variables or conjunctions: F(a) G(a, b) H(a, b, c) Atomic formulae These wffs are atomic formulae, but are not sentences (atomic or otherwise) because they include free variables: F(x) G(a, z) H(x, y, z) Compound sentences These wffs are compound sentences. They are sentences, but are not atomic sentences because they are not atomic formulae: ∀x (F(x)) ∃z (G(a, z)) ∃x ∀y ∃z (H(x, y, z)) ∀x ∃z (F(x) ∧ G(a, z)) ∃x ∀y ∃z (G(a, z) ∨ H(x, y, z)) Compound formulae These wffs are compound formulae. They are not atomic formulae but are built up from atomic formulae using logical connectives. They are also not sentences because they contain free variables: F(x) ∧ G(a, z) G(a, z) ∨ H(x, y, z) Interpretations A sentence is either true or false under an interpretation which assigns values to the logical variables. We might for example make the following assignments: Individual constants a: Socrates b: Plato c: Aristotle Predicates Fα: α is sleeping Gαβ: α hates β Hαβγ: α made β hit γ Sentential variables p: It is raining. Under this interpretation the sentences discussed above would represent the following English statements: p: "It is raining." F(a): "Socrates is sleeping." H(b, a, c): "Plato made Socrates hit Aristotle." ∀x (F(x)): "Everybody is sleeping." ∃z (G(a, z)): "Socrates hates somebody." ∃x ∀y ∃z (H(x, y, z)): "Somebody made everybody hit somebody." (They may not have all hit the same person z, but they all did so because of the same person x.) ∀x ∃z (F(x) ∧ G(a, z)): "Everybody is sleeping and Socrates hates somebody." ∃x ∀y ∃z (G(a, z) ∨ H(x, y, z)): "Either Socrates hates somebody or somebody made everybody hit somebody." Translating sentences from a natural language into an artificial language Sentences in natural languages can be ambiguous, whereas the languages of the sentential logic and predicate logics are precise. Translation can reveal such ambiguities and express precisely the intended meaning. For example, take the English sentence “Father Ted married Jack and Jill”. Does this mean Jack married Jill? In translating we might make the following assignments: Individual Constants a: Father Ted b: Jack c: Jill Predicates: Mαβγ: α officiated at the marriage of β to γ Using these assignments the sentence above could be translated as follows: M(a, b, c): Father Ted officiated at the marriage of Jack and Jill. ∃x ∃y (M(a, b, x) ∧ M(a, c,