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

e quite difficult to establish in general), then the solutions of the program will be correct, and
we will find all possible ones.
Properties of Calculi (Theoretical Logic)
Correctness: (provable implies valid)
H ⊢ B implies H |= B (equivalent: ⊢ A implies |=A)
Completeness: (valid implies provable)
H |= B implies H ⊢ B (equivalent: |=A implies ⊢ A)
Goal: ⊢ A iff |=A (provability and validity coincide)
To TRUTH through PROOF (CALCULEMUS [Leibniz ∼1680])
c○: Michael Kohlhase 181
Of course, the logics we have studied so far are very simple, and not able to express interesting
facts about the world, but we will study them as a simple example of the fundamental problem of
Computer Science: How do the formal representations correlate with the real world.
Within the world of logics, one can derive new propositions (the conclusions, here: Socrates is
mortal) from given ones (the premises, here: Every human is mortal and Sokrates is human).
Such derivations are proofs.
Logics can describe the internal structure of real-life facts; e.g. individual things, actions, properties.
A famous example, which is in fact as old as it appears, is illustrated in the slide below.
If a logic is correct, the conclusions one can prove are true (= hold in the real world) whenever
the premises are true. This is a miraculous fact (think about it!)
The miracle of logics
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