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## 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. We need to try to avoid skewed logic.**Logic**What is Logic? • Logic • For CS/Math Types: from Factasia • A formal language, given a precisely defined syntax and semantics, becomes a logic when rules for correct reasoning in the language are formally described • For Philosophers: from Factasia • Logic is the study of necessary truths and of formal systems for deriving them. • The study by which arguments are classified into good ones and bad ones. (Irving Copi)**Logical Systems**• There are actually many logical systems • The one we will examine in class is called RS1 (I think…it is a simple propositional calculus, in any event.) • 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.**Complex propositions**• Take for instance - US • If Weapons USAttack • If Weapons ~Insp • If ~Insp Weapons • If ~Insp USAttack • Take for instance - Iraq • If Insp ~Weapons • If Weapons ~USAttack • If Insp USAttack**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.