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Logic - PowerPoint PPT Presentation

Logic. A short primer on Deduction and Inference. We will look at Symbolic Logic in order to examine how we employ deduction in cognition. Logic. A short primer on Deduction and Inference. We need to try to avoid skewed logic. Logic. What is Logic?. Logic

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Presentation Transcript

A short primer on Deduction and Inference

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

A short primer on Deduction and Inference

We need to try to avoid skewed logic.

What is Logic?

• Logic

• The study by which arguments are classified into good ones and bad ones.

• 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

Compound Statements

• Conjunctions (Conjunction Junction)

• Two simple statements may be connected with a conjunction

• The conjunction “and”

• The disjunction “or”

• “and”

• Symbolized by “•”

• "Roses are Red and Violets are blue.“

• "Republicans are conservative and Democrats are liberal.“

• P • Q (P and Q)

• “or”

• Symbolized by “v”

• "Republicans are conservative or Republicans are moderate

• P v Q

• Not

• Symbolized by ~

• That is not a rose

• Bob is not a Republican

• ~A

• These may be used to symbolize complex statements

• The other symbol of value is

• Equivalence ()

• This is not quite the same as “equal to”.

• 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

• 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

• 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

• 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

• 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

• We use the conditional or implication a great deal.

• It is the core statement of the scientific law, and hence the hypothesis.

• 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

• In order to use these logical components, we have constructed “rules of Inference”

• These rules are essentially “how we think.”

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

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

• Classic reasoning

• All men are mortal.

• Socrates is a man.

• Therefore Socrates is mortal.

• 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.

• 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.

• 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.

• 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

• Epimenedes the Cretan says that all Cretans are liars.“

• < The next statement is true.

• < The previous statement is false.

• For further info

• 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.

• 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?

• It is possible for voting preferences to result in elections in which a less preferred candidate wins over a preferred one.

• 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.