Logic

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# Logic - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Logic' - gwydion

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Presentation Transcript
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
• The study by which arguments are classified into good ones and bad ones.
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
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.
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
• Epimenedes the Cretan says that all Cretans are liars.“
• < The next statement is true.
• < The previous statement is false.
• For further info
• 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.
• 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?