propositional

Itto be consistentyou memorize 17 x in the set a

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Unformatted text preview: t semantics of the language Propf is given by assigning ttruth values to t t t each proposition in Prop. Clearly, an arbitrary assignment of truth values is You should very carefully inspect everything is critical that with the meaning not interesting, since we would likethis table! Itto be consistentyou memorize 17 X in the set {∧, ∨, ⊕, →, ↔} a binary function MX : B × B → B. The truth tables of these connectives are as follows: P f f t t Semantics Q M∧ (P, Q) M∨ (P, Q) M⊕ (P, Q) M→ (P, Q) M↔ (P, Q) f f f f t t t f t t t f f f t t f f t t t f t t You should very carefully inspect this table! It is critical that you memorize and fully understand the meaning of each connective. The semantics of the language Prop is given by assigning truth values to each proposition in Prop. Clearly, an arbitrary assignment of truth values is not interesting, since we would like everything to be consistent with the meaning of the connectives that we have just learned. For example, if the propositions a and b have been assigned the value t, then it is reasonable to insist that a ∧ b be assigned the value t as well. Therefore, we will introduce the concept of a valuation, which models the semantics of Prop in an appropriate way. A valuation v : Prop → B is a function that assigns a truth value to each proposition in Prop such that V1. v ￿¬a￿ = M¬ (v ￿a￿) V2. v ￿(a ∧ b)￿ = M∧ (v ￿a￿ , v ￿b￿) V3. v ￿(a ∨ b)￿ = M∨ (v ￿a￿ , v ￿b￿) V4. v ￿(a ⊕ b)￿ = M⊕ (v ￿a￿ , v ￿b￿) V5. v ￿(a → b)￿ = M→ (v ￿a￿ , v ￿b￿) V6. v ￿(a ↔ b)￿ = M↔ (v ￿a￿ , v ￿b￿) holds for all propositions a and b in 18 Prop. The properties V1–V6 ensure and fully understand the meaning of each connective. The semantics of the language Prop is given by assigning truth values to each proposition in Prop. Clearly, an arbitrary assignment of truth values is not interesting, since we would like everything to be consistent with the meaning of the connectives that we have just learned. For example, if the propositions a and b have been assigned the value t, then it is reasonable to insist that a ∧ b be assigned the value t as well. Therefore, we will introduce the concept of a valuation, which models the semantics of Prop in an appropriate way. A valuation v : Prop → B is a function that assigns a truth value to each proposition in Prop such that Valuations V1. V2. V3. V4. V5. V6. v ￿¬a￿ = M¬ (v ￿a￿) v ￿(a ∧ b)￿ = M∧ (v ￿a￿ , v ￿b￿) v ￿(a ∨ b)￿ = M∨ (v ￿a￿ , v ￿b￿) v ￿(a ⊕ b)￿ = M⊕ (v ￿a￿ , v ￿b￿) v ￿(a → b)￿ = M→ (v ￿a￿ , v ￿b￿) v ￿(a ↔ b)￿ = M↔ (v ￿a￿ , v ￿b￿) holds for all propositions a and b in Prop. The properties V1–V6 ensure that the valuation respects the meaning of the connectives. We can restrict a valuation v to a subset of the set of proposition. If A and B are subsets of Prop such that A ⊆ B...
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