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## PowerPoint Slideshow about 'Logic' - gwydion

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Logic

A short primer on Deduction and Inference

We will look at Symbolic Logic in order to examine how we employ deduction in cognition.

Logical Systems

- There are actually many logical systems
- The one we will examine in class is called RS1 (I think)
- It is comprised of
- Statements
- "Roses are red“
- "Republicans are Conservatives“
- “P”
- Operators
- And
- Or
- not
- Some Rules of Inference

Logic

Compound Statements

- Conjunctions (Conjunction Junction)
- Two simple statements may be connected with a conjunction
- The conjunction “and”
- The disjunction “or”

The conjunction operator

- “and”
- Symbolized by “•”
- "Roses are Red and Violets are blue.“
- "Republicans are conservative and Democrats are liberal.“
- P • Q (P and Q)

The disjunction operator

- “or”
- Symbolized by “v”
- "Republicans are conservative or Republicans are moderate
- P v Q

Negation

- Not
- Symbolized by ~
- That is not a rose
- Bob is not a Republican
- ~A

Operators

- These may be used to symbolize complex statements
- The other symbol of value is
- Equivalence ()
- This is not quite the same as “equal to”.

Truth Tables

- Statements have “truth value”
- For example, take the statement P•Q:
- This statement is true only if P and Q are both true.

P Q P•Q

T T T

T F F

F T F

F F F

Truth Tables (cont)

- Hence “Republicans are conservative and Democrats are liberal.” is true only if both parts are true.
- On the other hand, take the statement PvQ:
- This statement is true only if either P or Q are true, but not both. (Called the “exclusive or”)

P Q PvQ

T T F

T F T

F T T

F F F

The Inclusive ‘or’

- Note that ‘or’ can be interpreted differently.
- Both parts of the disjunction may be true in the “inclusive or”. This statement is true if either or both P or Q are true.

P Q PvQ

T T T

T F T

F T T

F F F

The Exclusive ‘or’

- With the exclusive or, of p is true, than q cannot be.
- Only one part of the disjunction may be true in the “exclusive or”. This statement is true if either P is true or Q is true, but not both.

P Q PvQ

T T F

T F T

F T T

F F F

The Conditional

- The Conditional
- if a (antecedent)
- then b (consequent)
- It is also called the hypothetical, or implication.
- This translates to:
- A implies B
- If A then B
- A causes B
- Symbolized by A B

The Implication

- We use the conditional or implication a great deal.
- It is the core statement of the scientific law, and hence the hypothesis.

Equivalency of the Implication

- Note that the Implication is actually equivalent to a compound statement of the simpler operators.
- ~p v q
- Please note that the implication has a broader interpretation than common English would suggest

Rules of Inference

- In order to use these logical components, we have constructed “rules of Inference”
- These rules are essentially “how we think.”

Modus Ponens

- This is the classic rule of inference for scientific explanation.

Modus Tollens

- This reflects the idea of rejecting the theory when the consequent is not observed as expected.

Hypothetical Syllogism

- Classic reasoning
- All men are mortal.
- Socrates is a man.
- Therefore Socrates is mortal.

Logical Systems

- Logic gives us power in our reasoning when we build complex sets of interrelated statements.
- When we can apply the rules of inference to these statements to derive new propositions, we have a more powerful theory.

Tautologies

- Note that p v ~p must be true
- “Roses are red or roses are not red.” must be true.
- A statement which must be true is called a tautology.
- A set of statements which, if taken together, must be true is also called a tautology (or tautologous).
- Note that this is not a criticism.

Tautologous systems

- Systems in which all propositions are by definition true, are tautologous.
- Balance of Power
- Why do wars occur? Because there is a change in the balance of power.
- How do you know that power is out of balance? A war will occur.
- Note that this is what we typically call circular reasoning.
- The problem isn’t the circularity, it is the lack of utility.

Useful Tautologies

- Can a logical system in which all propositions formulated within be true have any utility?
- Try Geometry
- Calculus
- Classical Mechanics
- But not arithmetic
- Kurt Gödel & his Incompleteness Theorem

The Liars Paradox

- Epimenedes the Cretan says that all Cretans are liars.“
- The Paper Paradox (a variant of the Liar’s paradox)
- < The next statement is true.
- < The previous statement is false.
- For further info
- Russell’s Paradox
- The paradox arises within naive set theory by considering the set of all sets that are not members of themselves.
- Such a set appears to be a member of itself if and only if it is not a member of itself.
- Hence the paradox

Grelling’s Paradox

- Homological – a word which describes itself
- Short is a short word
- English is an English word
- Heterological – a word which does not describe itself
- German is not a German words
- Long is not a long word
- Is heterological heterological?

Paradox of voting

- It is possible for voting preferences to result in elections in which a less preferred candidate wins over a preferred one.
- See Paradox of Voting
- Suppose you have 3 individuals and candidates A, B and C
- Individual 1: A > B > C
- Individual 2: C > A > B
- Individual 3: B > C > A
- Now if these individuals were asked to make a group choice (majority vote) between A and B, they would chose A;
- If asked to make a group choice between B and C, they would chose B.
- If asked to make a group choice between C and A, they would chose C.
- So for the group A is preferred to B, B is preferred to C, but C is preferred to A! This is not transitive which certainly goes against what we would logically expect.

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