https://teddyrookbook.com/logic/symbolic-logic/Recent content in Symbolic Logic on Teddy Rook Bookhttps://teddyrookbook.com/img/share.jpghttps://teddyrookbook.com/img/share.jpgHugoenhttps://teddyrookbook.com/logic/symbolic-logic/simple-compound/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/simple-compound/A simple statement has no other statement as a part; a compound statement is built from simpler ones with words like not, and, or, if. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/connectives/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/connectives/Negation ¬, conjunction ∧, disjunction ∨, conditional → and biconditional ↔ build compound formulas from statement letters. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/main-connective/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/main-connective/Parentheses group a formula; the main connective is the one with the widest scope, applied last, and it decides what kind of statement the formula is. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/translate-not-and-or/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/translate-not-and-or/"But", "although" and "yet" are ∧; "neither p nor q" is ¬(p ∨ q) or ¬p ∧ ¬q; "not both" is ¬(p ∧ q). Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/translate-conditionals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/translate-conditionals/"If p, q", "q if p", "p only if q", "q provided that p", "p is sufficient for q" and "q is necessary for p" are all p → q; "p unless q" is ¬q → p, or p ∨ q. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/translate-biconditional/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/translate-biconditional/"p if and only if q" and "p just in case q" are p ↔ q; longer sentences are translated from the main connective inward. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/symbols-to-english/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/symbols-to-english/Read a formula aloud in plain English that keeps its grouping, using "either…or", "both…and" and "it is not the case that". Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/truth-functions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/truth-functions/Each connective is a truth function: the truth value of the compound depends only on the values of its parts. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/compute-truth-value/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/compute-truth-value/Given the values of the letters, work from the innermost parts outward to find the value of the whole formula. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/truth-tables/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/truth-tables/A formula with n letters has 2ⁿ rows; list them in a fixed order and fill a column for each part, ending with the main connective. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/tautology-contradiction/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/tautology-contradiction/A tautology is true in every row, a self-contradiction false in every row, and a contingent statement true in some rows and false in others. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/equivalence/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/equivalence/Two formulas are logically equivalent when they have the same truth value in every row; then their biconditional is a tautology. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/de-morgan/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/de-morgan/¬(p ∧ q) is equivalent to ¬p ∨ ¬q, and ¬(p ∨ q) to ¬p ∧ ¬q: moving a negation inside flips ∧ and ∨. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/conditional-family/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/conditional-family/p → q is equivalent to its contrapositive ¬q → ¬p, but not to its converse q → p or its inverse ¬p → ¬q. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/statement-relations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/statement-relations/Statements are consistent if some row makes them all true; two statements are contradictory if they have opposite values in every row. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/validity-truth-table/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/validity-truth-table/An argument is valid when no row makes every premise true and the conclusion false. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/short-truth-table/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/short-truth-table/Assume the premises true and the conclusion false, then work out the forced values; if that leads to a contradiction, the argument is valid. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/argument-forms/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/argument-forms/Modus ponens, modus tollens, hypothetical and disjunctive syllogism are valid forms; affirming the consequent and denying the antecedent are invalid. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/proofs-intro/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/proofs-intro/A proof is a numbered list of lines; each is a premise or follows from earlier lines by a rule, written beside it with the line numbers it uses. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/mp-mt/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/mp-mt/MP: from p → q and p, infer q. MT: from p → q and ¬q, infer ¬p. The letters can stand for any formulas. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/hs-ds/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/hs-ds/HS: from p → q and q → r, infer p → r. DS: from p ∨ q and ¬p, infer q. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/simp-conj-add/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/simp-conj-add/Simp: from p ∧ q, infer p. Conj: from p and q, infer p ∧ q. Add: from p, infer p ∨ q for any q at all. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/cd/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/cd/CD: from (p → q) ∧ (r → s) and p ∨ r, infer q ∨ s. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/substitution-instances/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/substitution-instances/A rule of inference applies only to whole lines whose main connectives match the rule's pattern, with any formulas in place of p, q, r and s. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/justify-lines/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/justify-lines/Name the rule and the earlier lines that justify each line of a finished proof. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/first-proofs/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/first-proofs/Supply the missing lines of short proofs, choosing each step by what the conclusion needs. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/proof-strategy/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/proof-strategy/Find the conclusion inside the premises, then plan which rule would produce it and what that rule needs. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/replacement-intro/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/replacement-intro/A replacement rule says two forms are equivalent, so either can replace the other anywhere, even inside a line, in either direction. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/dm-dn/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/dm-dn/DM: ¬(p ∧ q) :: ¬p ∨ ¬q and ¬(p ∨ q) :: ¬p ∧ ¬q. DN: p :: ¬¬p. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/com-assoc-dist/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/com-assoc-dist/Com swaps the sides of ∧ or ∨; Assoc regroups three parts joined by the same connective; Dist spreads ∧ over ∨ and ∨ over ∧. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/trans-impl/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/trans-impl/Trans: p → q :: ¬q → ¬p. Impl: p → q :: ¬p ∨ q. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/equiv-exp-taut/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/equiv-exp-taut/Equiv: p ↔ q :: (p → q) ∧ (q → p), or (p ∧ q) ∨ (¬p ∧ ¬q). Exp: (p ∧ q) → r :: p → (q → r). Taut: p :: p ∧ p and p :: p ∨ p. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/replacement-proofs/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/replacement-proofs/Combine rules of inference and replacement, often reshaping a line with replacement until an inference rule fits. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/conditional-proof/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/conditional-proof/To prove p → q, assume p, derive q inside an indented subproof, then close it and write p → q. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/indirect-proof/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/indirect-proof/To prove p, assume ¬p, derive a contradiction q ∧ ¬q, then close the subproof and write p. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/nested-subproofs/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/nested-subproofs/Subproofs can sit inside other subproofs; each must be closed, innermost first, and a theorem can be proved this way from no premises at all. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/predicates-constants/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/predicates-constants/A predicate letter stands for a property (F: is friendly) and a lowercase name for an individual (a: Ava); Fa says that Ava is friendly. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/quantifiers/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/quantifiers/∀x Fx says everything is F; ∃x Fx says at least one thing is F. A variable is bound when a quantifier covers it. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/translate-categorical/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/translate-categorical/A: ∀x(Sx → Px). E: ∀x(Sx → ¬Px). I: ∃x(Sx ∧ Px). O: ∃x(Sx ∧ ¬Px). Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/translate-harder/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/translate-harder/Handle "only", "none but", "all except", complex subjects (red cars) and complex predicates (fast or loud) inside one quantifier. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/quantifier-negation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/quantifier-negation/¬∀x Fx is equivalent to ∃x ¬Fx, and ¬∃x Fx to ∀x ¬Fx: moving a negation across a quantifier flips ∀ and ∃. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/truth-in-world/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/truth-in-world/In a world with a few named objects, ∀x Fx is true when every object is F, and ∃x Fx when at least one is. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/counterexample-interpretation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/counterexample-interpretation/Show an argument invalid by describing a small world where all its premises are true and its conclusion false. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/ui-eg/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/ui-eg/UI: from ∀x Fx, infer Fa for any name. EG: from Fa, infer ∃x Fx. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/ei/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/ei/EI: from ∃x Fx, infer Fa only for a name a that is new: not in any earlier line, the premises or the conclusion. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/ug/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/ug/UG: from Fy, infer ∀x Fx only when y is a variable standing for any object, not a name and not one introduced by EI. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/quantifier-proofs/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/quantifier-proofs/Strip the quantifiers (EI before UI), reason with the propositional rules, then put quantifiers back with UG or EG. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/quantifier-cp-ip/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/quantifier-cp-ip/CP, IP and quantifier negation (QN) work in predicate logic too, as long as each quantifier rule's restrictions are kept. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/relations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/relations/A relational predicate links two or more objects in order: Lab says a likes b, which is not the same as Lba. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/quantifier-order/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/quantifier-order/∀x∃y Lxy (everyone likes someone) differs from ∃y∀x Lxy (someone is liked by everyone); the order of mixed quantifiers matters. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/relation-properties/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/relation-properties/A relation can be reflexive (Rxx for all x), symmetric (Rxy → Ryx) and transitive (Rxy ∧ Ryz → Rxz), or fail any of them. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/identity/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/identity/a = b says a and b are the same object; identity lets us say "only", "someone other than" and "the same". Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/counting-identity/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/counting-identity/With identity we can count: "at least two F" is ∃x∃y(Fx ∧ Fy ∧ x ≠ y); "at most one" and "exactly one" follow the same pattern. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/identity-proofs/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/identity-proofs/The identity rules: a = a may be written anywhere, a = b gives b = a, and from Fa and a = b infer Fb. Symbolic Logic · Logic Ladderhttps://teddyrookbook.com/logic/symbolic-logic/soundness-completeness/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/logic/symbolic-logic/soundness-completeness/Provable (⊢) is about rules; valid (⊨) is about truth. A system is sound if everything provable is valid, and complete if everything valid is provable; our rules are both. Symbolic Logic · Logic Ladder