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diff --git a/mathematics/logic.md b/mathematics/logic.md index dca0cac..e19cb52 100644 --- a/mathematics/logic.md +++ b/mathematics/logic.md @@ -20,48 +20,60 @@ and many others. ### [propositional logic](https://ncatlab.org/nlab/show/propositional+logic) -**propositional logic** or *0th-order logic* deals with raw *propositions*. +**propositional logic** or **zeroth-order logic** deals with raw *propositions*. **propositions** are statements that *reduce* to a **truth value**. -truth values are classically either true or false. i'm not quite sure if there are alternative approaches that change this. +truth values are classically either true or false. in non-classical logics, this can differ. the basic foundations of propositional logic are as follows: - | -------|-------------- -p | a proposition -¬p | not p -p → q | p implies q -⊤ | true -⊥ | false -⊢ | derives/yields - -several logical concepts are *derivable* from the above: - - | -------|--------------------|---- -p ∨ q | ¬p → q | or -p ∧ q | ¬(p → ¬q) | and -p ⟺ q | (p → q) ∧ (q → p) | iff -p ⊕ q | (p ∨ q) ∧ ¬(p ∧ q) | xor - -to do anything interesting, several rules must exist: - -replacement rules | -------------------------------------|---------------- -((p ∨ q) ∨ r) → (p ∨ (q ∨ r)) | associative -(p ∧ q) → (q ∧ p) | commutative -((p ∧ q) → r) → (p → (q → r)) | exportation -(p → q) → (¬q → ¬p) | transposition -(p → q) → (¬p ∨ q) | implication -((p ∨ q) ∨ r) → ((p ∧ r) ∨ (q ∧ r)) | distributive -((p) ∧ (q)) → (p ∧ q) | conjunction -p ⟺ (¬¬p) | double negation - -double negation is not allowed in intuistionistic logic systems. +notation | definition +---------|-------------- +p | a *proposition* +¬p | *not* p +p → q | *if* p *then* q, p *implies* q +0 | *false* +1 | *true* + +several logical connectives are *derivable* from the above: + +notation | derivation | definition +------|---------------------|---- +p ∨ q | ¬p → q | p *or* q, *disjunction* +p ∧ q | ¬(p → ¬q) | p *and* q, *conjunction* +p → q | ¬p ∨ q | p *implies* q, (material) *implication* (again) +p ↔ q | (p → q) ∧ (q → p) | p *if and only if* q, p *iff* q +p ⊕ q | (p ∨ q) ∧ ¬(p ∧ q) | p *exclusively or* q, p *xor* q +p ↑ q | ¬(p ∧ q) | p *not both* q, p *nand* q +p ↓ q | ¬(p ∨ q) | *neither* p *nor* q, p *nor* q + +note that several of these definitions are circular. +our choice in $¬$ and $→$ as the primitive connectives is thus arbitrary. +interestingly, ↑ and ↓ are *functionally complete*: we may define all other connectives in terms of them. + +<details> +<summary>aside: nand and nor</summary> + +notation | definition +---------|----------- +¬p | p ↑ p +p → q | p ↑ ¬q +p ∨ q | ¬p ↑ ¬q +p ∧ q | (p ↑ q) ↑ (p ↑ q) +p ↔ q | (p ↑ q) ↑ (p ∨ q) + +notation | definition +---------|----------- +¬p | p ↓ p +p → q | (¬p ↓ q) ↓ (¬p ↓ q) +p ∨ q | (p ↓ q) ↓ (p ↓ q) +p ∧ q | ¬p ↓ ¬q +p ↔ q | ... + +</details> ### [predicate logic](https://ncatlab.org/nlab/show/predicate+logic) -**predicate logic** or *first-order logic* adds variables and quantifiers to propositions: +**predicate logic** or **first-order logic** adds variables and quantifiers to propositions: * ∀x: for all x * ∃y: there exists a y |