Atomic Sentence - Interpretations

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."