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Universal (Meta-)Logical Reasoning: Recent Successes

Christoph Benzm¨ullera,b

aFreie Universit¨at Berlin, FB Mathematik und Informatik, D-14195 Berlin, Germany

bUniversit´e du Luxembourg, FSTC, L-4365 Esch-sur-Alzette, Luxembourg

Abstract

Classical higher-order logic, when utilized as a meta-logic in which various other

(classical and non-classical) logics can be shallowly embedded, is suitable as a

foundation for the development of a universal logical reasoning engine. Such

an engine may be employed, as already envisioned by Leibniz, to support the

rigorous formalisation and deep logical analysis of rational arguments on the

computer. A respective universal logical reasoning framework is described in

this article and a range of successful ﬁrst applications in philosophy, artiﬁcial

intelligence and mathematics are surveyed.

Keywords: computational metaphysics, classical higher-order logic,

non-classical logics, automated reasoning

1. Introduction

The quest for a universal reasoning framework is very prominently repre-

sented in the works of Leibniz. He envisioned a scientia generalis founded on

acharacteristica universalis, that is, a universal formal language in which all

knowledge about the world and the sciences can be encoded. A quick study

of the survey literature on logical formalisms suggests that quite the opposite

to Leibniz’s dream has become reality. Instead of a characteristica universalis,

we are today facing a very rich and heterogenous zoo of diﬀerent logical sys-

tems, and instead of converging towards a single superior logic, this logic zoo is

further expanding, eventually even at accelerated pace. As a consequence, the

uniﬁed vision of Leibniz seems farther away than ever before. However, there

are also some promising initiatives to counteract these diverging developments,

and related works on unifying approaches to logic include categorial logic [1, 2],

algebraic logic [3] and coalgebraic logic [4, 5]. While some practical work has

been reported utilizing the algebraic logic approach [6, 7], these approaches

typically have had a strong emphasis on theory only.

IThis research was funded by the German National Research Foundation (DFG) under

Heisenberg grant BE 2501/9 (Studies in Computational Metaphysics) and by Volkswagen

Stiftung under grant CRAP (Consistent Rational Argumentation in Politics).

URL: http://christoph-benzmueller.de (Christoph Benzm¨uller)

Preprint submitted to Science of Computer Programming November 8, 2018

The solution presented here draws on another alternative at universal logical

reasoning: the shallow semantical embeddings (SSE) approach. This approach

has a very pragmatic motivation, foremost reuse of tools, simplicity and ele-

gance. It utilises classical higher-order logic (HOL) [8, 9] as a unifying meta-

logic in which the syntax and semantics of varying other logics can be explicitly

modeled and ﬂexibly combined. Oﬀ-the-shelf interactive theorem provers (ITPs)

and automated theorem provers (ATPs) for HOL [10] can then be employed to

reason about and within the shallowly embedded logics.

This survey article summarises and reﬂects upon the main results of the

research project Studies in Computational Metaphysics (CompMeta), conducted

from 2012 to 2017 at Freie Universit¨at Berlin and Stanford University. In this

project the SSE approach has been further developed and empirical studies

have been conducted in various disciplines, including philosophy, mathematics

and artiﬁcial intelligence. In philosophy/metaphysics, for example, an initial

focus has been on computer-supported assessments of modern variants of the

ontological argument for the existence of God, where the SSE approach has

been utilised in particular for automating variants of higher-order (multi-)modal

logics [11]. It is these sort of challenge applications of expressive logical reasoning

that the SSE approach is primarily addressing.

A most relevant aspect from the perspective of computer programming is

that the SSE approach in an elegant and theoretically well-founded manner

strives for maximal reuse of already existing theorem proving technology with

minimal coding eﬀort. The prototype systems that have been implemented in

the course of CompMeta performed surprisingly well in all the conducted case

studies, which provides good evidence for the practical relevance of the approach.

Note that the performance of these implemented systems will (to some extent)

naturally advance in the future without much eﬀort simply because the state-

of-the-art ATPs and satisﬁability modulo theories (SMT) solvers it integrates

will further improve in regular cycles.

The article is structured as follows. Section 2 outlines the SSE approach

and discusses its application with a challenge puzzle in epistemic reasoning: the

wise men puzzle. The presented solution puts a particular emphasis on the

adequate modeling of common knowledge. Section 3 presents the motivation

and objectives of the CompMeta project, in which the SSE approach has been

further explored and empirically assessed. The main results of CompMeta are

subsequently summarised and discussed in Sect. 4. Section 5 concludes the

article.

2. The Shallow Semantical Embeddings Approach

HOL has its roots in the logic of Frege’s Begriﬀsschrift [12]. However, the

version of HOL as addressed here is a (simply) typed logic of functions, which

has been proposed by Church [8]. It provides lambda-notation, as an elegant

and useful means to denote unnamed functions, predicates and sets (by their

characteristic functions). Types in HOL eliminate paradoxes and inconsisten-

cies. Russel’s paradox (the set of sets which do not contain themselves), for

example, which can be formalized in Frege’s logic, cannot be represented in

HOL due to type constraints. For more details and further references on HOL

and its automation we refer to the literature [9, 10]. Very relevant for the work

presented here is that the theory of HOL is well understood [13, 14] and that oﬀ-

the-shelf ATPs and ITPs for HOL exists, which can easily be reused. Respective

reasoning systems that are particularly relevant for the SSE approach include

the proof assistants Isabelle/HOL [15] and Coq [16], the ATPs LEO-II [17],

Leo-III [18] and Satallax [19], and the model ﬁnder Nitpick [20]. In the running

example discussed below the Isabelle/HOL proof assistant is used. There are

two main reasons for this choice: (i) The powerful graphical interface of Isa-

belle/HOL enables a particularly intuitive interaction with the SSE approach

in which the logic embeddings can be very elegantly displayed and edited, and

(ii) this system, via its Sledgehammer tool [21], integrates powerful ﬁrst-order

(FO) ATPs and SMT solvers, including E [22], CVC4 [23], Z3 [24], SPASS [25]

and Vampire [26], and it also connects with the HOL ATPs Leo-II and Satallax.

This combination makes it a most suitable environment for conducting mixed

interactive and automated experiments with the SSE approach.

In the remainder of this section the SSE approach will be outlined with the

help of a prominent puzzle in epistemic reasoning: the wise men puzzle (cf. its

discussion in the literature [27, 28]). A particular emphasis and novelty in the

formalisation below is the adequate modeling of the common knowledge of a

set of agents, which is deﬁned as the transitive closure of the agents mutual

knowledge. While the adequate encoding of the notion of transitive closure

poses a challenge for inexpressive knowledge representation frameworks, we here

utilise and showcase a particularly short and elegant solution in HOL (a single

line of code).

2.1. Outline of the SSE Approach

Let L be an object logic of interest, for example, higher-order (HO) modal

logic, which amongst others has prominent applications in metaphysics. Since

our running example below requires the combination of modalities, we will in

fact work with a HO multi-modal logic (HOMML).

The overall idea of the SSE approach is to provide a lean and elegant equa-

tional theory which interprets the syntactical constituents of logic L (in our case

HOMML) as lambda-terms of the meta-logic HOL.

An encoding of HOMML in HOL is presented in Fig. 1. Unlike the traditional

translation approach [29], the connection between HOMML and HOL, i.e. the

equational theory deﬁning the translation is itself formalised in HOL. Moreover,

in contrast to a deep logical embedding, where the syntax and the semantics

of logic L would be formalised in full detail (using structural induction and

recursion), only the crucial diﬀerences in the semantics of both, L and HOL, are

directly addressed in the equational theory, while the commonalities are shared

between both logics. HOMML and HOL, for example, share the domain of

individuals. A crucial diﬀerence, however, lies in the possible world semantics

on the side of HOMML. Hence, the equational theory deﬁning HOMML in HOL

provides an explicit modeling of this particular aspect of modal semantics. The

Figure 1: Shallow semantical embedding of HOMML in HOL.

central idea of this theory is to associate Boolean valued formulas ϕoof HOMML

with world-predicates (truth-sets characterised as lambda-abstractions) ϕi→oin

HOL, where istands for a reserved type for worlds (lines 5-7 in Fig. 1). The

predicate type i→ois abbreviated as σand αstands for the type i→i→oof

accessibility relations.

To establish our mapping it essentially suﬃces to equate the classical logical

connectives of HOMML with corresponding world-lifted predicates and relations

in HOL. For example, in line 13 in Fig. 1 the HOMML connective ∧is identiﬁed

with the lambda-term λϕ.λψ.λw.ϕw∧ψw, such that ϕ∧ψis mapped to the truth

set {x|ϕw ∧ψw}, which is denoted in HOL by the lambda-term λw.ϕw ∧ψw.

The indexed modal operators 2rof HOMML are identiﬁed with lambda-terms

λϕ.λw.∀v.rwv →ϕv, where argument symbol rdenotes an accessibility relation

between possible worlds. We may alternatively say that a parameterised, generic

2-operator is introduced here as λr.λϕ.λw.∀v.rwv →ϕv, such that 2rϕ is

mapped to the truth set λw.∀v.rwv →ϕv (line 21 in Fig. 1). This generic

2-operator can then be instantiated for concrete accessibility relations ras

required.

The mapping of constant symbols and variables from HOMML to HOL is

trivial, since only a type-lifting is required. Most importantly, the mapping

of HOMML to HOL can be deﬁned by a set of non-recursive equations (in

fact, abbreviations), in which the dependency of HOMML formulas on possible

worlds is made explicit, while other aspects and parameters of its semantic

interpretation, such as the underlying semantic domains, remain shared between

HOMML and HOL.

Another interesting and important aspect is that the SSE approach scales

well for FO and HO quantiﬁers. Analogous to the encoding of the proposi-

tional HOMML connectives, they can be introduced as simple abbreviations of

lambda-terms in HOL as well. The introduction of new binder mechanisms is

not required, since the already existing lambda-binder(s) in meta-logic HOL can

be elegantly reused. This is shown in line 17 in Fig. 1, where the HOL univer-

sal quantiﬁer Π(µ→o)→ois type-lifted to become a HOMML universal quantiﬁer

Π(µ→(i→o))→(i→o). In line 18 convenient binder notation is then deﬁned for

the lifted HOMML quantiﬁer, so that we may write ∀x.φx instead of the less

intuitive variant Π(λx.φx). Type polymorphism is employed in lines 17-18 to

avoid the otherwise required enumeration of such quantiﬁer-deﬁning equations

for diﬀerent argument types. The existential quantiﬁer for HOMML is intro-

duced analogously in lines 19-20.

The ﬁnal step is to provide a notion of validity for the type-lifted HOMML

formulas in HOL (see line 25 in Fig. 1): A type-lifted formula ϕ, is valid, denoted

here as bϕc, if and only if the application of ϕto wholds for all worlds w. In

addition, a notion of local validity, denoted here as bϕccw, can be introduced:

bϕccw is true if and only if ϕ cw holds, where cw is an uninterpreted constant

symbol representing the current world.

The presented equations thus characterise a fragment of HOL which, modulo

the above sketched type-lifting, corresponds to HOMML. The faithfulness of this

correspondence, that is, its soundness and completeness with respect to Henkin

semantics, can be established with pen and paper methods [11].1

2.2. Operations on (Accessibility) Relations, including Transitive Closure

Figure 2 presents some useful operations on (accessibility) relations. They

can e.g. be used to elegantly postulate an accessibility relation rto be reﬂexive,

transitive and euclidean, which are the semantic properties typically associated

with modal knowledge operators 2r(see also line 6 in Fig. 3). The expressivity

of HOL is particularly exploited in the single-line deﬁnition of the transitive clo-

sure operation tc in line 14 in Fig. 2, which expresses that two objects (worlds)

xand yare related in the transitive closure (tc R) of a relation Rif and only

if they are related in all transitive super-relations Qof R. Utilising this deﬁni-

tion, ATPs integrated with Isabelle/HOL via Sledgehammer can be employed

to prove some useful lemmata, including the transitivity of the transitive closure

1Further work will investigate whether such faithfulness proofs can eventually be formalised

as well in the approach presented here or whether e.g. a deep embedding is required in this

case.

Figure 2: Operations on (Accessibility) Relations, including Transitive Closure.

of any relation R(line 21) and the symmetry of the transitive closure of any

symmetric relation R(lines 27-28). In the lower window of the GUI in Fig. 2

we e.g. see that the FO ATPs E and SPASS quickly prove the latter lemma,

while the SMT solver CVC4 timesout (the SMT solver Z3 is still running here).

Sledgehammer determines the exact dependencies for the proven conjecture and

it identiﬁes a trusted proof tactic in Isabelle/HOL (here auto), which is capable

of reproving the lemma when the determined dependencies are preselected. The

reconstructed proof utilising the tactic auto is recognised by Isabelle’s inference

kernel. The lemma is then accepted as such by the system.

2.3. Example Application: Wise Men Puzzle

The wise men puzzle, a famous logic riddle whose formalisation has been

studied in some detail e.g. by Baldoni [27], is as follows: Once upon a time, a

king wanted to ﬁnd the wisest out of his three wisest men. He arranged them

in a circle and told them that he would put a white or a black spot on their

foreheads and that one of the three spots would certainly be white. The three

wise men could see and hear each other but, of course, they could not see their

faces reﬂected anywhere. The king, then, asked to each of them to ﬁnd out the

color of his own spot. After a while, the wisest correctly answered that his spot

was white.

An encoding of this epistemic puzzle scenario utilising the SSE approach is

presented in Fig. 3. As an improvement over related work and also over own pre-

vious experiments [30], an adequate modeling of mutual knowledge and common

knowledge is provided in Fig. 3 by following the suggestions of Sergot [31]. The

key idea is to model the knowledge of each wise men, say a, with the help of an

indexed KT45 (=S5) modal operator 2a. We thus introduce three accessibilty

relations a,band c(see line 3 in Fig.3) and instantiate the generic 2-operator

from Fig.1 accordingly to obtain the indexed knowledge operators 2a,2band

2c, one for each wise men (cf. their uses in lines 26, 28 and 30). The accessibilty

relations a,band care constrained in line 6 to obey reﬂexivity, transitivity and

euclideaness. This ensures that 2a,2band 2care KT45 knowledge operators

as intended. Following Sergot, the mutual knowledge of the wise men a,band

cis introduced next be deﬁning a relation Eabc as the union of the accessibility

relations a,band c(in line 8). However, the corresponding 2Eabc-operator

does not yet qualify as an operator for common knowledge, since it may fail to

be transitive. Hence, another relation Cabc is introduced as the transitive clo-

sure of Eabc. The ATPs integrated with Isabelle/HOL conﬁrm (in lines 12-14)

that Cabc is reﬂexive, transitive and euclidean, which means that 2Cabc is a

suitable encoding of the common knowledge of the wise men a,band c.

The formalisation of the epistemic puzzle scenario is continued in lines 16-

30 in Fig. 3.2In addition to the already declared constant symbols a,band

2This encoding still abstracts from the temporal dynamics of the scenario, and the adequate

inclusion of such aspects, for example by adopting and integrating an semantic embedding of

dynamic epistemic logic [32], is still ongoing work. The formalisation presented here never-

Figure 3: Wise Men Puzzle.

c, which denote the epistemic accessibility relations of the wise men in the

scenario, two further uninterpreted constants symbols are introduced in line 16

of Fig. 3. Predicate wise is used to identify and denote the set of wise men in

the scenario (see line 18), and the predicate ws expresses whether a wise man

has a white spot. Line 20 states that it is common knowledge of the wise men

a,band cthat at least one of them has a wite spot. Line 22 (respectively, line

24) then postulates that it is common knowledge that if one wise man has a

white spot (respectively, not a white spot), then the other wise men see and

thus know this. This information, which is implicit background knowledge that

is not explicitly stated in the puzzle itself, is nevertheless relevant for solving it.

Note in particular, how the exploitation of the meta-logic HOL here avoids the

otherwise required duplication of these axioms for diﬀerent combinations of x

and yin the object logic HOMML. Lines 26 and 28 encode the information that

the ﬁrst two wise men that are asked by the king (they are called aand bhere) do

not know whether they have a white spot. This again is postulated as common

knowledge of the wise men. Then, in line 30, the theorem is formulated that

the third wise men cnow knows that he has a white spot. This theorem can be

proven by the ATPs integrated with Isabelle/HOL via the Sledgehammer tool.

The lower part of the GUI window in Fig. 3 shows that the FO ATPs E and

SPASS succeed, and so does the SMT solver CVC4 (while Z3 is still running).

Moreover, in line 31, a consistency check for the entire formalisation of the

puzzle scenario is performed: the model ﬁnder Nitpick computes and presents a

model (not displayed here) that satisﬁes the presented axioms and deﬁnitions.

Note the elegance and minimal eﬀort with which an integrated interactive

and automated theorem proving environment for HOMML has been imple-

mented in Figures 1-3 on top of an existing theorem proving infrastructure

for HOL. In fact, the entire implementation of HOMML and its application to

the wise men puzzle did not require more than 94 line of Isabelle/HOL code,

including commentary. And still, a good degree of automation is achieved,

which signiﬁcantly beneﬁts from the existing ATPs and SMT solvers already

integrated with Isabelle/HOL via the Sledgehammer tool.

After this brief illustration of the SSE approach we now turn attention to

the CompMeta project and brieﬂy discuss its original motivation, its objectives,

and some relevant preceding work.

3. Ob jectives of the CompMeta Project and Preliminary Work

The CompMeta project has its intellectual roots in the author’s work, con-

ducted with colleagues since the mid-nineties, on the theory and practice of HO

theorem proving (cf. [14, 33, 34] and the references therein), on HO proof assis-

tants (cf. [35, 36] and the references therein), and on their applications in mathe-

theless already elegantly demonstrates some core advantages of the SSE approach, including

the already mentioned appropriate modeling of common knowledge based on the transitive

closure of mutual knoweldge.

matics, artiﬁcial intelligence and education (e.g. [37, 38, 39]). These research ac-

tivities inspired ﬁrst experiments towards the development of a universal (meta-)

logical reasoning framework based on the SSE approach [40, 41, 30, 11]. The

core motivation for the CompMeta project has been to further consolidate these

initial ideas and to assess the approach in empirical studies. The main ob jectives

thus included

1. to further explore the theoretical foundation of the SSE approach,

2. to exemplarily implement the approach for a range of challenge logics in

existing ATPs and ITPs for HOL,

3. to provide evidence for its universal logical reasoning capabilities within

exemplary case studies in metaphysics and beyond,

4. and to educate a new generation of students and researchers to master the

SSE approach.

Regarding (1) it was planned to study the faithfulness of the embedding of

further challenging quantiﬁed non-classical logics in HOL. The hypothesis has

been that in all cases the faithfulness, i.e. soundness and completeness of the

embedding, can be shown when a notion of Henkin semantics is assumed (on

the side of both logics). Regarding (2), a close collaboration with the projects

Leo-II and Leo-III, running in parallel at Freie Univerit¨at Berlin, was foreseen,

in addition to the use of the proof assistants Isabelle/HOL and Coq. With re-

spect to (3), a focus has been on applications in theoretical philosophy, resp. in

metaphysics, since there is a particular need for very expressive non-classical

logics in this area. For example, hyper-intensional second order modal logic is

utilised as the starting point in Zalta’s Principia Logico-Metaphysica [42], and

similarly expressive logics are studied in prominent recent textbooks by Stal-

naker [43] and Williamson [44]. Unfortunately, however, there had not been

any attempts prior to the CompMeta project to implement and automate such

challenging logic formalisms in computer systems. Mainstream knowledge rep-

resentation formalisms in computer science and artiﬁcial intelligence, including

e.g. semantic web technologies, typically fail to deliver (not only) in this ap-

plication context due to their lack of expressivity. The CompMeta project, in

contrast, intended to address this gap and to contribute to the pioneering of

the new area of computational metaphysics, which has its roots in the work of

Zalta and colleagues at Stanford University [45, 46, 47]. For the appropriate

modeling of foundational philosophical ontologies and for the formal analysis of

challenge arguments in philosophy (and beyond) a suitably expressive modeling

and reasoning framework was obviously required. With respect to (4), the goal

has been to design and oﬀer an interdisciplinary lecture course on computational

metaphysics, in which the active use of the CompMeta framework was intended

to play a central role in combination with the training of a new generation of

students to independently master the approach.

4. Results of the CompMeta Pro ject

The main results and highlights of the CompMeta project are summarised

in this section and references are given to the most important publications

stemming from the project.

4.1. Application Study I: Ontological Argument for the Existence of God

Diﬀerent modern variants of the ontological argument for the existence of

God, one of the still vividly debated masterpiece arguments in metaphysics

(see e.g. Sobel’s textbook [48] and the references therein), have been rigorously

analysed on the computer in the course of CompMeta. These contributions,

many of which were achieved in close collaboration with Bruno Woltzenlogel-

Paleo, received a media repercussion on a global scale.3

In the course of the conducted experiments [49, 50, 51, 52], the theorem

prover Leo-II detected a previously unknown inconsistency in Kurt G¨odel’s

prominent, HO modal logic variant [53] of the ontological argument, while Dana

Scott’s amendment [54] of it was veriﬁed for logical soundness in the interactive

proof assistants Isabelle/HOL [15] and Coq [55].4In Fig. 4 the axioms causing

the inconsistency in G¨odel’s manuscript are highlighted (see also the discussion

in Sect. 4.6.1). This inconsistency, which was missed by philosophers, is ex-

plained in detail in two conference papers [51, 50]. Further relevant insights

contributed or conﬁrmed by ATPs e.g. include the separation of relevant from

irrelevant axioms, the determination of mandatory properties of modalities, and

undesired side-implications of the axioms such as the modal collapse.5

Further variants of G¨odel’s axioms were proposed by Anderson, H´ajek and

Bjørdal [57, 58, 59, 60, 61, 62]. These variants have meanwhile also been for-

mally analysed, and ATPs have even contributed to the clariﬁcation of an un-

settled philosophical dispute between Anderson and H´ajek [63]. In the course

of this work, diﬀerent notions of quantiﬁcation (actualist and possibilist) have

been utilised and combined within the semantical embedding approach [64].

Moreover, the modal collapse, whose avoidance has been the key motivation for

the contributions of Anderson, Bjørdal and H´ajek (and many others), has been

further investigated [65].

A signiﬁcant further contribution has been achieved by David Fuenmayor,

a philosophy student recruited from in the computational metaphysics lecture

3See e.g. http://www.spiegel.de/international/germany/scientists-use-computer-to-

mathematically-prove-goedel-god-theorem-a-928668.html

4Scott was not aware of the inconsistency in G¨odel’s variant. Amongst others, he slightly

modiﬁed G¨odel’s deﬁnition of essence, which causes the inconsistency. Scott did so because

it felt natural to him to require that essential properties of an individual should actually be

possessed by that individual, and so he added a respective conjunct to the deﬁnition; cf. also

[50, 51].

5The modal collapse [56, 48] is a sort of constricted inconsistency at the level of possible

world semantics. The assumption that there may actually be more than one possible world

is refuted; this follows from G¨odel’s axioms as the ATPs quickly conﬁrm. In other words,

G¨odel’s axioms, as a side-eﬀect, imply that everything is determined (we may even say: that

there is no free will).

Figure 4: The axioms causing the inconsistency in G¨odel’s modal logic variant of the ontologi-

cal argument for the existence of God are highlighted in blue. The inconsistency was detected

by the HO ATP Leo-II. (Disclaimer: Unpublished works of Kurt G¨odel are Copyright Insti-

tute for Advanced Study and are used with permission. All rights reserved by Institute for

Advanced Study)

course at Freie Universit¨at Berlin (see Section 4.5). Fuenmayor, in a student

project [66, 67], formalised the most relevant parts of Fitting’s [68] textbook

Types, Tableaus, and G¨odel’s God. This book develops another interesting

emendation of the ontological argument, which — similar to other recent works

— aims at preserving the overall conclusion (necessary existence of God), while

at the same time getting rid of the modal collapse. Fitting’s means to achieve

this is by modifying the foundational logical system. Instead of an extensional

HO modal logic he employs a more expressive intensional HO modal logic, which

enables a diﬀerent, and as Fitting explains, more adequate interpretation of

e.g. the notion of positive poperties in G¨odel’s argument.

The so far mentioned studies only address a small portion of the entire

relevant literature on the ontological argument. By extending these studies, it

can be expected that many further issues in human refereed contributions can

be revealed. A follow-up project could thus try to develop an encompassing

map that rigorously distinguishes sound from unsound work in this area.

Summary of key insights. Variants of extensional and intensional HO modal

logics can easily be implemented in the SSE approach; a very good degree of

proof automation can be achieved this way, matching or exceeding the argumen-

tation granularity we typically ﬁnd in human authored publications on this sub-

ject; ﬂexible logic modiﬁcations and combinations are supported; the approach

is practically highly useful and it combines automated theorem proving with

model and countermodel ﬁnding (the latter well supports the detection of typos

and minor issues during the formalisation process); it has been demonstrated

how the approach supports a novel, experimental style of work in metaphysics.

4.2. Application Study II: Zalta’s Principia Logico-Metaphysica

Formalising and automating masterpiece rational arguments in philosophy

with the SSE approach on the computer is not trivial. However, it still leads

to comparably small corpora of axioms, lemmata and theorems, and, hence,

it does not provide reliable feedback on the scalability of the approach for

larger and more ambitious formalisations. For that reason another challenge

has been tackled in the CompMeta project: the Principia Logico-Metaphysica

(PLM) of Edward Zalta [42], which aims at a foundational logical theory for

metaphysics, mathematics and the sciences (PLM thus intends to subsume the

Principia Mathematica [69]). Zalta has chosen a hyper-intensional, relational

second-order modal logic S5 [70, 71] as the foundational logic for PLM. It has

thus been a challenge question for CompMeta, whether this non-trivial foun-

dational logic can still be suitably encoded and automated in the semantical

embedding approach. Besides hyper-intensionality, a particular challenge has

been to overcome the conceptional gap between the relational core of PLM

and the functional core of HOL, and to suitably handle the diﬀerent strengths

of comprehension principles supported in both logics that assert the existence

of relations and functions (the use of unrestricted comprehension principles in

PLM causes undesirable paradoxes and inconsistencies [72]). And, of course, a

main challenge has also been to deal with the comparably large size of PLM

in relation to the small axiom sets as studied in the context of the ontological

argument.

The author’s initial attempts, conducted during an extended research stay

at Stanford University in 2015/16, to semantically embed PLM’s base logic in

HOL by following a pure proof theoretic approach were unsuccessful. Later in

2016, in the course of the computational metaphysics lecture course in summer

2016 at Freie Universit¨at Berlin (see below), Zalta in an invited presentation

then outlined some ideas towards a set theoretical semantics for PLM, which

were suggested to him by Peter Aczel. This set theoretic perspective on PLM

subsequently enabled the development of a suitable shallow semantical embed-

ding of PLM in HOL. It was in fact Daniel Kirchner, a mathematics student

recruited the lecture course, who took on the challenge within an MSc thesis

project at Freie Universit¨at Berlin. Kirchner has meanwhile succeeded in for-

malising the PLM in Isabelle/HOL by suitably adapting the SSE approach so

that it soundly covers the base logic of PLM [73].

Kirchner’s work contributes various novel ideas and tools, including the pro-

vision of powerful automation means for PLM at diﬀerent, cross-linked levels of

abstraction. For example, he developed a direct, tactic-based theorem prover

for PLM in Isabelle/HOL, which, one-to-one, implements the proof theory of

PLM as developed by Zalta with pen and paper. This object-level theorem

prover for PLM is connected with the HOL meta-level in Kirchner’s work via the

speciﬁcally tailored shallow semantical embedding he developed, and this link

establishes an Isabelle/HOL-internal criterion, modulo expansion of the seman-

tical embedding, for the soundness of his novel prover. Further, similar provers

are provided by him at well-deﬁned, intermediate expansion levels. Kirchner’s

architecture thus provides multiple options for proof automation, ranging from

the full expansion of the semantical embedding (combined with calls to oﬀ-the-

shelf reasoning tools integrated with Isabelle/HOL via the Sledgehammer tool)

to the more intuitive, one-to-one automation of the proof theory of PLM within

Kirchner’s new tactic-based theorem prover.

An unexpected, but key result of Kirchner’s work has been the discovery

of a paradox in PLM [74, 73] (in the spirit of Russel’s paradox [75] for Frege’s

logic of the Begriﬀsschrift [76]): a deeply-rooted and known paradox is reintro-

duced in PLM, respectively, in the abstract object theory underlying the PLM,

when the logic of complex terms is simply adjoined to the frameworks specially-

formulated comprehension principle for relations. Kirchner’s result constitutes

a new and important paradox, given how much expressive and analytic power

is contributed by having the two kinds of complex terms in the system. The

results also provide a fresh perspective on the question of whether relational

type theory or functional type theory better serves as a foundation for logic and

metaphysics [72].

In close collaboration and supported by further experiments with Isa-

belle/HOL, possible emendations of PLM are currently being studied by Zalta

and Kirchner. The ongoing style of interaction well illustrates a new dynam-

ics in the scientiﬁc discovery process in metaphysics: rigorous experimentation

with implementations of foundational logical systems may quicken and inspire

the scientiﬁc discovery process in this area and also foster more reliable results.

Summary of key insights. The semantic embedding approach scales for ambi-

tious and large projects in metaphysics such as PLM; the approach is practically

applicable and already shows a good degree of automation, which will naturally

further improve (with the ATPs it relies upon); with the help of the implemented

framework new knowledge has been contributed; moreover, students can be well

motivated when using the approach to dive into complex, foundational questions

on the edge of current research in metaphysics in short time.

4.3. Application Study III: Free Logic and Axioms Systems for Category Theory

Partiality and undeﬁnedness are prominent challenges in various areas of

mathematics and computer science. Unfortunately, however, modern proof as-

sistant systems and ATPs based on traditional classical or intuitionistic log-

ics provide rather inadequate support for these challenge concepts. Free logic

[77, 78, 79, 80] oﬀers a theoretically appealing solution, but it has been consid-

ered as rather unsuited towards practical utilisation.

In collaboration with Dana Scott, a shallow embedding of free logic in HOL

has been developed and implemented in the CompMeta project. Just as for

the embeddings mentioned above, various state-of-the-art FO and HO ATPs

and model ﬁnders, which are integrated (modulo suitable logic translations)

with Isabelle/HOL via the Sledgehammer tool, can now be utilised to automate

reasoning in free logic. As a result we obtain an elegant and powerful imple-

mentation of an integrated interactive-automated theorem proving (and model

ﬁnding) environment for free logic.

To demonstrate the practical relevance of this new system, a series of axioms

systems for category theory has been systematically explored [81, 82, 83]. The

starting point has been a generalisation of the standard axioms for a monoid

to a partial composition operation. The purpose of this work has not been

to make or claim any contribution to category theory but rather to show how

formalisations involving the kind of logic required, in this case free logic, can be

implemented and validated within modern proof assistants such as Isabelle/HOL

when utilising the SSE approach.

Subsequently, the relation of the developed axiom systems to alternative pro-

posals from the literature has been studied within the framework. This includes

an axiom set proposed by Freyd and Scedrov in their textbook Categories, Al-

legories [84] for which we have revealed a technical ﬂaw. Either all operations,

e.g. morphism composition, are total in their theory or their axiom system is in-

consistent. This observation applies when a free logic reading of their axiomatic

theory is adopted, where the free variables are assumed to range over all objects,

including the “undeﬁned”. When adopting an algebraic reading of their axioms

system, where free variables range only over all deﬁned objects, then strictness

axioms or conditions are missing in their framework. Both readings have been

formalised in the SSE approach.

Thus, in interaction with the SSE based implementation of free logic in

Isabelle/HOL, a minor (one may say technical) but nevertheless relevant issue

in a mathematics textbook has been revealed that domain experts had missed

before. The repair for this problem is quite straightforward, however. The

solution essentially corresponds to a set of axioms proposed by Scott [85] in the

1970s.

In the experiments reported above, the exploration studies have been signif-

icantly supported by series of experiments in which automated reasoning tools

were called from within the proof assistant Isabelle/HOL via the Sledgehammer

tool. Moreover, very useful feedback was obtained at various stages from the

model ﬁnder Nitpick [20], saving us from making several mistakes.

At the conceptual level this work exempliﬁes a new style of explorative

mathematics which rests on fruitful human-machine interaction with integrated

interactive-automated theorem proving technology. The conducted experiments

were such that the required reasoning was often too tedious and time-consuming

for humans to be carried out repeatedly with highest level of precision. It is

here where cycles of formalisation and experimentation eﬀorts in Isabelle/HOL

provided signiﬁcant support. Moreover, the technical inconsistency issue for

the axiom system of Freyd and Scedrov was discovered by ATPs, which further

emphasises the added value of automated theorem proving in this area.

Summary of key insights. The SSE approach is applicable also to free and in-

clusive logics, which so far were believed to be too diﬃcult to automate and thus

of little practical relevance; quite to the contrary: as our experiments show, the

approach is indeed well suited for practical applications, e.g. for the exploration

of mathematical theories in domains such as category theory, where partiality

and undeﬁnedness play a central role; new knowledge can be discovered this

way.

4.4. Theoretical Study IV: Universal Cut-Elimination

The development of cut-free calculi for expressive logics, such as quantiﬁed

non-classical logics, is usually a non-trivial task. However, for a wide range of

logics there exists a surprisingly elegant and uniform solution: simply utilise

the SSE approach. More precisely, by modeling and studying these logics as

semantically embedded fragments of HOL (with Henkin semantics), existing

cut-elimination results for HOL (with Henkin semantics) may be reused. In the

course of the CompMeta project, this idea has been further explored and exem-

plarily applied for proving cut-elimination for quantiﬁed conditional logics [86].

Conditional logics [87, 88], known also as logics of normality or typicality,

have many applications, including counterfactual reasoning, default reasoning,

deontic reasoning, metaphysical modeling, action planning and reasoning about

knowledge. Moreover, it is well known that they subsume normal modal logics,

since the modal box operator can be deﬁned in terms of the more expressive

conditional operator. In contrast to the rather straightforward, Kripke-style

semantics of normal modal logics, conditional logics come e.g. with a HO selec-

tion function semantics, which makes them interesting objects of study. While

there is broad literature on propositional conditional logics, comparably few

authors have addressed FO extensions of conditional logics; those include Del-

grande [89, 90] and Friedman et al. [91].

The conditional logics studied in the course of the CompMeta project utilise

constant- and/or varying-domain FO quantiﬁers and they combine these with

further quantiﬁers for propositional variables. Such a rich combination has not

been adressed in the literature before. In particular, cut-elimination for these

logic(s) was still open (only for propositional conditional logics some related

results had been available [92, 93]; cf. also the references therein).

While earlier, practical work in the CompMeta project had already shown

that automation of quantiﬁed conditional logics is indeed feasible by utilising

the SSE approach [94], the second half of the project then switched the atten-

tion to the theoretical challenge of proving cut-elimination. It was then shown

[86], that by utilising the SSE approach for quantiﬁed (and non-quantiﬁed) con-

ditional logics, the question whether cut-elimination holds for them can in fact

be reduced to proving the faithfulness of their semantical embedding in HOL.

The latter task, however, constitutes a much simpler problem than proving

cut-elimination directly.

The exploited reduction principle is similarly applicable to other object logics

in the SSE approach, including many logics for which cut-elimination is still

open. However, special attention has to be payed to cut-simulation [86, 95],

which may render cut-elimination as a pointless criterion.

Key insights:. Cut-elimination of a given object logic can often be reduced to

showing the faithfulness of a shallow semantical embedding of this logic in HOL

(with Henkin semantics); the approach has been applied to prove cut-elimination

for some variants of quantiﬁed conditional logics, for which the question was still

open; it should be possible to obtain similar cut-elimination results for many

other challenging object logics by adopting the same reduction principle.

4.5. Educational Study V: Lecture Course on Computational Metaphysics

The early successes in the CompMeta project inspired the design of a world-

wide new lecture course on computational metaphysics [96, 97]. This lecture

course, which was set-up and held in collaboration with Alexander Steen and

Max Wisniewski, was awarded with the 2015/16 central teaching award of

Freie Universit¨at Berlin.6The course received substantial support from Jasmin

Blanchette (Amsterdam), Wolfgang Lenzen (Osnabr¨uck), Bruno Woltzenlogel-

Paleo (Canberra) and Edward Zalta (Stanford), who all contributed invited

guest lectures.

Students with heterogenous knowledge backgrounds from computer science,

mathematics, philosophy and physics attended the lecture course, and they came

from all three major universities of Berlin: Freie Universit¨at Berlin, Technical

6cf. http://www.fu-berlin.de/campusleben/lernen-und-lehren/2016/160428-lehrpreis/

University Berlin and Humboldt University Berlin. The attendance in the lec-

tures usually varied between 40 and 70 students. 36 students were formally

registered for the course and were graded.

The steep learning curves of nearly all students were astonishing, in partic-

ular in the second half of the course, when small, heterogeneous student groups

were formed to work each on an encoding and formal assessment of a diﬀerent

publication in philosophy or mathematics by adapting and utilising the SSE

approach within the Isabelle/HOL proof assistant. The heterogeneous group

compositions, the 24/7 feedback from the Isabelle/HOL environment, and the

motivating project topics were prime reasons, as the author believes, for the very

good overall results of the course. A selection of project results has meanwhile

been presented at conferences or published as book chapters or journal articles

[98, 66, 74, 73, 99, 100, 67, 101]. Several students picked up follow-up topics

and turned them into BSc or MSc thesis projects [102, 103, 104, 105, 106].

A key ingredient for the successful implementation of the course has been,

that a single methodology and overall technique (the SSE approach) was used

throughout, enabling the students to quickly adopt a wide range of diﬀerent logic

variants in short time within a single proof assistant framework (Isabelle/HOL).

The interdisciplinary course concept appears well suited to foster a much im-

proved logic education across disciplines.

Summary of key insights. The SSE approach is well suited to support a novel

form of university level logic education to heterogeneous groups of students;

excellent learning curves are possible; new teaching methods are enabled in

interaction with ITPs and ATPs for HOL.

4.6. Further Results and Comments

A range of related application studies (contributed partly also by collabora-

tors) has not been mentioned above. Amongst others, these works include se-

mantical embeddings of multivalued logic SIXTEEN [107], nominal logics [108],

temporal logics [109], paraconsistent logics, intuitionistic modal logics, etc. Also

the (partly ongoing) work of Streit, another student recruited from the compu-

tational metaphysics lecture course, on the formalisation of Boolos’ textbook

on provability logic [110] and on the formalisation of Bostrom’s simulation ar-

gument [111] has not been addressed above.

A relevant and challenging future application direction of the semantical

embedding approach lies in the modeling of legal, ethical, social and cultural

norms in intelligent machines [112]. To enable such applications, the author is

currently, in a collaboration with Leon van der Torre and Xavier Parent from

the University of Luxemburg, adapting the semantical embedding approach to

cover recent developments in the area of deontic logics [113]. Standard deon-

tic logic, which is just a normal modal logic, is obviously already covered by

the approach. More challenging has been the semantical embedding of e.g. in-

put/output logic [114] and dyadic deontic logic [115]. First results in this ap-

plication direction are promising [116, 117, 118, 119].

4.6.1. Note on Invention and Creativity in ATPs

As reported above, the theorem prover Leo-II detected the inconsistency of

the axioms in G¨odel’s original variant of the ontological argument; this incon-

sistency was not known to philosophers before. The clue in the proof of falsity

from the axioms [51, 50], is the empty essence lemma: from G¨odel’s [53] def-

inition of essential properties (essence, cf. Ess. in Fig. 4) it follows that the

empty property, i.e., the everywhere false property (alternatively we may pick

the property of being self-diﬀerent), is an essential property of every individual.

Dana Scott [54] slightly modiﬁed G¨odel’s deﬁnition of essence in his variant

of the ontological argument (for cosmetic reasons — the inconsistency was not

known to him at the time), with the eﬀect that the empty essence lemma is no

longer valid.

In its successful, automatic discovery of the inconsistency, the Leo-II prover

had to guess the instantiation of the empty property for a second-order variable

during proof search [51, 50]. This part in the proof is non-analytic: inspection

shows that one cannot synthesise the required instantiation by e.g. uniﬁcation

with existing information (terms) in the search space. In fact, blind guessing of

this instantiation seems unavoidable here. The author considers this as a small

but nevertheless very interesting example for a true discovery (based on guessing

and checking) by an ATP; and this discovery is philosophically relevant.

4.6.2. Improved Infrastructure for HO Interactive and Automated Reasoning

In the course of the above works, and in close collaboration with de-

velopment of Leo-III, the CompMeta project fostered the development of a

reusable theorem proving infrastructure for a range of non-classical logics

[120, 121, 122, 123, 18]. This includes various, reusable encodings in Isa-

belle/HOL syntax, Coq syntax and in TPTP syntax, which have been all made

publicly available.7Moreover, this includes a ﬂexible pre-processing module

[18, 124, 125] for the Leo-III prover (and any other TPTP THF [34] compli-

ant prover). This preprocessor turns Leo-III into a ﬂexible reasoner for a very

wide range of propositional and quantiﬁed modal logics. In fact, no other im-

plemented system is available today which covers a wider range of modal logic

variants than Leo-III, and this approach can easily be extended for many other

non-classical logics that have been mentioned above.

5. Conclusion

The presented reasoning framework based on shallow semantical embeddings

in HOL constitutes the most widely applied universal logical reasoning approach

available to date. There is, however, a signiﬁcant diﬀerence to Leibniz’ origi-

nal idea of a characteristica universalis and to various related proposals in the

literature. Instead of proposing a single, universal object-level formalism, the

7Cf. e.g. https://github.com/FormalTheology/GoedelGod

shallow semantical embedding approach supports many diﬀerent competing ob-

ject logics from the logic zoo. No ontological commitment is enforced at the

object logic level. For example, the approach well supports both classical and

intuitionistic object logics, and can even elegantly combine them. The concrete

portfolio of embedded object logics is determined by the speciﬁc requirements of

an application at hand. Only at meta-level a single, unifying logic is provided,

namely HOL (or any richer logic incorporating HOL). By unfolding the deﬁni-

tions of the logic embeddings, problem encodings utilising these object logics

are uniformly mapped to meta-logic HOL. This way Leibniz’ vision of a char-

acteristica universalis is realised in an indirect way in the presented approach:

universal logical reasoning is established (only) at the meta-level in HOL.

The presented universal logical reasoning framework has many challenging

applications in artiﬁcial intelligence, computer science, philosophy, mathematics

and natural language processing. A most relevant and timely application di-

rection concerns the application of the semantical embedding approach for the

modeling of ethical, legal, social and cultural norms in intelligent machines [116],

ideally in combination with the realisation of human-intuitive forms of rational

argumentation in machines complementing internal decision making means at

the level of statistical information and subsymbolic representations.

Acknowledgements. A big thanks (in alphabetical order) goes to all collabo-

rators and supporters of the CompMeta project, including (but not limited

to): Jasmin Blanchette, Harold Boley, Frode Bjørdal, Chad Brown, Maximilian

Claus, Ali Farjami, David Fuenmayor, Tobias Gleissner, Max Haslbeck, Daniel

Kirchner, Hanna Lachnitt, Wolfgang Lenzen, Tomer Libal, Irina Makarenko,

Paul Oppenheimer, Jens Otten, Xavier Parent, Larry Paulson, Florian Rabe,

Raul Rojas, Fabian Sch¨utz, Hans-J¨org Schurr, Dana Scott, Alexander Steen,

David Streit, Geoﬀ Sutcliﬀe, Leon van der Torre, Max Wisniewski, Bruno

Woltzenlogel-Paleo, Edward Zalta, Marco Ziener.

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