1. Propositional Logic
5) And
Copyright 2008, Scott Gray

2. Conjunctions Statements joined by AND
The component statements are called “conjuncts”
AND is represented by the ampersand: &
Copyright 2008, Scott Gray

3. English Equivalents Bear is black and Coda is yellow.
Bear is black, but Coda is yellow.
Bear is black, however Coda is yellow.
Bear is black, moreover Coda is yellow.
Bear is black although Coda is yellow.
Bear is black yet Coda is yellow.
Bear is black even though Coda is yellow.
All of these may be symbolized: B & Y
Copyright 2008, Scott Gray

4. AND Reminders We view the conjuncts as being commutative
There are occasions where you will need parentheses:
A & B → C does this mean
(A & B) → C or
A & (B → C)
Copyright 2008, Scott Gray

5. AND Gotchas Not every English “and” is a logical AND
Are the following two sentences equivalent?
Marvin and Nancy are cousins
Marvin is a cousin and Nancy is a cousin
Sometimes an English and should be represented as an arrow:
Give us the tools of war and we shall finish the job
You must think, not blindly see a word in English and use a logic operator which we have given the same name
Copyright 2008, Scott Gray

6. More for Your Notebook Common introduction to conclusions: thereforeso
hence
thus
consequently
it follows that
proves that
shows that
Copyright 2008, Scott Gray

7. More for Your Notebook, cont.
Common introduction to premises:
since
because
for
These follow conclusions and introduce premises – which means that sometimes the conclusion is stated before the premises
Copyright 2008, Scott Gray

8. Ampersand In You can form a conjunction statement from two statements
If you have a sentence Q and a sentence B, then you may write down Q & B
The justification column has the two conjunct lines and “&I”; i.e.: 2,6 &I
Copyright 2008, Scott Gray

9. Ampersand Out From a conjunction derive either conjunct
If you have a sentence Q & P, you may write down Q (you may also write down P)
The justification has the conjunction line and “&O”; i.e.: 3 &O
Copyright 2008, Scott Gray

10. Ampersand Notes A & (B & C) is equivalent to (A & B) & C
Can you prove this?
1 A & (B & C) A
2 A 1 &O
3 B & C 1 &O
4 B 3 &O
5 C 3 &O
6 A & B 2,4 &I
7 (A & B) & C 6,5 &I
Copyright 2008, Scott Gray

11. Sample Proofs P → Q, P → R, P ∴ R & Q 1 P → Q A 2 P → R A 3 P A
4 Q 1,3 →O
5 R 2,3 →O
6 R & Q 5,4 &I
Copyright 2008, Scott Gray

12. Sample Proofs, cont. (P & R) → (Q → T), Q → P, Q & (P → R) ∴ T
1 (P & R) → (Q → T) A
2 Q → P A
3 Q & (P → R) A
4 Q 3 &O
5 P → R 3 &O
6 P 2,4 →O
7 R 5,6 →O
8 P & R 6,7 &I
9 Q → T 1,8 →O
10 T 9,4 →O
Copyright 2008, Scott Gray

13. Sample Proofs, cont. P & Q, P → R, Q → T ∴ R & T 1 P & Q A 2 P → R A
3 Q → T A
4 P 1 &O
5 Q 1 &O
6 R 2,4 →O
7 T 3,5 →O
8 R & T 6,7 &I
Copyright 2008, Scott Gray

14. Sample Proofs, cont. (P & (P → R)) & (R → T) ∴ T
1 (P & (P → R)) & (R → T) A
2 P & (P → R) 1 &O
3 P &O
4 P → R &O
5 R ,3 →O
6 R → T &O
7 T ,5 →O
Copyright 2008, Scott Gray

15. Assignments Read Chapter 3 Do all of the exercises
Be sure to ask me questions if you don’t understand something or can’t solve a problem
Copyright 2008, Scott Gray