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List of axioms | A Wisdom Archive on List of axioms | | List of axioms A selection of articles related to List of axioms | |
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List of axioms
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| ARTICLES RELATED TO List of axioms | | | | | | | | |  List of axioms: Encyclopedia II - Controversy over Cantor's theory - Objections to Cantor's theoremAs shown above, most objections to Cantor's theorem (i.e. the theorem that no set can be correlated one-one with the set of all of its subsets) result from misunderstanding it (for it relies on mostly logical assumptions and steps).
Wittgenstein, however, disparages it as trivial, a result that might have been well known before the invention of set theory, "and familiar even to school-children". The child wonders, given a list of decimals, how to write a number different from any on the list. "The method says: Not at all: change the f ...
See also:Controversy over Cantor's theory, Controversy over Cantor's theory - Preface, Controversy over Cantor's theory - Introduction, Controversy over Cantor's theory - Cantor's argument, Controversy over Cantor's theory - Reception of the argument, Controversy over Cantor's theory - Naïve objections, Controversy over Cantor's theory - Objections to Cantor's theorem, Controversy over Cantor's theory - Objections to Hume's principle, Controversy over Cantor's theory - Objection to the axiom of infinity, Controversy over Cantor's theory - Objections to the power set axiom, Controversy over Cantor's theory - Footnote Read more here: » Controversy over Cantor's theory: Encyclopedia II - Controversy over Cantor's theory - Objections to Cantor's theorem |
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| | | | | List of axioms: Encyclopedia II - First-order logic - Defining first-order logic A predicate calculus consists of
formation rules (i.e. recursive definitions for forming well-formed formulas).
transformation rules (i.e. inference rules for deriving theorems).
a (possibly countably infinite) set of axioms or axiom schemata.
The axioms considered here are the logical axioms which are part of the predicate calculus. Further, non-logical axioms are added in specific first-order theories: these are not regarded as truths of l ...
See also:First-order logic, First-order logic - Defining first-order logic, First-order logic - Vocabulary, First-order logic - Formation rules, First-order logic - Equality, First-order logic - Inference rules, First-order logic - Quantifier axioms, First-order logic - The predicate calculus, First-order logic - Metalogical theorems of first-order logic, First-order logic - Comparison with other logics Read more here: » First-order logic: Encyclopedia II - First-order logic - Defining first-order logic |
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| | | | | List of axioms: Encyclopedia II - First-order logic - Vocabulary The "vocabulary" is composed of
A set of predicate variables (or relations) each with some valence ≥1, which are often denoted by uppercase letters P, Q, R,...
A set of constants, often denoted by lowercase letters a, b, c,... .
A set of functions, each of some valence ≥ 1, which are often denoted by lowercase letters f, g, h,... .
An infinite set of variables, often denoted by lowercase letters x, y, z,... .
Symbols denoting logical operators: ¬ (logica ...
See also:First-order logic, First-order logic - Defining first-order logic, First-order logic - Vocabulary, First-order logic - Formation rules, First-order logic - Equality, First-order logic - Inference rules, First-order logic - Quantifier axioms, First-order logic - The predicate calculus, First-order logic - Metalogical theorems of first-order logic, First-order logic - Comparison with other logics Read more here: » First-order logic: Encyclopedia II - First-order logic - Vocabulary |
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| | | | | List of axioms: Encyclopedia II - First-order logic - Inference rules The inference rule modus ponens is the only one required from propositional logic for the formalization given here. It states that if φ and φ→ψ are both proved, then one can deduce ψ.
The inference rule called Universal Generalization is characteristic of the predicate calculus. It can be stated as
where φ is supposed to stand for an already-proven theorem of predicate calculus.
Notice that Generalization is analogous to the Necessitati ...
See also:First-order logic, First-order logic - Defining first-order logic, First-order logic - Vocabulary, First-order logic - Formation rules, First-order logic - Equality, First-order logic - Inference rules, First-order logic - Quantifier axioms, First-order logic - The predicate calculus, First-order logic - Metalogical theorems of first-order logic, First-order logic - Comparison with other logics Read more here: » First-order logic: Encyclopedia II - First-order logic - Inference rules |
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| | | | | List of axioms: Encyclopedia II - First-order logic - Quantifier axioms The following four logical axioms characterize a predicate calculus:
PRED-1:
PRED-2:
PRED-3:
PRED-4:
These are actually axiom schemata: the expression W stands for any wff in which x is not free, and the expression Z(x) stands for any wff with the additional convention that Z(y) stands for the same wff with y replacing all free occurrences of x.
...
See also:First-order logic, First-order logic - Defining first-order logic, First-order logic - Vocabulary, First-order logic - Formation rules, First-order logic - Equality, First-order logic - Inference rules, First-order logic - Quantifier axioms, First-order logic - The predicate calculus, First-order logic - Metalogical theorems of first-order logic, First-order logic - Comparison with other logics Read more here: » First-order logic: Encyclopedia II - First-order logic - Quantifier axioms |
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| | | | | List of axioms: Encyclopedia II - First-order logic - The predicate calculus The predicate calculus is an extension of the propositional calculus that defines which statements of first order logic are provable. If the propositional calculus is defined with a suitable set of axioms and the single rule of inference modus ponens (this can be done in many different ways), then the predicate calculus can be defined by appending some additional axioms and the additional inference rule "universal generalization". More precisely, as axioms for the predicate calculus we take:
All tautologies from the propositiona ...
See also:First-order logic, First-order logic - Defining first-order logic, First-order logic - Vocabulary, First-order logic - Formation rules, First-order logic - Equality, First-order logic - Inference rules, First-order logic - Quantifier axioms, First-order logic - The predicate calculus, First-order logic - Metalogical theorems of first-order logic, First-order logic - Comparison with other logics Read more here: » First-order logic: Encyclopedia II - First-order logic - The predicate calculus |
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| | | | | List of axioms: Encyclopedia II - First-order logic - Formation rules The formation rules define the terms, formulas, and the free variables in them as follows.
The set of terms is recursively defined by the following rules:
Any constant is a term (with no free variables).
Any variable is a term (whose only free variable is itself).
Any expression f(t1,...,tn) of n≥1 arguments (where each argument ti is a term and f is a function symbol of valence n) is a term. I ...
See also:First-order logic, First-order logic - Defining first-order logic, First-order logic - Vocabulary, First-order logic - Formation rules, First-order logic - Equality, First-order logic - Inference rules, First-order logic - Quantifier axioms, First-order logic - The predicate calculus, First-order logic - Metalogical theorems of first-order logic, First-order logic - Comparison with other logics Read more here: » First-order logic: Encyclopedia II - First-order logic - Formation rules |
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| | | | | List of axioms: Encyclopedia II - List of cohomology theories - Ordinary homology theories These are the theories satisfying the "dimension axiom" of the Eilenberg-Steenrod axioms that the homology of a point vanishes in dimension other than 0. They are determined by an abelian coefficient group G, and denoted by H(X, G) (where G is sometimes omitted, especially if it is Z). Usually G is the integers, the rationals, the reals, the complex numbers, or the integers mod a prime p.
The cohomology functors of ordinary cohomology theories are represented by Eilenberg-Maclane spaces.
On simplicial ...
See also:List of cohomology theories, List of cohomology theories - Notation, List of cohomology theories - Ordinary homology theories, List of cohomology theories - Homology and cohomology with integer coefficients., List of cohomology theories - Homology and cohomology with rational or real or complex coefficients., List of cohomology theories - Homology and cohomology with mod p coefficients., List of cohomology theories - K-theories, List of cohomology theories - Real K-theory, List of cohomology theories - Complex K-theory, List of cohomology theories - Quaternionic K-theory, List of cohomology theories - K theory with coefficients, List of cohomology theories - Connective K-theories, List of cohomology theories - Self conjugate K-theory, List of cohomology theories - Morava K-theory, List of cohomology theories - Bordism and cobordism theories, List of cohomology theories - Stable homotopy and cohomotopy, List of cohomology theories - Unoriented cobordism, List of cohomology theories - Complex cobordism, List of cohomology theories - Oriented cobordism, List of cohomology theories - Special unitary cobordism, List of cohomology theories - Spin cobordism and variants, List of cohomology theories - Symplectic cobordism, List of cohomology theories - Clifford algebra cobordism, List of cohomology theories - PL cobordism and topological cobordism, List of cohomology theories - Brown-Peterson cohomology, List of cohomology theories - Theories related to elliptic curves, List of cohomology theories - Elliptic cohomology, List of cohomology theories - Topological modular forms Read more here: » List of cohomology theories: Encyclopedia II - List of cohomology theories - Ordinary homology theories |
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| | | | | List of axioms: Encyclopedia II - Propositional calculus - Alternative calculus It is possible to define another version of propositional calculus, which defines most of the syntax of the logical operators by means of axioms, and which uses only one inference rule.
Propositional calculus - Axioms.
Let φ, χ and ψ stand for well-formed formulas. (The wffs themselves would not contain any Greek letters, but only capital Roman letters, connective operators, and parentheses.) Then the axioms are
THEN-1: φ → (χ → φ)
THEN-2: (φ → (χ → ψ)) → ((Ï ...
See also:Propositional calculus, Propositional calculus - Grammar, Propositional calculus - Calculus, Propositional calculus - Axioms, Propositional calculus - Inference rules, Propositional calculus - Example of a proof, Propositional calculus - Soundness and completeness of the rules, Propositional calculus - Sketch of a soundness proof, Propositional calculus - Sketch of completeness proof, Propositional calculus - Alternative calculus, Propositional calculus - Axioms, Propositional calculus - Inference rule, Propositional calculus - Meta-inference rule, Propositional calculus - Example of a proof, Propositional calculus - Other logical calculi Read more here: » Propositional calculus: Encyclopedia II - Propositional calculus - Alternative calculus |
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| | | | | List of axioms: Encyclopedia II - Propositional calculus - Other logical calculi Propositional calculus is about the simplest kind of logical calculus in any current use. (Aristotelian "syllogistic" calculus, which is largely supplanted in modern logic, is in some ways simpler--but in other ways more complex--than propositional calculus.) It can be extended in several ways.
The most immediate way to develop a more complex logical calculus is to introduce rules that are sensitive to more fine-grained details of the sentences being used. When the "atomic sentences" of propositional logic are broken up into te ...
See also:Propositional calculus, Propositional calculus - Grammar, Propositional calculus - Calculus, Propositional calculus - Axioms, Propositional calculus - Inference rules, Propositional calculus - Example of a proof, Propositional calculus - Soundness and completeness of the rules, Propositional calculus - Sketch of a soundness proof, Propositional calculus - Sketch of completeness proof, Propositional calculus - Alternative calculus, Propositional calculus - Axioms, Propositional calculus - Inference rule, Propositional calculus - Meta-inference rule, Propositional calculus - Example of a proof, Propositional calculus - Other logical calculi Read more here: » Propositional calculus: Encyclopedia II - Propositional calculus - Other logical calculi |
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| | | | | List of axioms: Encyclopedia II - Propositional calculus - Grammar The language consists of:
The capital letters of the alphabet, standing as propositional variables. These are atomic formulas. Conventionally, either the Latin alphabet (A, B, C) or the Greek alphabet (χ, φ, ψ) is used, but the two are not mixed.
Symbols denoting the following connectives (or logical operators): ¬, ∧, ∨, →, ↔. (We may do with fewer operators (and thus symbols) by having some abbreviate others — e.g. P → Q is equivalent to ¬ P ∨ Q.)
The left ...
See also:Propositional calculus, Propositional calculus - Grammar, Propositional calculus - Calculus, Propositional calculus - Axioms, Propositional calculus - Inference rules, Propositional calculus - Example of a proof, Propositional calculus - Soundness and completeness of the rules, Propositional calculus - Sketch of a soundness proof, Propositional calculus - Sketch of completeness proof, Propositional calculus - Alternative calculus, Propositional calculus - Axioms, Propositional calculus - Inference rule, Propositional calculus - Meta-inference rule, Propositional calculus - Example of a proof, Propositional calculus - Other logical calculi Read more here: » Propositional calculus: Encyclopedia II - Propositional calculus - Grammar |
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| | | | | List of axioms: Encyclopedia II - Propositional calculus - Soundness and completeness of the rules The crucial properties of this set of rules are that they are sound and complete. Informally this means that the rules are correct and that no other rules are required. These claims can be made more formal as follows.
We define a truth assignment as a function that maps propositional variables to true or false. Informally such a truth assignment can be understood as the description of a possible state of affairs (or possible world) where certain statements are true and others are not. The semantics o ...
See also:Propositional calculus, Propositional calculus - Grammar, Propositional calculus - Calculus, Propositional calculus - Axioms, Propositional calculus - Inference rules, Propositional calculus - Example of a proof, Propositional calculus - Soundness and completeness of the rules, Propositional calculus - Sketch of a soundness proof, Propositional calculus - Sketch of completeness proof, Propositional calculus - Alternative calculus, Propositional calculus - Axioms, Propositional calculus - Inference rule, Propositional calculus - Meta-inference rule, Propositional calculus - Example of a proof, Propositional calculus - Other logical calculi Read more here: » Propositional calculus: Encyclopedia II - Propositional calculus - Soundness and completeness of the rules |
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| | | | | List of axioms: Encyclopedia II - First-order logic - Formation rules The set of terms is recursively defined by the following rules:
Any constant is a term.
Any variable is a term.
Any function symbol f(t1,...,tn) of n≥1 arguments (where each argument ti is a term and n is the valence of the function symbol) is a term.
Closure Clause: Nothing else is a term.
The set of well-formed formulas (usually called wffs or just formulas) is recursiv ...
See also:First-order logic, First-order logic - Defining first-order logic, First-order logic - Vocabulary, First-order logic - Formation rules, First-order logic - Equality, First-order logic - Inference rules, First-order logic - Quantifier axioms, First-order logic - The predicate calculus, First-order logic - Metalogical theorems of first-order logic, First-order logic - Comparison with other logics Read more here: » First-order logic: Encyclopedia II - First-order logic - Formation rules |
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| | | | | List of axioms: Encyclopedia II - Laws of Form - The book There are several editions of LoF, the first in 1969, the most recent (a German translation) in 1997. The mathematics fills only about 55pp and is not difficult. But LoF's mystical and declamatory prose style, and its love of paradox, make it a challenging read for mathematicians and non-mathematicians alike. In this and other respects, Spencer-Brown was much influenced by Wittgenstein and R. D. Laing. At the same time, LoF also echoes a number of themes from the work of Charles Peirce, Bert ...
See also:Laws of Form, Laws of Form - The book, Laws of Form - The Form, Laws of Form - The primary arithmetic and its axioms, Laws of Form - The notion of 'canon', Laws of Form - The primary algebra, Laws of Form - Applying the form to Boolean algebra and logic, Laws of Form - An example calculation, Laws of Form - A technical digression, Laws of Form - Resonances in religion philosophy and science, Laws of Form - Related work, Laws of Form - Footnotes Read more here: » Laws of Form: Encyclopedia II - Laws of Form - The book |
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| | | | | List of axioms: Encyclopedia II - Laws of Form - The Form The symbol:
also called the Mark or Cross, is the essence of the Laws of Form.
In Spencer-Brown's initimable and enigmatic fashion, the Mark symbolizes the root of cognition, i.e., the dualistic Mark indicates the capability of differentiating a "this" from a "that."
In LoF, a Cross denotes the drawing of a "distinction", and can be thought of as signifying the following, all at once:
The act of drawing a boundary around something, thus separating it fr ...
See also:Laws of Form, Laws of Form - The book, Laws of Form - The Form, Laws of Form - The primary arithmetic and its axioms, Laws of Form - The notion of 'canon', Laws of Form - The primary algebra, Laws of Form - Applying the form to Boolean algebra and logic, Laws of Form - An example calculation, Laws of Form - A technical digression, Laws of Form - Resonances in religion philosophy and science, Laws of Form - Related work, Laws of Form - Footnotes Read more here: » Laws of Form: Encyclopedia II - Laws of Form - The Form |
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| | | | | List of axioms: Encyclopedia II - Laws of Form - The primary arithmetic and its axioms Begin with the void, the only "atomic" expression. Then posit two inductive rules:
Given any expression, a Cross can be written over it;
Any two expressions can be concatenated.
Thus the syntax of the primary arithmetic, a Dyck language of order 1 with a null alphabet, and the simplest instance of a context-free language in the Chomsky hierarchy. LoF often uses the phrase calculus of indications in place of "primary arithmetic".
The primary arithmetic and algebra begin with a definitio ...
See also:Laws of Form, Laws of Form - The book, Laws of Form - The Form, Laws of Form - The primary arithmetic and its axioms, Laws of Form - The notion of 'canon', Laws of Form - The primary algebra, Laws of Form - Applying the form to Boolean algebra and logic, Laws of Form - An example calculation, Laws of Form - A technical digression, Laws of Form - Resonances in religion philosophy and science, Laws of Form - Related work, Laws of Form - Footnotes Read more here: » Laws of Form: Encyclopedia II - Laws of Form - The primary arithmetic and its axioms |
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