Logic has applications {logic applications}. People can describe how and why they accepted proposition. Sets and logical operations have equivalences. Union is equivalent to AND. Intersection is equivalent to OR. Universal set is equivalent to TRUE. Empty set is equivalent to FALSE. Complementary set is equivalent to NEGATION or NOT.
Logical Boolean algebra can determine proposition validity {proposition algebra} {algebra of propositions}.
Logic algebras {Boolean algebra}| {extensional logic} can depend on set theory {logic of classes}.
operations
Rules for operating on sets are the same as logic rules. Boolean algebra uses the number zero as empty set for false and the number one as universal set for true. Boolean addition is set-theory selection operation. Boolean-algebra negation is set complement: -a = ~a = NOT a. Boolean-algebra addition is set union {inclusive OR, logic}: a + b = (a | b) = a OR b. Boolean-algebra subtraction is set intersection: a - b = (a & b) = a AND b.
laws
Boolean algebra follows set-theory contradiction, commutation, association, and distribution laws. Boolean algebra uses excluded-middle law.
Number calculus is equivalent to logic calculus, if 0 equals truth-value false and 1 equals truth-value true {number, calculus}.
Statements using must and may have logic {deontic logic}.
Valid argument schemas {logical calculus} can use rules or syntax to move from simple valid arguments to complex ones, such as natural-deduction calculus (Gentzen) [1934] or tableau or truth-tree calculus (Beth) [1955]. For first-order logic, semantic proof is also syntactic proof {soundness theorem, logical calculus}.
Predicates can have calculus {predicate calculus, logic} {calculus of relations}.
laws
Predicate calculus uses contradiction law, excluded-middle law, detachment rule, tautology, addition, association, permutation, and summation. AND, NOT, OR, ALL, SOME, and EQUIVALENT have meaning.
variables
Predicates can have variables. Predicate can have more than one variable {n-place predicate}.
first-order
Variable can be terms {first-order predicate}. For first-order predicates, constants are proper nouns, and variables are pronouns or common nouns {term, predicate}.
First-order predicate calculus is complete.
first-order: quantifiers
First-order predicate calculus {functional calculus} {first-order logic} {restricted predicate calculus} can have quantifiers. Quantifiers can affect variables {bound variable} or not {free variable}.
If A implies B, if A and B have bound variables, and if B, then every A value has a B value {generalization rule} {rule of generalization}. If A implies B, if A and B have bound variables, and if B implies A, then an A value exists {specification rule} {rule of specification}.
second-order
Variables can be predicates {second-order predicate}.
second-order: recursion
Predicates can contain themselves {recursive predicate}. Recursive predicates can assume existence of a set that does not actually exist and so have contradiction.
True formula {string, logic} sequences {canonical proof} {proof, logic} {demonstration, logic} go from premises to conclusions without errors. Canonical proofs establish mathematical-statement meaning. Non-canonical proofs establish canonical-proof possibility.
Formal logic {propositional calculus, logic} {sentential calculus} can be about statements {proposition} that have one subject, one predicate, constants, and variables. Statements are propositional functions, with variable for subject and subject property for predicate. Assertion that proposition is true is a different statement than the proposition itself.
connectives
Propositional calculus uses NOT, OR, AND, IMPLIES or IF/THEN, and IF AND ONLY IF connectives. AND, NOT, and OR are constant operators.
quantifiers
Subject can have universal quantifier: for any x, subject has the property. Subject can have existential quantifier: at least one x has the property.
instantiation
Variables have possible-value sets {class, propositional calculus}. Values can substitute for subject variables {universal instantiation}. If arbitrarily selected values have a predicate property, class has property {universal generalization}. If class has property, value has property {existential instantiation}. If value has property, at least one thing in class has property {existential generalization}.
Put all arguments into chain of syllogisms {Scholastic method}| to prove or refute answers.
Valid argument schemas can use semantics and try to find counterexamples. If argument finds none, it is proof {semantic proof}. For first-order logic, semantic validity is also syntactic validity {soundness theorem, semantic proof}. For first-order logic, if one can find semantic counterexample, syntactic calculus cannot prove argument. Second-order, monadic, modal, and temporal logics use semantic argument proofs.
Natural deduction has calculus {sequent calculus}. Statement sequence gives reasoning chain and conclusion {sequent, reasoning}. Stating simple statements {basic sequent} in natural deductions starts premise or conclusion.
rules
Rules {introduction rule} can allow operation to make more complex formulas from simpler ones. Rules {elimination rule} can allow inference from complex formulas to simpler formulas. The reductio-ad-absurdum rule eliminates hypotheses. Sequent-calculus proofs can be truth-trees or truth-tables, which always eliminate formulas {cut elimination theorem, sequent}.
Sets and function-of-sets sets have object types {theory of types, logic} {type theory}.
purpose
Distinguishing between types avoids set-theory paradoxes.
types
Sets about objects have type 0. Sets about functions of type-0 sets have type 1. Sets about type-1 sets have type 2. Type-n sets are sets about type n-1 sets.
reducibility
For any type, an equivalent type-0 propositional function exists {axiom of reducibility, type theory} {reducibility axiom, type theory}. Equivalent type-0 propositional functions {relation, type} have classes as members. Classes have object sets. For example, functions can have two variables, and its class can have variable pairs as members.
class
Classes are similar if they have one-to-one correspondence. They are then reflexive, symmetric, and transitive.
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Date Modified: 2022.0225