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

least study the interplay of mathematical statements (which can be true or false) with the copula
“and”, “or” and “not”.
Semantic Properties of Boolean Expressions
Definition 301 Let M := 〈U, I〉 be our model, then we call e
true under ϕ in M, iff Iϕ(e) = T (write M |= ϕ e)
false under ϕ in M, iff Iϕ(e) = F (write M |= ϕ e)
satisfiable in M, iff Iϕ(e) = T for some assignment ϕ
valid in M, iff M |= ϕ e for all assignments ϕ (write M |= e)
falsifiable in M, iff Iϕ(e) = F for some assignments ϕ
unsatisfiable in M, iff Iϕ(e) = F for all assignments ϕ
Example 302 x ∨ x is satisfiable and falsifiable.
Example 303 x ∨ ¬x is valid and x ∧ ¬x is unsatisfiable.
Notation 304 (alternative) Write [e] M
ϕ for Iϕ(e), if M = 〈U, I〉.
(and [e] M , if e is ground, and [e], if M is clear)
Definition 305 (Entailment) (aka. logical consequence)
We say that e entails f (e |= f), iff Iϕ(f) = T for all ϕ with Iϕ(e) = T
(i.e. all assignments that make e true also make f true)
c○: Michael Kohlhase 175
Let us now see how these semantic properties model mathematical practice.
In mathematics we are interested in assertions that are true in all circumstances. In our model
of mathematics, we use variable assignments to stand for circumstances. So we are interested
in Boolean expressions which are true under all variable assignments; we call them valid. We
often give examples (or show situations) which make a conjectured assertion false; we call such
examples counterexamples, and such assertions “falsifiable”. We also often give examples for
certain assertions to show that they can indeed be made true (which is not the same as being
valid yet); such assertions we call “satisfiable”. Finally, if an assertion cannot be made true in any
circumstances we call it “unsatisfiable”; such assertions naturally arise in mathematical practice in
the form of refutation proofs, where we show that an assertion (usually the negation of the theorem
we want to prove) leads to an obviously unsatisfiable conclusion, showing that the negation of the
theorem is unsatisfiable, and thus the theorem valid.
Example: Propositional Logic with ADT variables
Idea: We use propositional logic to express things about the world
(PLNQ ˆ= Predicate Logic without Quantifiers)
Example 306 Abstract Data Type: 〈{B, I}, {. . ., [love: I × I → B], [bill: I], [mary: I], . . .}〉
ground terms:
g1 := love(bill, mary) (how nice)
g2 := love(mary, bill) ∧ ¬love(bill, mary) (how sad)
g3 := love(bill, mary) ∧ love(mary, john) ⇒ hate(bill, john) (how natural)
Semantics: by mapping into known stuff, (e.g. I to persons B to {T, F})
91