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Introductory Logic Lecture Notes, Schemes and Mind Maps of Logic

Harvard UniversityLogic

These lecture notes cover the basics of introductory logic, including the components of an argument, deductive arguments, valid and invalid arguments, and formal logic. The notes provide examples and mini quizzes to test understanding. subject to revision up until a certain date. The notes are likely intended for a philosophy or logic course at a university level.

Typology: Schemes and Mind Maps

2022/2023

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PHI 201, Introductory Logic
Lecture Notes∗
Feb 2, 2004
∗Note: These are subject to revision up until 10am on Feb 3.
0-0
•In order to make an argument, you have to make a claim (the conclusion)
and you have to give some evidence for the claim (the premises).
“Bush tried to justify the war with Iraq by citing the danger of WMDs.
But now we’ve found out that there were no WMDs. So, either Bush
lied, or he got bad intelligence. If Bush got bad intelligence, then he
made some really terrible appointments to top posts in the CIA.
Therefore, either Bush is a liar, or he’s incompetent.”
•Two components of an argument:
1. Conclusion
2. Premises
1
pf3
pf4
pf5
pf8

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PHI 201, Introductory Logic

Lecture Notes∗

Feb 2, 2004

∗Note: These are subject to revision up until 10am on Feb 3.

  • In order to make an argument , you have to make a claim (the conclusion ) and you have to give some evidence for the claim (the premises ). “Bush tried to justify the war with Iraq by citing the danger of WMDs. But now we’ve found out that there were no WMDs. So, either Bush lied, or he got bad intelligence. If Bush got bad intelligence, then he made some really terrible appointments to top posts in the CIA. Therefore, either Bush is a liar, or he’s incompetent.”
  • Two components of an argument:
    1. Conclusion
    2. Premises
  • The components of arguments are all statements — something that can be true or false. (“bivalence”) Examples: - “The ten millionth digit in the decimal expansion of π is 2.” - “The helium atom has a single electron.”
  • Not all sentences are statements. For example: - “Let’s watch Hasselhoff’s Berlin wall video again!”

2

Deductive Arguments

  • In a deductive argument, the intention is to show that the conclusion follows from the premises with absolute certainty. - Conclusion follows from the premises. - Conclusion is entailed by the premises. - Conclusion is a logical consequence of the premises. - The premises imply the conclusion. - The premises justify the conclusion.
  • Deductive arguments occur in the wild, and can be spotted in mathematics and computer science departments, and occasionally in some philosophy departments.
  • Most academic £elds of study are descriptive : They tell us how nature and people do, in fact, behave. (e.g., chemistry, biology, psychology, anthropology, physics)
  • Philosophy (and speci£cally logic) aspires to be prescriptive , or normative. - In ethics, we study how people ought to behave, not how they do in fact behave. - In logic, we study how people ought to think, not how they do in fact think.

6

We will replace the vague word “good” with a precisely de£ned word: sound. A sound argument has two features.

  1. It uses good evidence — i.e., its premises are all true.
  2. The conclusion follows logically from the premises.

De£nition: An argument is valid if its conclusion follows logically from its premises — if it is impossible for the premises all to be true, but for the conclusion to be false.

Nota Bene: A valid argument need not have true premises, nor a true conclusion — validity requires only that if the premises were true then the conclusion would also be true.

Examples of Valid Arguments

  1. A majority of Americans believe that it is wrong to cheat on your taxes.
  2. If the majority of Americans believe it, then it must be true.
  3. It is wrong to cheat on your taxes.
  4. It is morally wrong to kill an innocent human being.
  5. A fetus is an innocent human being.
  6. It is morally wrong to kill a fetus.

8

Examples of Invalid Arguments

NB: The statements in an invalid argument can have any combination of truth values.

  1. If God does not exist, then life is meaningless.
  2. God exists.
  3. Life is meaningful.
  4. Only people born in the US are eligible to become President.
  5. John Edwards was born in the US.
  6. John Edwards is eligible to become President.

Formal Logic

  • Starting Assumption of Formal Logic: Whether or not an argument is valid depends only on its form ; its content is irrelevant to validity. (Logic is “blind to the actual facts.”) “Logic contains no matter at all, only form of thought.” (Immanuel Kant)
  • If two arguments have the same form, either they are both valid, or they are both invalid.
  • But arguments don’t wear their form on their sleeve.

12

  1. Either God exists or everything is permitted. 1. Either G or E
  2. God does not exist. 2. not- G
  3. Everything is permitted. 3. E
  1. If God doesn’t exist then there aren’t any 1. If not- G then not- M moral rules.
  2. God exists. 2. G
  3. There are moral rules. 3. M
  4. If PU is not in the US then it isn’t in CA. 1. If not- U then not- C
  5. PU is in the US. 2. U
  6. PU is in California. 3. C

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Argument Forms

  • An argument A is an instance of the form F just in case A results from a uniform replacement of words/sentences for blanks in F.
  • A counterexample to an argument A is a an argument A′^ that has the same form as A, and where A′^ has (actually) true premises and a false conclusion.
  • An argument A is valid just in case there is no counterexample to A.
  • For thought: Can we ever know for sure that an argument is valid? How can we be sure that someone won’t £nd a counterexample?