Download Introduction to Proofs, Rules of Equivalence, Rules of

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 nonvalid, 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 nonvalid (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).