Download Discrete Computational Structures (CS 225) Definition of Formal Proof

Discrete Computational Structures (CS 225)
Definition of Formal Proof
A formal proof in our system of propositional logic is a series of statements, leading from some
premises to some conclusion, where each step in the series is logically justified in some way. More
precisely, if we have premises, consisting of a set of logical statements:
P = {P1 , P2 , . . . , Pk }
and a conclusion, consisting of a logical statement, C, then a proof of C from P is a finite
sequence of logical statements:
S1
S2
..
.
Sn
where the last line is the conclusion, Sn = C, and each statement Si is either:
1. A member of the premise set, Si = Pj ∈ P.
2. A result of applying one of the logical equivalency rules (text, p. 35) to a previous statement
in the proof.
3. A result of applying one of the valid argument forms (text, p. 61) to one or more previous
statements in the proof.
When we write a proof, we will number each line, and justify it by citing its source. When
justifying the application of equivalency rules and argument forms, we will give the name of the
logical principle we are using, along with the line number(s) to which the rule or form is applied.
For example, we can easily prove the conclusion, p∧q, from the premise set {∼∼p, ∼∼q} as follows:
1. ∼∼p
[Premise]
2. ∼∼q
[Premise]
3. ∼∼p ∧ ∼∼q
[Conjunction: 1, 2]
4. ∼∼p ∧ q
[Double Negative: 3]
5. p ∧ q
[Double Negative: 4]
(Note that this uses the valid argument form, Conjunction, from the text, page 61, and the logical
equivalence, Double Negative, from page 35.)
1
2