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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. 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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Logic l.jpg
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 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


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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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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 isn’t 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 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


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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?


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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 > 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.