Brief Introduction to Logic

Outline

- Historical View
- Propositional Logic Syntax
- Propositional Logic Semantics
- Satisfiability
- Natural Deduction Proofs.

Historical view

- Philosophical Logic
- 500 BC to 19th Century
- Symbolic Logic
- Mid to late 19th Century
- Mathematical Logic
- Late 19th to mid 20th Century
- Logic in Computer Science

Philosophical Logic

- 500 B.C 19th Century
- Logic dealt with arguments in the natural

language used by humans. - Example
- All men are motal.
- Socrates is a man
- Therefore, Socrates is mortal.

Philosophical Logic

- Natural language is very ambiguous.
- Eric does not believe that Mary can pass any

test. - I only borrowed your car.
- Tom hates Jim and he likes Mary.
- It led to many paradoxes.
- This sentence is a lie. (The Liars Paradox)

The Sophists Paradox

- A Sophist is sued for his tuition by the school

that educated him. He argued that he must win,

since, if he loses, the school didnt educated

him well enough, and doesnt deserve the money.

The school argue that he must loss, since, if he

win, he was educated well enough therefore should

pay for it.

Symbolic Logic

- Mid to late 19th Century.
- Attempted to formulate logic in terms of a

mathematical language - Rules of inference were modeled after various

laws for manipulating algebraic expressions.

Mathematical Logic

- Late 19th to mid 20th Century
- Frege proposed logic as a language for

mathematics in 1879. - With the rigor of this new foundation, Cantor was

able to analyze the notion of infinity in ways

that were previously impossible. (2N is strictly

larger than N) - Russells Paradox
- T S S ? S

Logic in Computer Science

- In computer science, we design and study systems

through the use of formal languages that can

themselves be interpreted by a formal system. - Boolean circuits
- Programming languages
- Design Validation and verification
- AI, Security. Etc.

Logics in Computer Science

- Propositional Logic
- First Order Logic
- Higher Order Logic
- Theory of Construction
- Real-time Logic, Temporal Logic
- Process Algebras
- Linear Logic

Syntax

- The symbol of the language.
- Propositional symbols A, B, C,
- Prop set of propositional symbols
- Connectives ? (and), ? (or), ? (not), ?

(implies), ? (is equivalent to), ? (false). - Parenthesis (, ).

Formulas

- Backus-Naur Form
- Form Prop (?Form) (Form o Form).
- Context-Free Grammar
- Form ? Prop,
- Form ? (? Form),
- Form ? (Form o Form)

Formulas (2)

- The set of formulas, Form, is defined as the

smallest set of expressions such that - Prop ? Form
- p?Form ? (?p)?Form
- p,q ?Form ? (p o q) ? Form

Formulas (3)

- Examples
- (?A)
- (?(?A))
- (A ? (B ? C))
- (A? (B? C))
- Correct expressions of Propositional Logic are

full of unnecessary parenthesis.

Formulas (4)

- Abbreviations. Let o?, ?, ?. We write
- A o B o C o
- in the place of
- (A o (B o (C o )))
- Thus, we write
- A ? B ? C, A?B?C,
- in the place of
- (A ? (B ? C)), (A? (B? C))

Formulas (5)

- We omit parenthesis whenever we may restore them

through operator precedence - ? binds more strictly than ?, ?, and ?, ? bind

more strictly than ?, ?. - Thus, we write
- ??A for (?(?A)),
- ?A ?B for ((?A ) ?B)
- A ?B? C for ((A?B) ? C),

Semantics

- Def) A truth assignment, ?, is an elements of

2Prop(I.e., ? ? 2Prop). - Two ways to think of truth assignment
- 1) X ? Prop
- 2) ? Prop ? 0,1
- Note These notions are equivalence.

Philosophers view

- ? p means
- ? satisfies p or
- ? is true of p or
- p holds at ? or
- ? is a model of p

Satisfaction Relation

- Def 1) ? (2Prop x Form)
- ? A if ? (A) 1 (or, A ? ?)
- ? ?p if it is not the case ? p.
- ? p?q if ? p and ? q
- ? p ? q if ? p or ? q
- ? p ? q if ? p implies ? q
- ? p ? q if ? p iff ? q

Satisfaction Relation

- A,B A ? B
- Iff A,B A and A,B B
- Iff A ? A,B and B ? A,B

Electrical Engineers view

- A mapping of voltages on a wire ? Prop ? 0,1
- ? 0,1 ? 0,1
- ?(0) 1 and ?(1) 0
- ? 0,12 ? 0,1
- ?(0,0) ?(0,1) ?(1,0)0 and ?(1,1)1
- ? 0,12 ? 0,1
- ?(1,1) ?(0,1) ?(1,0)1 and ?(0,0)0

Semantics

- Def 2)
- A(?) ?(A)
- (?p)(?) ?(p(?))
- (p o q)(?) o(p(?), q(?))
- Lemma) Let p ? Form and ? ? 2Prop, then ? p

iff p(?) 1.

Software Engineers view

- Intuition a formula specifies a set of truth

assignments. - Def 3) Function Models From ? 22Prop
- models(A) ? ?(A) 1, A ? Prop
- models(?p) 2Prop models(p)
- models(p?q) models(p) ? models(q)
- models(p?q) models(p) ? models(q)
- models(p?q) (2Prop models(p)) ? models(q)

Theorem

- Let p ? Form and ? ? 2Prop, then the following

statements are all true - 1. ? p
- 2. p(?) 1
- 3. ? ? models(p)

Relevance Lemma

- Lets use AP(p) to denote the set of all

propositional symbols occurred in p. Let ?1, ?2 ?

2Prop, p?Form. - Lemma) if ?1AP(p) ?2AP(p) , then
- ?1 p iff ?2 p
- Corollary) ? p iff ?AP(p) p

Algorithmic Perspective

- Truth Evaluation Problem
- Given p?Form and ? ? 2AP(p), does

? p ? Does p(?) 1 ? - Eval(p, ?)
- If p ? A, return ?(A).
- If p ? (?q), return ?(Eval(q, ?))
- If p ? (q o r), return o(Eval(p), Eval(q))
- Eval uses polynomial time and space.

Extension of

- Let T ? 2Prop, ? ? Form
- Def) T p if T ? models(p)
- i.e., ? 22Prop X Form
- Def) T ? if T ? models(?)
- models(?) ?p?? models(p)
- I.e., ? 22Prop X 2Form

Extension of

- ? 2Form x 2Form
- Def) ?1 ?2
- iff models(?1) ? models(?2)
- Iff for all ? ? 2Prop
- if ? ?1 then ?

?2

Semantic Classification

- A formula p is called valid if models(p) 2Prop.

We denote validity of the formula p by p - A formula p is called satisfiable if models(p) ?

?. - A formula is not satisfiable is called

unsatisfiable or contradiction.

Semantic Classification(II)

- Lemma
- A formula p is valid iff ?p is unsatifiable
- p is satisfiable iff ?p is not valid
- Lemma
- p q iff (p ? q)

Satisfiability Problem

- Given a p, is p satisfiable?
- SAT(p)
- B0
- for all ? ? 2AP(p)
- B B ? Eval(p,?)
- end
- return B
- NP-Complete

Proofs

- Formal Proofs. We introduce a notion of formal

proof of a formula p Natural Deduction. - A formal proof of p is a tree whose root is

labeled p and whose children are assumptions p1,

p2, p3, of the rule r we used to conclude p.

Proofs

- Natural Deduction Rules. For each logical symbol

o?, ?, ?, ?, and each formula p with outermost

connective o, we give - A set of Introduction rules for o, describing

under which conditions p is true - A set of Elimination rules for o, describing what

we may infer from the truth of p.

Proofs

- Natural Deduction notations for proofs.
- Let p be any formula, and ? be a set of formulas.

We use the notation - ?
- p
- abbreviated by ?- p, for
- there is a proof of p whose assumptions are

included in ?.

Proofs

- Natural Deduction assumptions of a proof
- p1 p2 p3
- r --------------------------------
- p
- are inductively defined as
- all assumptions of proofs of p1, p2, p3, , minus

all assumptions we crossed.

Proofs

- Identity Principle The simplest proof is
- p
- -----
- p
- having 1 assumption, p, and conclusion the same

p. - We may express it by ?-p, for all p??
- We call this proof The Identity Principle (from

p we derive p).

Proofs

- Rules for ?
- Introduction rules none (? is always false).
- Elimination rules from the truth of ? (a

contradiction) we derive everything - ?
- ----
- p
- If ?- ?, then ?-p, for all p

Proofs

- Rules for ?
- Introduction rules
- p q
- --------
- p ? q
- If ?- p and ?- q then ?- p ? q

Proofs

- Elimination rules
- p ? q p ? q
- -------- -------
- p q
- If ?- p ? q, then ?- p and ?- q

Proofs

- Rules for ? Introduction rule
- p
- q
- --------
- p?q
- If ?,p - q, then ?-p?q
- We may drop any number of assumptions equal to p

from the proof of q.

Proofs

- Elimination rule
- p?q p
- ----------------
- q
- If ?-p?q and ?-p, then ? - q.

Proofs

- The only axiom not associated to a connective,

nor justified by some Introduction rule, is

Double Negation - ?p
- .
- ?
- ---
- p
- If ?, ?p- ?, then ?-p
- We may drop any number of assumptions equal to ?p

from the proof of q.

Soundness

- - p then p

Completeness

- p then - p