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

2.6 Propositional Logic
2.6.1 Boolean Expressions and Propositional Logic
We will now look at Boolean expressions from a different angle. We use them to give us a very
simple model of a representation language for
• knowledge — in our context mathematics, since it is so simple, and
• argumentation — i.e. the process of deriving new knowledge from older knowledge
Still another Notation for Boolean Expressions
Idea: get closer to MathTalk
Use ∨, ∧, ¬, ⇒, and ⇔ directly (after all, we do in MathTalk)
construct more complex names (propositions) for variables
(Use ground terms of sort B in an ADT)
Definition 299 Let Σ = 〈S, D〉 be an abstract data type, such that B ∈ S and
[¬: B → B], [∨: B × B → B] ∈ D, then we call the set T g
B (Σ) of ground Σ-terms of sort
B a formulation of Propositional Logic.
We will also call this formulation Predicate Logic without Quantifiers and denote it with
PLNQ.
Definition 300 Call terms in T g
B (Σ) without ∨, ∧, ¬, ⇒, and ⇔ atoms. (write A(Σ))
Note: Formulae of propositional logic “are” Boolean Expressions
replace A ⇔ B by (A ⇒ B) ∧ (B ⇒ A) and A ⇒ B by ¬A ∨ B. . .
Build print routine ˆ· with A ∧ B = A ∗ B, and ¬A = A and that turns atoms into
variable names. (variables and atoms are countable)
c○: Michael Kohlhase 173
Conventions for Brackets in Propositional Logic
we leave out outer brackets: A ⇒ B abbreviates (A ⇒ B).
implications are right associative: A 1 ⇒ · · · ⇒ A n ⇒ C abbreviates A 1 ⇒
(· · · ⇒ (· · · ⇒ (A n ⇒ C)))
a stands for a left bracket whose partner is as far right as is consistent with existing brackets
(A ⇒ C ∧ D = A ⇒ (C ∧ D))
c○: Michael Kohlhase 174
We will now use the distribution of values of a Boolean expression under all (variable) assignments
to characterize them semantically. The intuition here is that we want to understand theorems,
examples, counterexamples, and inconsistencies in mathematics and everyday reasoning 5 .
The idea is to use the formal language of Boolean expressions as a model for mathematical
language. Of course, we cannot express all of mathematics as Boolean expressions, but we can at
5 Here (and elsewhere) we will use mathematics (and the language of mathematics) as a test tube for understanding
reasoning, since mathematics has a long history of studying its own reasoning processes and assumptions.
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