Logic

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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.
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Logic
A short primer on Deduction and Inference
We need to try to avoid skewed logic.
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Logic
What is Logic?

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

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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

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Logic
Compound Statements

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

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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)

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The disjunction operator

• or
• Symbolized by v
• "Republicans are conservative or Republicans are
  moderate
• P v Q

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Negation

• Not
• Symbolized by
• That is not a rose
• Bob is not a Republican
• A

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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.

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Truth Tables

• Statements have truth value
• For example, take the statement PQ
• This statement is true only if P and Q are both
  true.
• P Q PQ
• T T T
• T F F
• F T F
• F F F

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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

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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

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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

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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

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The Implication

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

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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

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Rules of Inference

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

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Modus Ponens

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

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Modus Tollens

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

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Disjunctive Syllogism
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Hypothetical Syllogism

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

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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.

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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.

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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 isnt the circularity, it is the lack
  of utility.

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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

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• The Liars Paradox
• Epimenedes the Cretan says that all Cretans are
  liars.
• The Paper Paradox (a variant of the Liars
  paradox)
• lt The next statement is true.
• lt The previous statement is false.
• For further info
• Russells 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

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Grellings 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?

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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 gt B gt C
• Individual 2 C gt A gt B
• Individual 3 B gt C gt 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.