Logic: Lecture 1, 07 August 2012
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What are we going to do in this course?

Structure of logical arguments

Syntax and semantics

  All men are mortal.
  Socrates is a man.
  Therefore, Socrates is mortal.

What's important?  "All"?  "Mortal"?

  Borogoves are mimsy whenever it is brillig.
  It is now brillig and this thing is a borogove.
  Hence this thing is mimsy.

Validity of the whole statement is independent of the meaning of
the parts.

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Two broad streams of study in logic

Proof theory: Study of formal systems of reasoning, structure
              of proofs

Model theory: Logic as a language to describe properties of
              mathematical structures, what can and cannot be
              expressed, what properties are algorithmically
              verifiable etc

The focus of this course will primarily be model theory
 
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Propositional logic;

Syntax

  Infinite set P of atomic propositions/statements/facts

  Formulas: p in P, not A, A or B  [Notation: ~A for not A]

  Structural induction principle

Semantics

  Valuation: v: P -> {tt, ff}

  Extend v (uniquely) to all formulas
    - defines our interpretation of not and or

Definition: satisfiability and validity

  A is satisfiable if there is a valuation v such that v(A) = tt
  A is valid if for every valuation v, v(A) = tt

  A is valid iff ~A is not satisfiable

Vocabulary 

  Voc(A) = set of propositions appearing in A

  Valuations that agree on Voc(A) assign same value to A

Hence, build finite truth table to decide if a given formula A
is valid/satisifiable --- effective algorithm

Validity is not always algorithmically decidable, but we may be
able to enumerate all valid formulas (semi-decision procedure)

Preview of axiomatizations
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