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.