He lived 1815 to 1864 and studied symbolic logic and logic of classes or extensional logic. Arithmetic and algebras have axioms and theorems allowing independent term or variable meanings. Axioms and theorems can be statements, sets, classes, events, or durations. Syllogisms can use arithmetic notation, and algorithm can prove them {Boolean algebra, Boole}. Boolean algebra has sets, union operation, intersection operation, complement operation, zero element, and unit element. Arithmetic axioms hold for elements and operations.
Epistemology
Mind has ability to conceive class, designate individual class members by common name, perform other logical tasks, and think logically {laws of thought, Boole}. Thought laws are innate and inherited.
He lived 1822 to 1900. Because circle chords can have varying angles to tangents, for example perpendicular to radius and parallel to tangent, different ways of selecting chords lead to different probabilities that chord is less than inscribed-equilateral-triangle side {Bertrand's paradox}.
He lived 1832 to 1898 and studied symbolic logic. Assuming inference rule is not the same as assuming conditional statement.
He lived 1862 to 1956. Integers are describable in words with a finite number of letters. An integer exists that is the least integer not describable in 100 or less letters. However, that phrase has less than 100 letters {Richard's paradox} [1905].
He lived 1881 to 1966, tried to define numbers, and helped develop quantum logic. He helped develop the idea that mathematics requires mental constructions for truth {intuitionism, Brouwer} [1924]. Unconstructed and non-existent things cannot be the basis for truth. Infinities cause excluded-middle-law contradiction, so mathematics cannot use this law.
He lived 1861 to 1947 and was idealist. He studied logical analysis, axiomatized logic, and developed logicism. Events can relate {process, Whitehead}. Relations and events transform object properties. Objects are always changing properties or property values. Reality is about such changes {process philosophy, Whitehead}. Since no properties exist for significant times, processes and relations are more important than matter, time, and position. All things interconnect and continually adjust to environment {philosophy of organism, Whitehead}. Higher properties emerge from lower systems. God is always becoming, and this unifies universe. Qualities are not substances but are mind-activity results.
He lived 1880 to 1940 and helped develop three-valued logic [1910 to 1913].
He lived 1883 to 1964. Elements {Sheffer stroke element} can equal "Not AND" and fire if either, but not both, of two input elements fire. Sheffer-stroke-element combinations can make OR element, AND element, and NOT element. Using many Sheffer stroke elements creates devices whose output fires if and only if most inputs fire.
Skolem lived 1887 to 1963. Lowenheim lived 1878 to 1957. If countable sets have formal models, domain is countable {Löwenheim-Skolem theorem}, as proved by Löwenheim [1915] and Skolem [1920]. However, real numbers are not countable {Skolem paradox}. Models {nonstandard model} can have elements that are not countable.
He lived 1886 to 1939 and invented definition theory. He helped develop quantum logic, based on equivalence {protothetic logic, Lesniewski}, abstract quantifiers {ontology logic, Lesniewski}, and part and whole relations {mereology, Lesniewski}. Logic is not about real world, only about statements. Wholes are not just sets or sums of parts, because parts relate. Because living things can replace parts, modal or temporal logic can maintain integrated wholes by maintaining relations among replaced parts.
He lived 1897 to 1954. Symbol strings can substitute other symbol strings {Post grammar, Post} {Post machine} [1936], to make formal systems. Start with long symbol string and substitute, using symbol-string precedence rules.
Logic can be three-valued {many-valued logic}. Many-valued logic can use cyclic negation, so next truth-value negates previous one. Such systems include all finite-valued logics. Such logics can represent switching circuits with many inputs and/or outputs.
He lived 1891 to 1953, studied analytic philosophy, and helped develop quantum logic. Spaces and times are relative. Probability depends on frequency. Induction depends on frequency. The geometry people use for universe is just conventional, not real, because instruments can systematically alter from expectations.
He lived 1883 to 1964, helped develop modal or relevance logic, developed implication requiring necessity {strict implication, Lewis}, and studied phenomenalism.
He lived 1898 to 1980 and helped develop quantum logic [1930].
He lived 1906 to 1965 and invented natural deduction and worked with infinite-valued logic [1934 to 1936].
He lived 1909 to 1945. He developed formal first-order logic {natural deduction, Gentzen} [1935], which only assumes inference laws. One rule uses premises and operator to make compound statement {introduction rule, Gentzen}. Another rule uses compound statement and statement to make statement. Statements depend on simple and compound sequent statements. Sequent-calculus proofs can be truth-trees or truth-tables {cut elimination theorem, Gentzen}, which eliminate formulas. Natural deduction led to proof theory.
He lived 1902 to 1983, founded modern logical theory, studied part and whole relations {mereology, Tarski}, helped develop quantum logic, and invented Banach-Tarski theorem.
Convention establishes basic-linguistic-element use and meaning {basic vocabulary}, which can construct complex term and sentence meanings {compositional semantics} {recursive semantics}.
Formal languages have consistent syntax, in which sentences form correctly or not. Formal language uses objects to replace language variables and predicates to replace language functions {interpretation, Tarski}. Truth is about interpretation {semantic theory of truth}. Determining truth requires defining what constitutes satisfying interpretation {satisfaction, Tarski}, which requires metalanguage {Tarski's theorem}. Formal languages have true interpretations {model, Tarski}. Premise sets can be models. If premise model is sentence model, sentences are premise-set consequences {theory of logical consequence} {logical consequence theory}.
For two sentence systems, sentences in one system can derive from sentences in other system {equipollence, Tarski}.
He lived 1886 to 1941. Self-applicable can mean thing expresses property that it has. Self-applicable can mean expression applies to itself. If heterological means not-self-applicable, then heterological is both self-applicable and not-self-applicable {Grelling's paradox} {Weyl's paradox}.
He lived 1903 to 1995, studied denotation, and helped develop quantum logic. Symbol strings can represent numbers and functions. Using functions on input function and data strings makes output function and data strings {lambda calculus, Church}. Lambda acts on variable or function, or variable and function combination, which is second-function dummy variable: lambda(x(f(x))) = f, lambda(x(f(x)))(a) = f(a), lambda(f(f(f(x)))) = lambda(f(lambda(x)(f(f(x))))) = lambda(f(x)(f(f(x)))). This expression is a function and precedes a value, which substitutes into function. In particular, after lambda, expressions can have variable zero times, function of variable one time, function of function of variable two times, and so on: 0 = lambda(f(x)(x)), 1 = lambda(f(x)(f(x))), 2 = lambda(f(x)(f(f(x)))). Function of function equals lambda and function of function {abstraction, lambda calculus}: f(f(x)) = lambda(x)(f(f(x))). Really, symbols are functions. Lambda calculus represents recursion, iteration, and algorithm loops. Recursive functions can be equation sets. Recursive functions are computable {Church's theorem}. Functions are computable if they are recursive {Church's thesis, recursion}. Recursive functions can be lambda calculus. Lambda calculus is equivalent to Post grammar and Turing machine and so can express all algorithms. LISP computer language depends on lambda calculus.
Epistemology
Formal systems can prove most theorems {effectively calculable} {computability}. Lambda calculus shows that it is impossible to prove some valid theorems in most formal systems, including arithmetic.
He lived 1909 to 1994, studied recursion theory and formal logic, and added subtraction to lambda calculus. At least one mathematical truth is true intuitionistically but not Platonically [Kleene, 1952].
She lived 1921 to ? and studied modal logic. The possibility that something has an attribute implies that something exists that possibly has the attribute {Barcan formula}, assuming that possible worlds overlap.
He lived 1921 to ? and invented fuzzy-set theory or fuzzy logic.
He lived 1923 to ? and developed laws of form {calculus of indications, Spencer-Brown}, based on differences. Autopoietic theory references his work.
They helped develop relevance logic, modal logic, deontic logic, and logical connectives.
Granting permission for two things can sound like permitting first or second, and so like exclusive OR, but is actually conjunction {confectionary fallacy, Jennings}. It is a deduction fallacy.
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Date Modified: 2022.0225