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

instrumental to the simplification of arithmetic mentioned above. When the arithmetical calculi
were sufficiently well-understood and in principle a mechanical procedure, and when the art of
clock-making was mature enough to design and build mechanical devices of an appropriate kind,
the invention of calculating machines for arithmetic by Wilhelm Schickard (1623), Blaise Pascal
(1642), and Gottfried Wilhelm Leibniz (1671) was only a natural consequence.
We will see that it is not only possible to calculate with numbers, but also with representations
of statements about the world (propositions). For this, we will use an extremely simple example;
a fragment of propositional logic (we restrict ourselves to only one logical connective) and a small
calculus that gives us a set of rules how to manipulate formulae.
A simple System: Prop. Logic with Hilbert-Calculus
Formulae: built from prop. variables: P, Q, R, . . . and implication: ⇒
Semantics: Iϕ(P ) = ϕ(P ) and Iϕ(A ⇒ B) = T, iff Iϕ(A) = F or Iϕ(B) = T.
K := P ⇒ Q ⇒ P , S := (P ⇒ Q ⇒ R) ⇒ (P ⇒ Q) ⇒ P ⇒ R
A ⇒ B A
MP
B
A
[B/X](A) Subst
Let us look at a H 0 theorem (with a proof)
C ⇒ C (Tertium non datur)
Proof:
P.1 (C ⇒ (C ⇒ C) ⇒ C) ⇒ (C ⇒ C ⇒ C) ⇒ C ⇒ C
(S with [C/P ], [C ⇒ C/Q], [C/R])
P.2 C ⇒ (C ⇒ C) ⇒ C (K with [C/P ], [C ⇒ C/Q])
P.3 (C ⇒ C ⇒ C) ⇒ C ⇒ C (MP on P.1 and P.2)
P.4 C ⇒ C ⇒ C (K with [C/P ], [C/Q])
P.5 C ⇒ C (MP on P.3 and P.4)
P.6 We have shown that ∅ ⊢ H 0 C ⇒ C (i.e. C ⇒ C is a theorem) (is is also valid?)
c○: Michael Kohlhase 178
This is indeed a very simple logic, that with all of the parts that are necessary:
• A formal language: expressions built up from variables and implications.
• A semantics: given by the obvious interpretation function
• A calculus: given by the two axioms and the two inference rules.
The calculus gives us a set of rules with which we can derive new formulae from old ones. The
axioms are very simple rules, they allow us to derive these two formulae in any situation. The
inference rules are slightly more complicated: we read the formulae above the horizontal line as
assumptions and the (single) formula below as the conclusion. An inference rule allows us to derive
the conclusion, if we have already derived the assumptions.
Now, we can use these inference rules to perform a proof. A proof is a sequence of formulae that
can be derived from each other. The representation of the proof in the slide is slightly compactified
to fit onto the slide: We will make it more explicit here. We first start out by deriving the formula
(P ⇒ Q ⇒ R) ⇒ (P ⇒ Q) ⇒ P ⇒ R (2.1)
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