# 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