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

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+# 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