Today’s Topics

• Introduction to Proofs

• Rules of Inference

• Rules of Equivalence

Truth Tables Allow One to Determine Whether an Argument is Valid or Not. BUT:

• Truth tables are unwieldy (remember, the number of rows in a truth table increases exponentially. R = 2n where n is the number of variables).

• Truth tables work only for sentential logic.

We need another, more powerful, test for validity: Proofs

• Remember, validity is truth preserving. Validity guarantees that if you start with true claims, and reason according to legitimate rules, you MUST end up with true claims.

• A proof demonstrates validity by giving a set of instructions on how to get from the premises to the conclusion.

Deductive Validity

• A characteristic of arguments in which the truth of the premises guarantees the truth of the conclusion. It is impossible for the premises of a valid argument to be true and its conclusion to be false.

• A proof establishes that an argument is valid.

A proof is a finite set of formulae, beginning with the premises of an argument and ending with its conclusion, in which each formula following the premises is derived from the preceding formulae according to established rules of inference and equivalence.

NOTE: Constructing a proof definitively establishes that an argument is valid.

• HOWEVER, failure to construct a proof proves nothing. It may be because the argument is non-valid, or it may be because YOU can’t construct the proof.

• A Proof Procedure is only one half of a Decision Procedure.

Arguments and Argument Forms

• An argument is a set of statements, one of which (the conclusion) is supposed to follow from the others (the premises).

• All statements have a statement form, i.e., they are substitution instances of one of our 5 basic statement forms.

Argument Form

• An argument form set of statement forms

• Every substitution instance of an argument form is an argument

• Argument forms are either valid or non-valid (truth tables verify the validity of valid argument forms)

Valid Argument Forms and Rules• Valid argument forms allow us to establish

legitimate patterns of inference, legitimate rules.

• If a pattern of inference is valid, we can rest assured that any instance of that pattern is valid without having to the instance.

• Never forget: Validity is a matter of form.• Recognizing and applying valid rules

(patterns) simplifies the task of establishing validity.

Valid Argument Forms Justify Two Types of RULES

• Rules of Inference – Operate on whole lines only– Generate new lines with unique truth values

• Rules of Equivalence– Operate on whole lines or parts of lines– Generate lines equivalent to the original lines

Rules of Inference

• Generate new lines whose truth value follows from, but is not identical to, the truth of the source lines.

• Operate on lines whose statement forms match the statement forms of the lines in the argument form of the rule.

• Can be applied ONLY to entire lines, not parts of lines.

Choosing Rules

• Our system of logic uses 18 rules of inference (8) and equivalence (10).

• Why 18? Well, there is a trade off having a set of rules most people can master (1,000 rules would be a tad too many) and having a system that is not too cumbersome (we could make do with one rule, but using it is a real pain).