127 lines
4.6 KiB
Markdown
127 lines
4.6 KiB
Markdown
# Propositions
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- Declarative sentences with truth values (T or F)
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- Atomic propositions (basic building blocks)
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- Compound propositions (combinations of atomic propositions)
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### Argument Form: Modus Ponens (Affirming the Antecedent)
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- Content: If it’s raining, then the ground is wet. The ground is wet. Therefore, it was raining.
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### Argument Form: Disjunctive Syllogism
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- Content: Either it’s raining or the plants need water. It’s not raining. Therefore, the plants need water.
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## Propositional Connectives
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- Negation (¬): p is true if and only if ¬p is false
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- Conjunction (∧): p ∧ q is true if and only if both p and q are true
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- Disjunction (∨): p ∨ q is true if and only if at least one of p or q is true
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- Implication (⇒): p ⇒ q is false if and only if p is true and q is false
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- Equivalence (⇔): p ⇔ q is true if and only if p and q have the same truth value
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## Precedence Order of Connectives
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1. Negation (¬)
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2. Conjunction (∧)
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3. Disjunction (∨)
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4. Implication (⇒)
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5. Equivalence (⇔)
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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).
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### Propositions and Connectives (Examples)
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#### Atomic Propositions:
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- p: The cat is on the mat.
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- q: The dog is sleeping.
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#### Compound Propositions Using Connectives:
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- p ∧ q: The cat is on the mat and the dog is sleeping.
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- ¬p: It’s not the case that the cat is on the mat.
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- p ∨ q: Either the cat is on the mat or the dog is sleeping (or both).
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- p ⇒ q: If the cat is on the mat, then the dog is sleeping.
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- p ⇔ q: The cat is on the mat if and only if the dog is sleeping.
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#### Natural Language Statements
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- Natural language statement: If you study hard, then you will pass the exam.
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- Formalized as: p ⇒ q
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- Natural language statement: Either you will go to the party or stay home and study.
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- Formalized as: p ∨ ¬q
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#### Logic Formulae
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- Formula: (p ∧ q) ⇒ ¬r
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- Interpretation: If both p and q are true, then r is false.
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- Formula: ¬(p ∨ ¬q)
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- Interpretation: It’s not the case that either p is true or q is false.
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## Truth Tables for Connectives
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- Negation: T|F, F|T
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- Conjunction: T&T|TT, F&F|FT, TT|T
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- Disjunction: T∨T|TT, F∨F|FF, TF|TF
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- Implication: T⇒T|TT, F⇒T|FT, TT|F
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- Equivalence: T⇔T|TT, F⇔F|FF, TT|T
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#### Truth Table for P ⇒ Q:
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| P | Q | P ⇒ Q |
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| --- | --- | ----- |
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| T | T | T |
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| T | F | F |
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| F | T | T |
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| F | F | T |
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## Classes of Propositions
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- Tautologies: Always true (e.g., p ∨ ¬p)
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- Contradictions: Always false (e.g., p ∧ ¬p)
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- Contingent propositions: Neither tautology nor contradiction
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### Logical Equivalence (≡)
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- Two formulae are logically equivalent if their equivalence is a tautology
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#### Examples
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- p ∧ q ≡ q ∧ p
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- ¬(p ∨ q) ≡ ¬p ∧ ¬q
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### Logical Implication or Entailment (|=)
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- Formula p entails q if and only if the implication p ⇒ q is a tautology
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#### Examples
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- p ∧ q |= q
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- ¬(p ∨ ¬q) |= ¬p ⇒ q
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# Ambiguous and Vague Sentences
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- Ambiguity: A sentence with multiple distinct meanings.
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- Ambiguous sentence: “I want to have dinner with you or your friend.”
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- Interpretation 1: You can choose between having dinner with me or my friend.
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- Interpretation 2: I want to have dinner with you and your friend together.
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- Vagueness: A sentence with only one meaning, but the distinction between truth and falsity is unclear.
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- Vague sentence: “The book is heavy.”
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- Vague because no quantitative measure of heaviness is provided.
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# Logic as a Formal Language
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- Alphabet: Symbols for denoting propositions, identifiers, punctuation symbols ((), propositional connectives).
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- Syntax: Rules defining the order of symbols in sentences, precedence order of connectives.
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- Semantics: Assignment of meaning to correctly written sentences.
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### Examples of Logical Structures
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- Argument: If the train has six carriages and serves a rural community, then it is not overcrowded.
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- Formalized as: p ∧ q ⇒ ¬r
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- 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.
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- Formalized as: p ⇒ q ≡ ¬q ⇒ ¬p
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# Summary
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- Logicians focus on argument form
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- Deduction involves justifying conclusions based solely on premises
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- Connectives join atomic propositions to form compound propositions
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