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| '''Q<sub>0</sub>''' is Peter Andrews' formulation of the [[Simply typed lambda calculus|simply-typed lambda calculus]],
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| and provides a foundation for mathematics comparable to first-order logic plus set theory. | |
| It is a form of [[higher-order logic]] and closely related to the logics of the
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| [[HOL theorem prover]] family.
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| The theorem proving systems [http://gtps.math.cmu.edu/tps.html TPS and ETPS]
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| are based on Q<sub>0</sub>. In August 2009, TPS won the first-ever competition
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| among higher-order theorem proving systems.<ref>[http://www.cs.miami.edu/~tptp/CASC/22/ The CADE-22 ATP System Competition (CASC-22)]</ref>
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| == Axioms of Q<sub>0</sub> ==
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| The system has just five axioms, which can be stated as:
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| <math>(1)</math> <math>g_{oo}T \and g_{oo}F = \forall x_o \centerdot g_{oo}x_o</math>
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| <math>(2^\alpha)</math> <math>[x_\alpha = y_\alpha] \supset \centerdot \, h_{o\alpha}x_\alpha = h_{o\alpha}y_\alpha</math>
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| <math>(3^{\alpha\beta})</math> <math>f_{\alpha \beta} = g_{\alpha \beta} = \forall x_\beta \centerdot f_{\alpha \beta}x_{\beta} = g_{\alpha \beta}x_\beta</math>
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| <math>(4)</math> <math>[\lambda \bold{x_\alpha} \bold{B}_\beta] \bold{A}_\alpha
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| = \bold{S}^{\bold x_\alpha}_{A_\alpha}\bold{B}_\beta</math>
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| <math>(5)</math> <math>\iota_{i(oi)}[\text{Q}_{oii}y_i] = y_i\,</math>
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| (Axioms 2, 3, and 4 are axiom schemas—families of similar axioms. Instances of Axiom 2 and
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| Axiom 3 vary only by the types of variables and constants, but instances of Axiom 4 can have
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| any expression replacing '''A''' and '''B'''.)
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| The subscripted "''o''" is the type symbol for boolean values, and subscripted
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| "''i''" is the type symbol for individual (non-boolean) values. Sequences of these
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| represent types of functions, and can include parentheses to distinguish different function
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| types. Subscripted Greek letters such as α and β are syntactic variables for type
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| symbols. Bold capital letters such as {{math|'''A'''}}, {{math|'''B'''}}, and {{math|'''C'''}}
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| are syntactic variables for WFFs, and bold lower case letters such as
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| {{math|'''x'''}}, {{math|'''y'''}} are syntactic variables for variables.
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| {{math|'''S'''}} indicates syntactic substitution at all free occurrences.
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| The only primitive constants are {{math|Q<sub>((oα)α)</sub>}}, denoting equality
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| of members of each type α, and {{math|''ι<sub>(i(oi))</sub>''}}, denoting a
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| description operator for individuals, the unique element of a set containing exactly one individual.
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| The symbols λ and brackets ("[" and "]") are syntax of the language.
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| All other symbols are abbreviations for terms containing these, including quantifiers ∀ and ∃.
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| In Axiom 4, {{math|'''x'''}} must be free for {{math|'''A'''}} in {{math|'''B'''}},
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| meaning that the substitution does not cause any occurrences of
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| free variables of {{math|'''A'''}} to become bound in the result of the substitution.
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| === About the axioms ===
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| <ul> | |
| <li>Axiom 1 expresses the idea that {{math|''T''}} and {{math|''F''}} are the only boolean values. | |
| <li>Axiom schemas 2<sup>''α''</sup> and 3<sup>''αβ''</sup>
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| express fundamental properties of functions.
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| <li>Axiom schema 4 defines the nature of λ notation.
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| <li>Axiom 5 says that the selection operator is the inverse of the equality
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| function on individuals. (Given one argument, {{math|Q}} maps that individual to
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| the set/predicate containing the individual. In '''Q<sub>0</sub>''', {{math|x {{=}} y}} | |
| is an abbreviation for {{math|Qxy}}, which is an abbreviation for {{math|(Qx)y}}.)
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| </ul>
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| In {{harvnb|Andrews|2002}}, Axiom 4 is developed in five subparts that break
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| down the process of substitution. The axiom as given here is discussed as an
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| alternative and proved from the subparts.
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| == Inference in Q<sub>0</sub> ==
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| Q<sub>0</sub> has a single rule of inference.
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| '''Rule R.''' From {{math|'''C'''}} and
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| {{math|'''A'''<sub>''α''</sub> {{=}} '''B'''<sub>''α''</sub>}}
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| to infer the result of replacing one
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| occurrence of {{math|'''A'''<sub>''α''</sub>}} in {{math|'''C'''}} by an occurrence of
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| {{math|'''B'''<sub>''α''</sub>}},
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| provided that the occurrence of {{math|'''A<sub>α</sub>'''}} in {{math|'''C'''}}
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| is not (an occurrence of a variable) immediately preceded by {{math|λ}}.
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| Derived rule of inference '''R′''' enables reasoning from a set of hypotheses ''H''.
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| '''Rule R′.''' If
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| {{math|''H'' ⊦ '''A'''<sub>α</sub> {{=}} '''B'''<sub>α</sub>}},
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| and {{math|''H'' ⊦ '''C'''}}, and {{math|'''D'''}} is obtained from {{math|'''C'''}}
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| by replacing one occurrence of {{math|'''A'''<sub>α</sub>}} by an occurrence
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| of {{math|'''B'''<sub>α</sub>}}, then
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| {{math|''H'' ⊦ '''D'''}}, provided that:
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| * The occurrence of {{math|'''A'''<sub>α</sub>}} in {{math|'''C'''}} is not an occurrence of a variable immediately preceded by {{math|λ}}, and
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| * no variable free in {{math|'''A'''<sub>α</sub> {{=}} '''B'''<sub>α</sub>}} and a member of {{math|''H''}} is bound in {{math|'''C'''}} at the replaced occurrence of {{math|'''A'''<sub>α</sub>}}.
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| Note: The restriction on replacement of {{math|'''A'''<sub>α</sub>}} by
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| {{math|'''B'''<sub>α</sub>}} in {{math|'''C'''}} ensures that any variable
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| free in both a hypothesis and {{math|'''A'''<sub>α</sub> {{=}} '''B'''<sub>α</sub>}}
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| continues to be constrained to have the same value in both after the replacement
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| is done.
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| The Deduction Theorem for Q<sub>0</sub> shows that proofs from hypotheses using Rule R′
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| can be converted into proofs without hypotheses and using Rule R.
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| Unlike some similar systems, inference in '''Q<sub>0</sub>''' replaces a subexpression at any depth
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| within a WFF with an equal expression. So for example given axioms:
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| 1. {{math|∃x Px}} <br>
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| 2. {{math|Px ⊃ Qx}}
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| and the fact that {{math|A ⊃ B ≡ (A ≡ A ∧ B)}}, we can proceed without removing the quantifier:
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| 3. {{math|Px ≡ (Px ∧ Qx)}} instantiating for A and B<br>
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| 4. {{math|∃x (Px ∧ Qx)}} rule R substituting into line 1 using line 3.
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| == Notes ==
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| {{Reflist}}
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| == References ==
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| *{{Cite book
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| | last = Andrews
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| | first = Peter B.
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| | title = An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof
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| | edition = 2nd
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| | publisher = [[Kluwer Academic Publishers]]
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| | location = Dordrecht, The Netherlands
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| | year = 2002
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| | isbn = 1-4020-0763-9 }} See also http://www.springer.com/mathematics/book/978-1-4020-0763-7.
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| *{{Cite journal
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| | last = Church
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| | first = Alonzo
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| | authorlink = Alonzo Church
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| | title = A Formulation of the Simple Theory of Types
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| | journal = Journal of Symbolic Logic
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| | volume = 5
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| | pages = 56–58
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| | year = 1940
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| }}
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| == Further reading ==
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| A [http://plato.stanford.edu/entries/type-theory-church/#ForBasEqu description of Q<sub>0</sub>]
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| in more depth; part of an article on
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| [http://plato.stanford.edu/entries/type-theory-church/ Church's Type Theory]
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| in the Stanford Encyclopedia of Philosophy.
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| | |
| [[Category:Logic in computer science]]
| |
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