T F F
F T T
F F T
Truth Table connectives: Biconditional:
The characteristic truth-table of the material conditional, 𝑨 biconditional B
𝑨 𝑩 𝑨 biconditional 𝑩
T T T
T F F
“If it either rains or snows, then there will be neither a picnic nor a football game.”
(R∨ S ) → ¬(P∨ G )
The antecedent is clearly a disjunction:
R∨ S The consequent is a negated disjunction: ¬(P∨ G )
A but B (TFL)
A ∧ B
Neither A nor B (TFL)
¬ A ∧ ¬ B ¬ ( A ∨ B )
Not both A and B (TFL)
¬( A ∧ B) ¬ A ∨ ¬ B
A if B (TFL)
B → A
A only if B (TFL)
A → B
‘A unless B ’: (TFL)
¬ B → A A ∨ B
A is a sufficient condition for B (TFL)
A→ B
A is a necessary condition for B (TFL)
B→ A
