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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
Logic

A short primer on Deduction and Inference

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

logic2
Logic

A short primer on Deduction and Inference

We need to try to avoid skewed logic.

logic3
Logic

What is Logic?

  • Logic
    • The study by which arguments are classified into good ones and bad ones.
logical systems
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
logic5
Logic

Compound Statements

  • Conjunctions (Conjunction Junction)
    • Two simple statements may be connected with a conjunction
      • The conjunction “and”
      • The disjunction “or”
the conjunction operator
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
The disjunction operator
  • “or”
  • Symbolized by “v”
    • "Republicans are conservative or Republicans are moderate
    • P v Q
negation
Negation
  • Not
  • Symbolized by ~
    • That is not a rose
    • Bob is not a Republican
    • ~A
operators
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
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
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
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
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
  • 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
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
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
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
Modus Ponens
  • This is the classic rule of inference for scientific explanation.
modus tollens
Modus Tollens
  • This reflects the idea of rejecting the theory when the consequent is not observed as expected.
hypothetical syllogism
Hypothetical Syllogism
  • Classic reasoning
    • All men are mortal.
    • Socrates is a man.
    • Therefore Socrates is mortal.
logical systems22
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
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
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
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
slide26
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
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
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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