# [isabelle] Implementing the higher-order logic Q0 within the Isabelle proof assistant software

```Dear Members of the Research Community,

For the purpose of implementing Peter B. Andrews' logic Q0 as presented in his
standard work on higher-order logic titled "An Introduction to Mathematical
Logic and Type Theory: To Truth Through Proof" within the Isabelle proof
assistant software, I am looking for experts who are familiar with Isabelle and
who would be interested in this project.

The higher-order logic Q0 has an extremely high level of
formalization/mechanization (the rules applied in each single step are
explicitly specified [cf. Andrews, 2002, pp. 215 ff.]), and virtually all of
mathematics is reduced to formal logic according to Russell's and Whitehead's
idea of logicism. The universality of Q0 as a foundation of mathematics is
preserved by its independence of philosophical assumptions such as the semantic
approach of model theory, as the single rule of inference (substitution, from
which the rule of modus ponens is derived) is a purely syntactical rule.
Technically, Q0 is typed lambda calculus in the form of a simple type theory
(i.e., without type variables) and an axiomatic (Hilbert-style) deductive
system with identity (equality) as the main notion, hence an improved
formulation of Church's type theory [Church, 1940; cf. Andrews, 2006], which is
known for its "precise formulation of the syntax" [Paulson, 1989, p. 5].
Featuring lambda calculus with the single variable binder lambda and "only four
separate kinds of primitive terms: variables, constants, function applications
and [lambda]-abstractions" [Gordon, 2000, p. 179], Q0 requires only two basic
types (individuals and truth values) and only two basic constants
(identity/equality and its counterpart, description) in order to obtain
definability of all of the propositional connectives, as well as all of the
quantifiers (universal, existential and uniqueness quantifier) and provability
of elementary logic on the basis of only five logical axioms, and formalized
number theory (with a non-logical axiom of infinity), thus reducing the
language of formal logic and mathematics to a minimal set of basic notions.

The general intent is to obtain a system with the highest level of
formalization and accuracy and with the expressiveness required for
formalization (of most or all) of mathematics such that the mathematician,
logician or philosopher can easily work with it whilst avoiding the burden of
technical details (i.e., software configuration or programming languages)
without compromising logical necessity or otherwise weakening logical rigor.

The implementation of Q0 should be exactly as specified in [Andrews, 2002, pp.
210-215] (as a Hilbert-style system). A short description is available online
at [Andrews, 2006]:
http://plato.stanford.edu/entries/type-theory-church/#ForBasEqu

According to a recent e-mail by Lawrence C. Paulson, an implementation of Q0 as
a Hilbert-style system (as a special case within natural deduction) in Isabelle
should be possible.

The paper "A Formulation of the Simple Theory of Types (for Isabelle)" by
Paulson, in which Q0 is mentioned ("Andrews [1] presents a formulation based on
equality." [Paulson, 1989, p. 14]) and in which simple type theory is
implemented, but as a natural deduction system, may serve as a basis.

website (see below).

Ken Kubota

References

Andrews, Peter B. (2002), An Introduction to Mathematical Logic and Type
Theory: To Truth Through Proof. Second edition. Dordrecht / Boston / London:
Kluwer Academic Publishers. ISBN 1-4020-0763-9. DOI: 10.1007/978-94-015-9934-4.

Andrews, Peter B. (2006), "Church's Type Theory". In: Stanford Encyclopedia of
Philosophy. Ed. by Edward N. Zalta. Available online at
http://plato.stanford.edu/entries/type-theory-church/ (July 25, 2015).

Church, Alonzo (1940), "A Formulation of the Simple Theory of Types". In:
Journal of Symbolic Logic 5, pp. 56-68.

Gordon, Mike (2000), "From LCF to HOL: A Short History". In: Proof, Language,
and Interaction. Ed. by Gordon Plotkin, Colin Stirling, and Mads Tofte.
Cambridge, MA et al.: MIT Press, pp. 169-185.

Paulson, Lawrence C. (1989), "A Formulation of the Simple Theory of Types (for
Isabelle)". Available online at
https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-175.pdf (July 25, 2015).

____________________

Ken Kubota
doi: 10.4444/100
http://dx.doi.org/10.4444/100

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