The Theory of Abstract ObjectsE. Badstubner, D. Kirchner, P. Schießl

Project of Computational Metaphysics 2016C. Benzmuller, A. Steen, M. WisniewskiFreie Universitat Berlin, Institute of Computer Science, Germany

IntroductionThe Theory of Abstract Objects is the core ofEdward Zalta’s upcoming Principia Metaphys-ica [2], a foundational theory of metaphyiscs inthe spirit of Whitehead and Russel’s PrincipiaMathematica.

Figure 1: Principia Mathematica and the current draft of Prin-cipia Metaphysica

The theory postulates the existence of ab-stract objects (e.g. mathematical objects likenumbers) that can encode properties ’xF ’in contrast to concrete ordinary objects (e.g.people, trees, posters) that can only exemplifyproperties ’Fx’.Exemplification is used for classical predica-

tion, e.g. ’John is happy’, whereas encodingis a second mode of predication used for ab-stract objects (e.g the fictional character Sher-lock Holmes).The idea is that if an object exemplifies a

property it must have a spatiotemporal loca-tion, a body with a shape, a mass, etc. Noneof that is true for the fictional character Sher-lock Holmes. Therefore the property of ’be-ing a detective’ is not exemplified by SherlockHolmes. On the other hand ’being a detec-tive’ is a property we use to identify SherlockHolmes and distinguish him from other fic-tional characters. To account for that we saythat Sherlock Holmes encodes the property ofbeing a detective.

The Theory of Abstract Objects

(I) ∃x(A!x & ∀F (xF ≡ Φ))

(II) x = y ≡ �∀F (xF ≡ yF )

The equations above are the two most impor-tant principles of the Theory of Abstract Ob-jects.

Figure 2: Metaphysics goes beyond real-world experience

Their intention can informally be read as:

(I) For each group of properties, there is anabstract object that encodes exactly theproperties in that group.

(II) Two abstract objects are identical if andonly if they (necessarily) encode the sameproperties.

Principia Metaphysica constructs a formal ax-iomatic system around these principles thatmakes it possible to describe and analyse allkinds of abstract objects within a single frame-work.

This includes theoretical mathematical ob-jects such as natural numbers, as well asphilosophical objects such as Forms (Plato),concepts (Leibniz), possible worlds (Leibniz),senses (Frege), the world as state of affairs(Wittgenstein), etc.

AutomationThe theory is a viable option for a foundationaltheory not only of metaphysics, but also ofmathematics. Its automation in an computer-assisted reasoning system is therefore a highlyinteresting challenge. Fig. 2 displays an exem-plary formalization within the interactive proofassistant Isabelle/HOL.

Figure 3: Proof of the theorem ”Possible worlds are maximal” inIsabelle/HOL.

Relational vs. Functional Type TheoryThe Theory of Abstract Objects is formulatedusing relational type theory, whereas higher or-der reasoning systems such as Isabelle/HOLare based on a functional type theory. The di-rect translation between the two is notoriouslyproblematic [1].As an example, the Theory of Abstract

Objects does not allow encoding subfor-mulas in lambda expressions (the firstprinciple mentioned above with Φ beingF = [λx∃F (xF & ¬Fx)] would lead to acontradiction similar to Russel’s paradox). Re-producing this restriction in functional typetheory is challenging.

Hyper-IntensionalityThe Theory of Abstract Object uses a definedequality for objects and relations, e.g. twoone-place relations are considered equal if theyare encoded by the same objects. This leads toa hyper-intensional logic for which the mate-rial equality for properties (two properties areequal if and only if the same objects exemplifythem) no longer holds. In the absence of en-coding formulas, on the other hand, the logicstill collapses to a classical extensional logic.As Boolean extensionality is a built-in featureof systems like Isabelle/HOL, this representsanother challenge.

Summary and ResultsSeveral solutions to the challenges mentionedabove were analysed and found to be unsat-isfactory (i.e. they either resulted in inconsis-tencies or showed major deviations from theintended logic of PM).A promising new approach for a complete em-

bedding of the theory based on a set-theoreticmodel has been proposed and is currently de-veloped further in the context of a master the-sis.

References[1] P. E. Oppenheimer and E. N. Zalta. Relations versus

functions at the foundations of logic: Type-theoreticconsiderations. Journal of Logic and Computation,21(2):351–374, Jun 2010.

[2] Edward Zalta. Principia logico-metaphysica(draft/excerpt). https://mally.stanford.edu/principia.pdf, 2016. [Online; accessed 30-November-2016].

Leibniz: Calculemus!

Computational Metaphysics is a interdisciplinary lecture course designed for advanced students ofcomputer science, mathematics and philosophy. The main objective of the course is to teach thestudents how modern proof assistants based on expressive higher-order logic support the formalanalysis of rational arguments in philosophy (and beyond). In our first course in Summer 2016 thefocus has been on ontological arguments for the existence of God. However, some students pickedformalisation projects also from other areas (including maths).

Computational Metaphysics was awarded the Central Teaching Award 2015 of the FU Berlin.