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

Example 309 A ⊢ B ⇒ A
Derivations and Proofs
Ax
A ⇒ B ⇒ A A ⇒E
B ⇒ A
c○: Michael Kohlhase 179
Definition 310 A derivation of a formula C from a set H of hypotheses (write H ⊢ C) is
a sequence A1, . . . , Am of formulae, such that
Am = C (derivation culminates in C)
for all (1 ≤ i ≤ m), either Ai ∈ H (hypothesis)
or there is an inference rule Al1 · · · Alk
, where lj < i for all j ≤ k.
Ai
Example 311 In the propositional calculus of
natural deduction we have A ⊢ B ⇒ A: the
sequence is A ⇒ B ⇒ A, A, B ⇒ A
Ax
A ⇒ B ⇒ A A ⇒E
B ⇒ A
Observation 312 Let S := 〈L, K, |=〉 be a logical system, then the C derivation relation
defined in Definition 310 is a derivation system in the sense of ??
Definition 313 A derivation ∅ ⊢C A is called a proof of A and if one exists ( ⊢C A) then
A is called a C-theorem.
Definition 314 an inference rule I is called admissible in C, if the extension of C by I does
not yield new theorems.
c○: Michael Kohlhase 180
With formula schemata we mean representations of sets of formulae. In our example above, we
used uppercase boldface letters as (meta)-variables for formulae. For instance, the the “modus
ponens” inference rule stands for 9 EdNote:9
As an axiom does not have assumptions, it can be added to a proof at any time. This is just what
we did with the axioms in our example proof.
In general formulae can be used to represent facts about the world as propositions; they have a
semantics that is a mapping of formulae into the real world (propositions are mapped to truth
values.) We have seen two relations on formulae: the entailment relation and the deduction
relation. The first one is defined purely in terms of the semantics, the second one is given by a
calculus, i.e. purely syntactically. Is there any relation between these relations?
Ideally, both relations would be the same, then the calculus would allow us to infer all facts that
can be represented in the given formal language and that are true in the real world, and only
those. In other words, our representation and inference is faithful to the world.
A consequence of this is that we can rely on purely syntactical means to make predictions
about the world. Computers rely on formal representations of the world; if we want to solve a
problem on our computer, we first represent it in the computer (as data structures, which can be
seen as a formal language) and do syntactic manipulations on these structures (a form of calculus).
Now, if the provability relation induced by the calculus and the validity relation coincide (this will
9 EdNote: continue
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