Logic 2 validity, other notions

1. Sentences
O Recall: an argument – is a series of
sentences.
O Sentences – contains info that can be T or
F.

2. What does not count as a
sentence?
O Questions. “What did you have for
breakfast?”
O Exclamations: “Ouch!”
O Commands (Imperatives): “Eat your
oatmeal!”

3. Exercise
O Sentence or not sentence?

4. 2 ways when we know that the
argument is bad:
O If it’s:
O 1. Invalid
O 2. Its premises are False

5. What does it mean “valid”?
DEF. An argument is valid if and
only if it’s impossible for premises
to be true and for the conclusion to
be false.
1. That is, for a valid argument, when
the premises are true, it is impossible
for the conclusion to be false.
2. In a valid argument conclusion
follows from the premises.
3. Validity is about form of the
argument.

6. Example
All men are mortal.
Socrates is a men.
Socrates is mortal.
1. Does the conclusion follow from the
premises? Then, it is valid.
2. Are premises true?

7. Example
(1) It is raining heavily.
(2) If you do not take an umbrella, you will
get soaked.
=> You should take an umbrella.
Are premises true? If not  bad argument
Even if true, does conclusion follow from the
premises?

8. All men are mortal. All M are R.
Socrates is a men. S is M
 Socrates is mortal. S is M
___________________________________
All apples are mammals.
Socrates is an apple.
 Socrates is mammal.
Premises true?
Does conclusion follow from premises?

9. O So validity is about the form of the
argument (it’s structure). It’s not about the
truth of the premises!

10. How to precisely identify if an
argument is valid?
O That’s what this course is about!
O Once again, a valid argument is an argument
when it’s impossible for its Premises to be
true, and at the same time for its conclusion
to be false.
O That is, if we identify that it’s possible for the
premises to be true in an argument, and for
the conclusion to be false, we know that such
argument is invalid.

11. Example
All men are mortal.
Socrates is a men.
Socrates is mortal.
In this kind of argument, if you substitute
premises with any kind of other true premises,
the conclusion will be always true. That’s
because this argument has a valid form!

12. Another example:
All popes reside at the Vatican.
John Paul II resides at Vatican.
Therefore, John Paul II is a pope.
All P are V
J is V
J is P
Is such case possible: when the premises are T, and
the conclusion is F?

13. All P are V
J is V
J is P
All basketballs are round.
Earth is round.
 Earth is a basketball.
True premises & false conclusion => the
argument is invalid.

14. SOUND ARGUMENT
DEF. SOUND ARGUMENT is a valid
argument with true premises.
If an argument is Valid, and its premises are true –
it’s SOUND.

15. DEDUCTIVE and INDUCTIVE
ARGUMENTS
O Deductive argument: deduces information
“from general to particular”
O Inductive argument: deduces information
“from particular to general”

16. Example of deductive
argument
O All men are mortal.
Socrates is a men.
Socrates is mortal.
O Everyone who eats carrots is a
quarterback.
John eats carrots.
Therefore, John is a quarterback.

17. Example of inductive
argument
In January 1997, it rained in San Diego.
In January 1998 it rained in San Diego.
In January 1999, it rained in San Diego.
=> It rains every January in San Diego.

18. The concept of validity applies only to
deductive arguments.
That is, Only a deductive argument can be
VALID (or invalid).

19. O In this course we focus only on deductive
arguments – which could be VALID or
INVALID.

20. Other logical notions
I. A sentence. A sentence could be:
1) Tautology
2) Contradiction
3) Contingent
II. Relation between sentences. Sentences
could be:
1) Consistent
2) Inconsistent
3) Logically equivalent

21. 1. TAUTOLOGY
A tautology is a sentence that
must be true, as a matter of logic
Example: Either it’s raining or it’s not.
There is either hat on my head, or there is
no hat on my head.

22. A CONTRADICTION
2.Def. A contradiction is a sentence that
must be false, as a matter of logic.
Ex.: The restaurant opens at 5pm, and it
begins serving between 4 and 9pm.
Peter is married to Marry, but Marry is not
married to Peter.
I’m hungry and I’m not hungry.

23. CONTINGENT SENTENCE
O A contingent sentence is neither a
tautology nor a contradiction.
Ex. It’s raining.
I haven’t head my coffee yet.
There is going to be RAT #2 soon.

24. RELATION BETWEEN
SENTENCES:
Ex.
1. John went to the store after he washed dishes.
John washed the dishes before he went to the
store.
2. If Julie sings, everyone is happy.
If someone is unhappy, Julie doesn’t sing.
3. It’s not true that all apples are red.
Some apples are not red.
O Def. Two sentences are logically
equivalent if they necessarily have the
same truth values.

25. O A set of sentences is logically
consistent, if it’s possible for all
members of the set to be true at the
same time. It is inconsistent otherwise.
Ex. 1. My only brother is taller than I am.
My only brother is shorter than I am.
2. Everybody left the room. John is still in the
room.
3. Lincoln is taller than Jones. Jones is taller
than Shorty. Shorty is taller than Lincoln.

26. HW for Thu:
1. P. 15&16, Part B, C. D
2. Read Appendix I “Some Common
Fallacies”