Discrete Structures

# Lecture Notes, 1 Oct 2010

 Last Update: 1 October 2010 Note: or material is highlighted

### §1.5: Rules of Inference (cont'd)

1. Rule of Inference vs. Tautologies:

1. Recall that MP is the rule of inference:

From                A
&                    (AB)
you may infer B

2. It looks like the proposition:

(*)        (A ∧ (AB)) → B)

But it isn't!

3. Differences:

1. MP is a sequence of 3 propositions;
it is a valid argument form

2. (*) is a single proposition;
it is a tautology

4. But they are related:

MP is valid iff (*) is a tautology

5. In general, a rule of inference

From A
&      B
infer C

is valid iff ((AB) → C) is a tautology

• This is called the "Deduction Theorem", and requires proof
(but we won't prove it)

2. More Rules of Inference:

1. Disjunctive Syllogism:

(AB)
¬A
B

2. Modul Tollens (method of denying (the consequent)):

(AB)
¬B
¬A

A
(AB)

4. Simplification:

(AB)
A

5. Conjunction:

A
B
(AB)

6. There is a chart of some rules of inference on p. 66 (Table 1).

1. In general, for each connective, there are "introduction" and "elimination" rules,
2. some of which require nested subproofs.

• An "introduction" rule for a connective "*" tells you how to "introduce" as a line of a proof a new proposition whose principal connective is "*".
• An "elimination" rule for a connective "*" tells you how to "eliminate" a connective to create a new proposition for a line of a proof.

3. Rules of inference are "atomic" valid argument forms.

• Longer arguments are valid if their "sub-arguments" are valid;
a sub-argument is valid if its sub-arguments are valid;
and so on, stopping at "atomic" arguments that are rules of inference.
• This is another example of what I have been calling "recursion"

3. The Resolution Rule of Inference:

1. Resolution is a single rule of inference that can do the work of all of the others.

• ∴, it's useful for computational implementations of FOL.
• The programming language Prolog is based on FOL and resolution.

2. Consider the following chart of rules of inference.
• The familiar rules are in the first column.
• The second column rewrites the rule using only ¬ & ∨
(which we know are functionally complete!)

 MP A → B A          B ¬A ∨ B A          B MT A → B ¬B          ¬A ¬A ∨ B ¬B          ¬A HS A → B B → C A → C ¬A ∨ B ¬B ∨ C ¬A ∨ C DS A ∨ B ¬A        B (already uses only ¬,∨)

3. Generalizing this pattern, we have Resolution:

AB1 ∨ … ∨ Bn
¬AC1 ∨ … ∨ Cm

(B1 ∨ … ∨ Bn) ∨ (C1 ∨ … ∨ Cm)

or, more simply:

AB
¬AC
BC

• To use this as the only rule,
we need to represent all propositions in "clause form"
(or "conjunctive normal form")
i.e., using only ¬, ∧, & ∨

4. More examples of proofs:

1. Prove that this argument is valid:

P1:   Lynn works part time or full time.
P2:   If Lynn doesn't play on the team,
then she doesn't work part time.
P3:   If Lynn plays on the team,
then she's busy.
P4:   Lynn doesn't work full time.
C :   ∴ Lynn is busy.

2. Syntax & Semantics of Representation:

Let pt = Lynn works part time.
Let ft = Lynn works full time.
Let plays = Lynn plays on the team.
Let busy = Lynn is busy.

3. Translation of argument:

P1:    pt &or ft
P2:    ¬plays → ¬pt
P3:    plays → busy
P4:    ¬ft
C :    ∴ busy

4. We want to prove that this is a valid argument

• without using truth tables

5. How? Next time!