G4G0-2/AI & Data Mining/Week 20/Week 20 - Intro to Propositional Logic.md

Propositions

• Declarative sentences with truth values (T or F)
• Atomic propositions (basic building blocks)
• Compound propositions (combinations of atomic propositions)

Argument Form: Modus Ponens (Affirming the Antecedent)

• Content: If it’s raining, then the ground is wet. The ground is wet. Therefore, it was raining.

Argument Form: Disjunctive Syllogism

• Content: Either it’s raining or the plants need water. It’s not raining. Therefore, the plants need water.

Propositional Connectives

• Negation (¬): p is true if and only if ¬p is false
• Conjunction (∧): p ∧ q is true if and only if both p and q are true
• Disjunction (∨): p ∨ q is true if and only if at least one of p or q is true
• Implication (⇒): p ⇒ q is false if and only if p is true and q is false
• Equivalence (⇔): p ⇔ q is true if and only if p and q have the same truth value

Precedence Order of Connectives

1. Negation (¬)
2. Conjunction (∧)
3. Disjunction (∨)
4. Implication (⇒)
5. Equivalence (⇔)

This means that in a formula without parentheses, ¬ takes precedence over ∧ and ∨, ∧ and ∨ have the same precedence but associativity to the left, and ⇒ and ⇔ also have the same precedence but associativity to the right. For example, p ∧ q ⇒ r is equivalent to (p ∧ q) ⇒ r, not p ∧ (q ⇒ r).

Propositions and Connectives (Examples)

Atomic Propositions:

• p: The cat is on the mat.
• q: The dog is sleeping.

Compound Propositions Using Connectives:

• p ∧ q: The cat is on the mat and the dog is sleeping.
• ¬p: It’s not the case that the cat is on the mat.
• p ∨ q: Either the cat is on the mat or the dog is sleeping (or both).
• p ⇒ q: If the cat is on the mat, then the dog is sleeping.
• p ⇔ q: The cat is on the mat if and only if the dog is sleeping.

Natural Language Statements

• Natural language statement: If you study hard, then you will pass the exam.
  • Formalized as: p ⇒ q
• Natural language statement: Either you will go to the party or stay home and study.
  • Formalized as: p ∨ ¬q

Logic Formulae

• Formula: (p ∧ q) ⇒ ¬r
  • Interpretation: If both p and q are true, then r is false.
• Formula: ¬(p ∨ ¬q)
  • Interpretation: It’s not the case that either p is true or q is false.

Truth Tables for Connectives

• Negation: T|F, F|T
• Conjunction: T&T|TT, F&F|FT, TT|T
• Disjunction: T∨T|TT, F∨F|FF, TF|TF
• Implication: T⇒T|TT, F⇒T|FT, TT|F
• Equivalence: T⇔T|TT, F⇔F|FF, TT|T

Truth Table for P ⇒ Q:

P	Q	P ⇒ Q
T	T	T
T	F	F
F	T	T
F	F	T

Classes of Propositions

• Tautologies: Always true (e.g., p ∨ ¬p)
• Contradictions: Always false (e.g., p ∧ ¬p)
• Contingent propositions: Neither tautology nor contradiction

Logical Equivalence (≡)

• Two formulae are logically equivalent if their equivalence is a tautology

Examples

• p ∧ q ≡ q ∧ p
• ¬(p ∨ q) ≡ ¬p ∧ ¬q

Logical Implication or Entailment (|=)

• Formula p entails q if and only if the implication p ⇒ q is a tautology

Examples

• p ∧ q |= q
• ¬(p ∨ ¬q) |= ¬p ⇒ q

Ambiguous and Vague Sentences

• Ambiguity: A sentence with multiple distinct meanings.
  • Ambiguous sentence: “I want to have dinner with you or your friend.”
    • Interpretation 1: You can choose between having dinner with me or my friend.
    • Interpretation 2: I want to have dinner with you and your friend together.
• Vagueness: A sentence with only one meaning, but the distinction between truth and falsity is unclear.
  • Vague sentence: “The book is heavy.”
    • Vague because no quantitative measure of heaviness is provided.

Logic as a Formal Language

• Alphabet: Symbols for denoting propositions, identifiers, punctuation symbols ((), propositional connectives).
• Syntax: Rules defining the order of symbols in sentences, precedence order of connectives.
• Semantics: Assignment of meaning to correctly written sentences.

Examples of Logical Structures

• Argument: If the train has six carriages and serves a rural community, then it is not overcrowded.
  • Formalized as: p ∧ q ⇒ ¬r
• Argument: If Bob eats carrots, then he will be able to see in the dark. Therefore, if Bob can’t see in the dark, then he hasn’t eaten carrots.
  • Formalized as: p ⇒ q ≡ ¬q ⇒ ¬p

Summary

• Logicians focus on argument form
• Deduction involves justifying conclusions based solely on premises
• Connectives join atomic propositions to form compound propositions