Discrete Structures

# Lecture Notes, 4 Oct 2010

 Last Update: 4 October 2010 Note: or material is highlighted

### §1.5: Rules of Inference (cont'd)

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.

1. Syntax & semantics of representation of atomic propositions:

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

2. Translation of argument:

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

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

• without using truth tables

4. Strategy:

Want to show:    busy (i.e., C)

Can derive C from P3 via MP if we can show: plays
Can derive "plays" from P2 via MT if we can show: pt
Can derive "pt" from P1 via DS if we can show: ¬ft
But we have ¬ft = P4

So, "unwind" this strategy to write the proof:

use that & DS on P1 to derive "pt"
use that & MT on P2 to derive "plays"
use that & MP on P3 to derive "busy"
QED!

5. Here's the proof:

 1 ¬ft : P4 2 pt ∨ ft : P1 3 pt : 1,2; DS 4 ¬plays → ¬pt : P2 5 plays : 3,4; MT 6 plays → busy : P3 7 busy : 5,6; MP

2. Digression: Was that really a legitimate use of MT?

1. After all, MT says:

From:                    AB
&:                       ¬B
you may infer:    ¬A

2. But the application above seems be of the form:

From:               ¬A¬B
&:                       B
you may infer:    A

3. Reply: We can handle this in several ways:

1. We could say (as I implicitly did above) that both of these are MT:

• After all, at their core, they follow the "same" pattern:

• From a conditional and the "opposite" of its consequent,
you may infer the "opposite" of its antecedent.

• But how would a computer know that?

2. We could say that there are two versions of MT,
namely, the two in IIA and IIB, above.

• That's another way to interpret what I did in the example.

3. We could prove that version B is derivable from version A; here's how:

1. ¬A¬B : premise
2. B               : premise
3. ¬¬B           : b; DN+SL

• (i.e., line c follows from line b by Double Negation
(which tells us that they are logically equivalent)
together with Shakespeare's Law
(which tells us that we can replace one with the other))

4. ¬¬A           : a,c; MT (version A)
5. A               : d; DN+SL

Now that we have this proof, we can appeal to MT (version B) in the future.

• It's like a "macro" in a computer program;
we can "expand" the macro whenever we need it.

3. Is This Argument Valid?

P1:   I play golf or tennis.
P2:   If it's not Sunday, I play golf and tennis.
P3:   If it's Saturday or Sunday, I don't play golf.
C :   ∴ I don't play golf.

1. Syntax & semantics of representation of atomic propositions:

g = "I play golf"
t = "I play tennis"
sun = "It's Sunday"
sat = "it's Saturday"

2. Translation of argument:

P1:  (g ∨ t)
P2:  (¬sun → (g ∧ t))
P3:  ((sat ∨ sun) → ¬g)
C :  ¬g

3. Strategy:

1. Try to show ¬g (i.e., C)
Can derive C from P3 via MP, if we can show (sat ∨ sun)
Can derive (sat ∨ sun) from "sun" via Addition, if we can show "sun"
Can derive "sun" from ¬(g ∧ t) via MT, if we can show ¬(g ∧ t)
Can derive ¬(g ∧ t) from (¬g ∨ ¬t) via DM+SL,
if we can show (¬g ∨ ¬t)
Can derive (¬g ∨ ¬t) either from ¬g
Can we derive either one of those?

2. Deriving ¬g won't help; that's what we're trying to do in the first place!
(This is called "begging the question" or arguing in a circle.)

3. Can we derive ¬t?

• P1 contains "t", but there's no rule that would allow us to derive ¬t from P1
• P2 contains "t", but there's no rule that would allow us to derive ¬t from P2

• So we probably can't derive ¬t.

4. So, let's revert to truth tables to see if the argument is valid or not.

• This exercise is left to the reader :-)
• Construct a truth table with 24=16 rows
& find a row with T premises but F conclusion!

• (The answer is online at the book's website!)

• Hint: Make
tval(g) = T
tval(sat) = tval(sun) = F
tval(t) = T
• e.g., it's Monday & I play golf and tennis.