Download Logic in Proofs (Valid arguments) A theorem is a hypothetical

Logic in Proofs (Valid arguments)
A theorem is a hypothetical statement of the form H 6 C, where H is a (compound) statement which is
taken as being true, and C is a statement which follows from H by logical reasoning.
Example: [(p 6 q) v (q 6 r) v (¬ r)] 6 (¬ p)
An argument in logic is a way to reach a conclusion based on prior statements.
An argument which has no counterexamples is said to be valid.
Rules of inference correspond to tautologies
Modus ponens: (p v (p6q)) 6 q is a tautology.
For compound propositions P and Q, (P v (P6Q)) 6 Q is also a tautology.
This rule may be expressed in short form as modus ponens: P and P 6 Q ˆ Q
Hypothetical Syllogism [(p6q) v (q6r)] 6 (p6r) is a tautology
In short form, hypothetical syllogism: P 6 Q and Q 6 R ˆ P 6 R.
Modus tollens ((p6q) v ¬q) 6 ¬p is a tautology.
In short form, modus tollens: P 6 Q and ¬Q ˆ ¬P
Other Rules of Inference in short form:
Addition: P ˆ P w Q
simplification: P v Q ˆ P
disjunctive syllogism: P w Q and ¬P ˆ Q
conjunction: P and Q ˆ P v Q
A formal proof is a sequence of propositions P1, P2, ..., P n, C in which each proposition is either a
hypothesis, a tautology, or a consequence of previous members of the chain by using an allowable rule
of inference.
In creating a formal proof we use Substitution Rules
Names don’t matter in a tautology (only the form)!
Equivalences do not change truth value!
Consider a proof of [(p 6 q) v (q 6 r) v (¬ r)] 6 (¬ p)
A sequence of statements that does not correspond to a formal proof is called a fallacy.
Fallacies often imitate the form of valid rules of inference, but will have a counterexample.
That is, there is at least one instance of truth values of the component variables which allows for
true hypotheses and a false conclusion.
Common fallacies:
Affirming the conclusion (reasoning by converse)
[(p6q)vq]6p is not a tautology, so the argument P6 Q / Q /ˆ P is a fallacy.
Denying the hypothesis (reasoning by inverse)
[(p6q)v¬p]6¬q is not a tautology, so the argument P6 Q / ¬P /ˆ ¬Q is a fallacy.
Circular reasoning (begging the question) occurs when an implicit assumption is strong enough
to get the desired conclusion.
[L6(A6L), (Assume L) then (A6L)] is fallacious because L could be false.
Jumping to conclusion (hasty generalization):
[P6Z, Q6Z, ...,Y6Z; ˆZ] is fallacious because all of P, Q, ..., Y could be false.