[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.

If you would like to find out more about this project, please contact me via my 
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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