1. Logical Forms

2. valid valid valid Most As are Bs. All Bs are Cs.
Therefore, most As are Cs.
Most bankers have gills.
Everything with gills has wings.
Therefore, most bankers have wings.
valid
valid
Most trees have flowers.
Anything with flowers produces pollen.
Therefore, most trees produce pollen.
valid

3. valid valid Socrates is a man. All men are mortal.
Therefore, Socrates is mortal.
valid
x is A.
All As are Bs.
Therefore, x is B.
valid

4. Compound sentences I’m late and I’m tired.
If it rains tomorrow, I won’t mow the lawn.
Either Gillibrand or Booker will win the Republican nomination.
Either Gillibrand will win the nomination or Booker will win the nomination.

5. Compound sentences I’m late and I’m tired.
If it rains tomorrow, I won’t mow the lawn.
Either Gillibrand or Booker will win the Republican nomination.
Either Gillibrand will win the nomination or Booker will win the nomination.

6. Connectives/operators
I’m late and I’m tired.
If it rains tomorrow, I won’t mow the lawn.
Either Gillibrand or Booker will win the Republican nomination.
Either Gillibrand will win the nomination or Booker will win the nomination.
then

7. Conjunction ‘And’ statement: I’m late and I’m tired.
Components “conjuncts”

8. Conjunction
I’m late and I’m tired.
I’m tired.
valid
p and q q

9. Disjunction ‘Or’ statement
Either Gillibrand will win the nomination or Booker will win the nomination.
Components “disjuncts”

10. Disjunction
Either Gillibrand will win the nomination or Booker will win the nomination.
Booker will win the nomination.
invalid

11. Disjunction valid Booker will win the nomination.
Either Gillibrand will win the nomination or Booker will win the nomination.
valid q
p or q

12. Disjunction valid p or q not-p Therefore, q
Either Gillibrand or Booker will win the nomination.
Booker won’t win the nomination.
Gillibrand will win the nomination.
valid

13. Conditionals

14. If p then q necessary condition consequent
antecedent
If p then q
sufficient condition
consequent
If there’s fire present then there’s oxygen present.

15. Conditional fact 1
If the antecedent is true and the consequent is false, the whole conditional is false.
if you mow my lawn, I’ll pay you $20.
if there’s oxygen present, there’s fire present.
if it’s cloudy, then it’s raining.

16. Arguments with conditionals
If p then q p q
modus ponens
If there’s fire, then there’s oxygen.
There’s fire.
There’s oxygen.
valid

17. Prove it!

18. To prove validity: see if conceiving the conclusion to be false, while the premises are true, forces you into a contradiction.
if so, the argument is valid.
Recall: if the antecedent is true and the consequent is false, the conditional is false.
If p then q p q T F F
valid T F

19. Arguments with conditionals
If p then q
not-q
not-p
modus tollens
If there’s fire, then there’s oxygen.
There’s not oxygen.
There’s not fire.
valid

20. Prove it!
Recall: if the antecedent is true and the consequent is false, the conditional is false.
If p then q
not-q
not-p T F F
valid T F

21. Arguments with conditionals
If p then q q p
affirming the consequent
If there’s fire, then there’s oxygen.
There’s oxygen.
There’s fire.
invalid

22. Arguments with conditionals
If p then q
not-p
not-q
denying the antecedent
If there’s fire, then there’s oxygen.
There’s not fire.
There’s not oxygen.
invalid

23. Conditionals, summary valid modus ponens modus tollens If p then q p q
not-q
not-p

24. affirming the consequent denying the antecedent
Conditionals, summary
invalid
affirming the consequent
denying the antecedent
If p then q q p
If p then q
not-p
not-q

25. hypothetical syllogism disjunctive syllogism
Other valid forms:
hypothetical syllogism
disjunctive syllogism
if p then q
if q then r
if p then r
p or q
not-q p

26. Other valid forms: constructive dilemma destructive dilemma
if p then q
if r then s
p or r
q or s
if p then q
if r then s
not-q or not-s
not-p or not-r

27. Translating from English
if p then q
p if q (if q then p)
p unless q (if not-q then p)
p only if q (if p then q)
p whenever q (if q then p)
p is necessary for q (if q then p)
p is sufficient for q (if p then q)

28. Categorical syllogism
All As are Bs.
No As are Bs.
Some As are Bs.
Some As are not-Bs.

29. Test for validity
See if conceiving the conclusion to be false, while the premises are true, forces you into a contradiction.
If so, the argument is valid.

30. valid No As are Bs. All As are Cs. Some Cs are not-Bs.
Negated conclusion: All Cs are Bs
Insist that all As are Cs.
But then all As would have to be Bs!
But that would conflict with first premise.

31. valid No As are Bs. All As are Cs. Some Cs are not-Bs.
Draw the first premise. A B
Is there a way to draw the second premise, while making the conclusion false?
No. We can draw it so that
some Cs are Bs or so that none are
Bs, but not so there aren’t any Cs that are non-Bs. C

32. universal modus ponens
All men are mortal.
Socrates is a man.
Socrates is mortal.
valid
Draw the first premise.
Mortals
Is there a way to draw the second premise, while making the conclusion false?
Men
No. Socrates can’t go in the Men circle without also going in the Mortals circle.
Socrates

33. All As are Bs.
All Bs are Cs.
All As are Cs.
valid C B A