General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

Iϕ(love(mary, bill) ∨ love(john, mary)) = T but Iϕ(love(john, mary)) = F .
c○: Michael Kohlhase 203
Obviously, the tableau above is saturated, but not closed, so it is not a tableau proof for our initial
entailment conjecture. We have marked the literals on the open branch green, since they allow us
to read of the conditions of the situation, in which the entailment fails to hold. As we intuitively
argued above, this is the situation, where Mary loves Bill. In particular, the open branch gives us
a variable assignment (marked in green) that satisfies the initial formula. In this case, Mary loves
Bill, which is a situation, where the entailment fails.
Practical Enhancements for Tableaux
Propositional Identities
Definition 340 Let ⊤ and ⊥ be new logical constants with I(⊤) = T and I(⊥) = F for
all assignments ϕ.
We have to following identities:
Name for ∧ for ∨
Idenpotence ϕ ∧ ϕ = ϕ ϕ ∨ ϕ = ϕ
Identity ϕ ∧ ⊤ = ϕ ϕ ∨ ⊥ = ϕ
Absorption I ϕ ∧ ⊥ = ⊥ ϕ ∨ ⊤ = ⊤
Commutativity ϕ ∧ ψ = ψ ∧ ϕ ϕ ∨ ψ = ψ ∨ ϕ
Associativity ϕ ∧ (ψ ∧ θ) = (ϕ ∧ ψ) ∧ θ ϕ ∨ (ψ ∨ θ) = (ϕ ∨ ψ) ∨ θ
Distributivity ϕ ∧ (ψ ∨ θ) = ϕ ∧ ψ ∨ ϕ ∧ θ ϕ ∨ ψ ∧ θ = (ϕ ∨ ψ) ∧ (ϕ ∨ θ)
Absorption II ϕ ∧ (ϕ ∨ θ) = ϕ ϕ ∨ ϕ ∧ θ = ϕ
De Morgan’s Laws ¬(ϕ ∧ ψ) = ¬ϕ ∨ ¬ψ ¬(ϕ ∨ ψ) = ¬ϕ ∧ ¬ψ
Double negation ¬¬ϕ = ϕ
Definitions ϕ ⇒ ψ = ¬ϕ ∨ ψ ϕ ⇔ ψ = (ϕ ⇒ ψ) ∧ (ψ ⇒ ϕ)
c○: Michael Kohlhase 204
We have seen in the examples above that while it is possible to get by with only the connectives
∨ and ¬, it is a bit unnatural and tedious, since we need to eliminate the other connectives first.
In this section, we will make the calculus less frugal by adding rules for the other connectives,
without losing the advantage of dealing with a small calculus, which is good making statements
about the calculus.
The main idea is to add the new rules as derived rules, i.e. inference rules that only abbreviate
deductions in the original calculus. Generally, adding derived inference rules does not change the
derivability relation of the calculus, and is therefore a safe thing to do. In particular, we will add
the following rules to our tableau system.
We will convince ourselves that the first rule is a derived rule, and leave the other ones as an
exercise.
Derived Rules of Inference
Definition 341 Let C be a calculus, a rule of inference A1 . . . An
is called a derived
C
inference rule in C, iff there is a C-proof of A1, . . . , An ⊢ C.
Definition 342 We have the following derived rules of inference
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