Logic: Lecture 1, 04 August 2015
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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 words are 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
properties are algorithmically verifiable etc

e.g.
Forall x, exists y, x < y) : there must be infnitely many
 elements (assuming < is irreflexive)

Forall x, exists y, x != y : only guarantees two elements

We will do some basic proof theory (sound and complete
axiomizations), but the focus of this course will primarily be
model theory.
 
Our main focus will be first-order logic (with quantifiers,
relations etc) that talks about mathematical structures.  To set
the stage, we start with a simpler logic of "atomic facts".

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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]

Derived connectives

  and, implies (->), iff (<->), ..

  Truth tables and "functional completeness" of not and or

Semantics

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

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

Principle of structural induction:

  Any function/predicate defined on P and, by induction, for
  "not" and "or", extends uniquely to a function over all
  formilas.

Definition: satisfiability and validity

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

Fact:
  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, can 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 using axioms and inference
rules. 

Preview of axiomatizations
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