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Universal Reasoning, Rational Argumentation and Human-Machine Interaction
Christoph Benzmüller
University of Luxemburg & Freie Universität Berlin
christoph.benzmueller@uni.lu | c.benzmueller@fu-berlin.de
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
well suited for realising a universal logic reasoning
approach. Universal logic reasoning in turn, as en-
visioned already by Leibniz, may support the rigor-
ous formalisation and deep logical analysis of ratio-
nal arguments within machines. A respective uni-
versal logic reasoning framework is described and
a range of exemplary applications are discussed. In
the future, universal logic reasoning in combination
with appropriate, controlled forms of rational argu-
mentation may serve as a communication layer be-
tween humans and intelligent machines.
1 Rational Argumentation – Communication
Interface between Humans and Machines
The ambition to understand, model and implement rational
argumentation and universal logical reasoning independent
of the human brain has a long tradition in the history of hu-
mankind. It reaches back at least to the prominent study of
syllogistic arguments by Aristoteles. Today, with the event of
increasingly intelligent computer technology, the question is
more topical than ever: if humans and intelligent machines
are supposed to amicably coexists, interact and collaborate,
appropriate forms of communication between them are re-
quired. For example, machines should be able to depict, as-
sess and defend their (options for) actions and decisions in
a form that is accessible to human understanding and judge-
ment. This will be crucial for achieving a reconcilable and
socially accepted integration of intelligent machines into ev-
eryday (human) life. The communication means between ma-
chines and humans should ideally be based on human-level,
rational argumentation, which since ages forms the funda-
ment of our social, juridical and scientific processes. Cur-
rent developments in artificial intelligence, in contrast, put
a strong focus on statistical information, machine learning
and subsymbolic representations, all of which are rather de-
tached from human-level rational explanation, understanding
and judgement. The challenge thus is to complement and en-
hance these human-unfriendly forms of reasoning and knowl-
edge representation in todays artificial intelligence systems
with suitable explanations amenable to human cognition, that
is, rational arguments. Via exchange of rational arguments
at human-intuitive level the much needed mutual understand-
ing and acceptance between humans and intelligent machines
can eventually be guaranteed. This is particularly relevant for
the assessment of machine actions in terms of legal, ethical,
moral, social and cultural norms purported by humans. But
what formalisms are available that could serve as a most gen-
eral basis for the modeling of human-level rational arguments
in machines?
2 Leibniz’ Vision
The quest for a most general framework supporting univer-
sal reasoning and rational argumentation is very prominently
represented in the works of Gottfried Wilhelm Leibniz (1646-
1716). He envisioned a scientia generalis founded on a char-
acteristica universalis, that is, a most universal formal lan-
guage in which all knowledge (and all arguments) about the
world and the sciences can be encoded. This universal logic
framework should, so Leibniz, be complemented with a cal-
culus ratiocinator, an associated, most general formal calcu-
lus in which the truth of sentences expressed in the character-
istica universalis should be mechanically assessable by com-
putation.1Leibniz’ envisioned, for example, that disputes be-
tween philosophers could be resolved by formalisation and
computation: “If this is done, whenever controversies arise,
there will be no more need for arguing among two philoso-
phers than among two mathematicians. For it will suffice to
take the pens into the hand and to sit down by the abacus,
saying to each other (and if they wish also to a friend called
for help): Let us calculate.” (Leibniz 1690, translation by
Lenzen (2004, p. 1)).2
Leibniz’ visionary proposal, which became famous under
the slogan Calculemus!: “Let us Calculate.”, is very ambi-
tious and far reaching: “If we had it [a characteristica uni-
1Leibniz characteristica universalis and calculus ratiocinator are
prominently discussed in the numerous philosophy books and pa-
pers. Recommended texts include Lenzen (2004) and Peckhaus
(2004).
2Quo facto, quando orientur controversiae, non magis disputa-
tione opus erit inter duos philosophos, quam inter duos Computis-
tas. Sufficiet enim calamos in manus sumere sedereque ad abacos,
et sibi mutuo (accito si placet amico) dicere: calculemus. (Leibniz
1684; cf. Gerhardt (1890, p. 200)).
arXiv:1703.09620v1 [cs.AI] 28 Mar 2017
versalis], we should be able to reason in metaphysics and
morals in much the same way as in geometry and analysis.”
(Leibniz 1677, Leter to Gallois; translation by Russell).3
From the perspective of the initially depicted challenge, an
obvious proposal hence is to extend and adapt Leibniz pro-
posal in particular to disputes (and interaction in general) be-
tween humans and intelligent machines. But how realistic is
a characteristica universalis and an associated calculus ratio-
cinator? What has modern logic to offer?
3 Zoo of Logical Formalisms
A quick study of the survey literature on logical formalisms4
suggest that quite the opposite to Leibniz’ dream has become
todays reality. Instead of a characteristica universalis,
a most general universal formalism supporting rigorous
formalisations across all scientific disciplines, we are today
actually facing a very rich and heterogenous zoo of different
logical systems. Their development is typically motivated
by e.g. different practical applications, different theoretical
properties, different practical expressivity, or different
schools of origin. Some exemplary species in the logic zoo
are briefly outlined:
On the side of classical logics there are propositional, first-order,
second-order and full higher-order logics. When rejecting cer-
tain basic assumptions, such as the law of excluded middle, we
arrive at intuitionistic and constructive logics, where we may
again distinguish propositional, first-order and higher-order vari-
ants. Higher-order logics, classical or non-classical, are typically
typed (to rule out paradoxes and inconsistencies) and different
type systems have been developed. This brings us in the area of
type theories (some proof assistants may (additionally) apply the
propositions as types paradigm and encode theorems as types
and proofs as terms.) Then there are numerous, so called non-
classical logics, including modal logics and conditional logics,
logics of time and space, provability logics, multivalued logics,
free logics, to name just a few examples. Deferring the explosion
principle (from falsity anything follows) we arrive at paraconsis-
tent logics. Moreover, various special purpose logics, e.g. seper-
ation logics and security logics, have recently been developed for
particular applications. Many of the mentioned logic species, e.g.
modal logics, have again a wide range of subspecies (e.g. logics
K, KB, KT, S4, S5 and different domain conditions for quantified
modal logics, etc.). And, to further complicate matters, certain
practical applications may even require flexible combinations of
logics.
Many of the above logic formalisms have their origin in
philosophy and they have then been picked up and further de-
veloped in e.g. computer science, artificial intelligence, com-
putational linguistics and mathematics. Instead of converg-
ing towards a single superior logic, the logic zoo is obviously
3Car si nous l’avions telle que je la concois, nous pourrions
raisonner en metaphysique et en morale au peu pres comme en Ge-
ometrie et en Analyse, . . . (Leibniz, Leter to Gallois, 1677; cf. Ger-
hardt (1890, p. 21)).
4See for example various handbooks on logical formalisms such
as Gabbay et al. (2004 2014); van Benthem and ter Meulen (2011);
Gabbay and Guenthner (2001 2014); Abramsky et al. (1992 2001);
Gabbay et al. (1993 1998); Blackburn et al. (2006).
further expanding, eventually even at accelerated pace. As a
consequence, the unified vision of Leibniz seems further re-
mote from todays reality than ever before.
However, there are also some promising initiatives to coun-
teract these diverging developments. Attempts at unifying ap-
proaches to logic include categorial logic (Lambek and Scott,
1986; Jacobs, 1999), algebraic logic (Andreka et al., 2017)
and coalgebraic logic (Moss, 1999; Rutten, 2000). Generally,
these approaches have a strong emphasis on theory. However,
some promising practical work has recently been reported uti-
lizing the algebraic logic approach (Guttmann et al., 2011;
Foster and Struth, 2015).
This paper defends another alternative at universal logical
reasoning. This approach has a very pragmatic motivation,
foremost reuse of tools, simplicity and elegance. It utilises
classical higher-order logic5(HOL) 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 higher-order interactive and automated theorem provers
can then be employed to reason about and within the shal-
lowly embedded logics. This way Leibniz vision can (at least
partially) be realised.
However, note the difference to Leibniz original idea: In-
stead of a single, universal logic formalism, the semantical
embedding approach supports different competing object log-
ics from the logic zoo. They are selected according to the spe-
cific requirements of particular applications, and, if needed,
they may be combined. Only at meta-level a single, unify-
ing logic is provided: HOL (or any richer logic incorporat-
ing HOL, provided that strong automation tools for it exist).
By unfolding the object logic encodings, problem representa-
tions are uniformly mapped to HOL. This way Leibniz vision
is realized in an indirect way: universal logical reasoning is
established at the meta-level in HOL.
4 HOL as Unifying Meta-Logic
Translations between logic formalisms are not new. For ex-
ample, by suitably encoding Kripke style semantics (possible
world semantics) many propositional modal logics (PMLs)
can be translated to classical first-order logic (FOL) (Ohlbach
et al., 2001; Schmidt and Hustadt, 2013). Modulo such trans-
formations, a range of PMLs can thus be uniformly char-
acterized as particular fragments of FOL. Moreover, with
the help of respective (external) logic translation tools im-
plementing these mappings, off-the-shelf theorem provers
for FOL have been turned into a practical reasoning sys-
tems for PMLs. A reasoning tool based on this idea is
MSPASS (Hustadt and Schmidt, 2000). Related approaches
5Classical higher-order logic has its roots in the logic of Frege’s
Begriffsschrift (Frege, 1879). However, the version of HOL as used
here is a (simply) typed logic of functions, which has been proposed
by Church (1940). 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 inconsistencies: e.g. the well known Russel paradox (set of
sets which do not contains themselves), which can be formalized
in Frege’s logic, cannot be represented in HOL due to type con-
straints. More information on HOL and its automation is provided
by Benzmüller and Miller (2014).
at generic theorem proving for different non-classical log-
ics include the tableau-based theorem provers LoTReC (Gas-
quet et al., 2005), MeTTeL (Tishkovsky et al., 2013) and the
tableau workbench (Abate and Goré, 2003). These systems
allow the syntax and proof rules of the logic of interest to
be explicitly specified in a respective interface from which
they then generate a custom-tailored, tableau based theorem
prover on the fly.6However, they are typically restricted to
propositional non-classical logics only, which significantly
limits their range of applications. In particular, non-trivial
rational arguments in philosophy and metaphysics are clearly
beyond their scope. Fact is: There are numerous reasoning
tools available for PMLs, but only a handful implemented
systems for first-order modal logics (FOML) (Benzmüller et
al., 2012). And, prior to the semantical embedding approach,
there was not a single, practically available theorem prover
for higher-order modal logics (HOML).
In the translation approach, which is generally not re-
stricted to PMLs and FOL, the external transformation tool
typically embodies and expands the semantics of the source
(aka object) logic which it then translates into the target logic.
The target logic is assumed to have equal or higher expressiv-
ity than the source logic, and the external transformation tool
operates at an (extra-logical) meta-level in which a semanti-
cally justified bridge is established between the former and
the latter. But do we actually need to segregate all these com-
ponents? Why not realising the very same basic idea within
one and the same logic framework, so that the source logic,
the target logic and the meta-level are all “living” in the same
space, and so that the logic transformations can themselves
be explicitly specified and verified by logical means?
This question has inspired research on shallow seman-
tical embeddings in HOL (Benzmüller and Paulson, 2008;
Benzmüller, 2010, 2011; Benzmüller and Paulson, 2010),
where the HOL meta-level is utilized to explicitly model
the source and target logic, and the mapping between them.
Moreover, in contrast to related work, the approach does not
stop at the level of propositional non-classical logics, but
rather puts an emphasis on first-order and higher-order quan-
tified non-classical logics to render it amenable for more am-
bitious applications, including rational arguments in meta-
physics, where e.g. higher-order modal logics play an im-
portant role. The choice of HOL at the meta-level is thereby
not by accident, but motivated as follows:
(A) For most logics in the logic zoo formal notions of se-
mantics have been depicted based on set theoretical means.
Examples are the Tarskian style semantics of classical pred-
icate logic and the Kripke style semantics of modal logic.
HOL which, thanks to its λ-notation, allows sets (e.g. {x|
p(x)}) to be modeled by their corresponding characteristic
functions (e.g. λx.p(x)), is well suited to elegantly encode
many such set theoretic notions of semantics explicitly in
form of a simple equational theory. The fact that HOL is
sufficiently expressive is actually not so surprising when not-
ing that an (informal) notion of classical higher-order logic
is typically also the meta-logic of coice in most contemporary
6Further related systems and tools are described and linked on-
line at http://www.cs.man.ac.uk/~schmidt/tools/
logic or maths textbooks.
(B) Interactive theorem proving in HOL is already well
supported in practice. Powerful interactive provers have
been developed over the past decades, including e.g. Is-
abelle/HOL (Nipkow et al., 2002), HOL4 (Gordon and Mel-
ham, 1993), HOL light (Harrison, 2009) and PVS (Owre and
Shankar, 2008). They often come with comfortable user-
interfaces and intuitive user interaction support. Related
proof assistants, which can also be turned into reasoners for
HOL, include Coq (Bertot and Casteran, 2004), Nuprl (Allen
et al., 2006) and Lean (de Moura et al., 2015). Note that
proof assistants have recently attracted lots of attention in
mathematics, for example, in the context of Hales’ success-
ful verification of his proof of the Kepler conjecture7. While
human experts alone had previously failed to fully assess his
proof (this has happened for the first time in history) his for-
mal verification attempt within the proof assistant HOL light
succeeded (Hales and others, 2015).8We will be facing an in-
creasing number of analogous situations in the future: human
and machine interactions will generate increasingly complex
artefacts across all sciences, which, due to their sheer com-
plexity and/or reasoning depth, will be deprived of traditional
means of human assessment. We instead need new forms and
means of scientific judgement, which again employ computer
technology to overcome these challenges. However, ideally
this computer technology is trusted (e.g. verified) and/or de-
livers rational arguments back in a form amenable to human
understanding and judgement.
(C) Also automated theorem proving in HOL has recently
made significant progress. Theorem provers such as LEO-
II (Benzmüller et al., 2008), Satallax (Brown, 2012) and the
model finder Nitpick (Blanchette and Nipkow, 2010) have
been successfully applied in a range of applications. More-
over, new reasoning systems, such as the Leo-III prover (Wis-
niewski et al., 2015) are currently under development.
(D) Interactive and automated reasoning in HOL has
recently been well integrated. Proof “Hammering” tools
(Blanchette et al., 2016), such as Sledgehammer (Blanchette
et al., 2013)and Hol(y)Hammer (Kaliszyk and Urban, 2015),
are now available. They allow the interactive users of proof
assistants such as Isabelle/HOL and HOL light to conve-
niently call FOL and HOL reasoners in the background (even
in parallel and remotely over the internet). Suitable logic
transformations are realized within these systems and results
are appropriately mapped back into trusted proofs the hosting
proof assistants. Further projects have recently been funded
in this area, including Matryoshka9, AI4REASON10 and
SMART11 . These projects, which (partly) integrate latest ma-
chine learning techniques, will significantly further improve
proof automation of routine tasks in interactive proof assis-
7Johannes Kepler (1571-1630) stated the conjecture that the most
dense way of stapling cannon balls (or oranges and alike) is the form
of a pyramid; the conjecture can be generalized beyond 3 dimen-
sional space.
8See also the following articles in New Scientist: http://tinyurl.
com/gvxzx42 and http://tinyurl.com/jr8rdfq.
9http://matryoshka.gforge.inria.fr
10http://ai4reason.org/
11http://cordis.europa.eu/project/rcn/206472_en.html
tants, with the effect that users can better concentrate on chal-
lenge aspects only.
So, how does the semantical embedding approach work?
Let L be an object logic of interest, for example, HOML as
often required in metaphysics. The overall idea is to pro-
vide a lean and elegant equational theory which interprets
the syntactical constituents of logic L as terms of the target
(and meta-)logic HOL. Different to the traditional translation
approach, this connection, i.e. the equational theory, is itself
formalized in HOL. Moreover, in contrast to a deep logical
embedding, where (the syntax and) the semantics of L would
be formalized in full detail, only the crucial differences in
the semantics of both are addressed in the equational theory
and the commonalities, such as the notions of domains, are
shared. Regarding the HOML L and HOL, for example, a
crucial difference lies in the possible world semantics of L,
and, hence, the equational theory provides an explicit model-
ing of this particular aspect of modal semantics. More con-
cretely, it associates the Boolean valued formulas ϕoof L
with world-predicates (λ-abstractions) ϕi→oin HOL (where
istands for a reserved type for worlds). To establish such a
mapping it essentially suffices to equate the logical connec-
tives of L (e.g. ∧and 2) with corresponding world-lifted
predicates and relations in HOL (e.g. λϕ.λψ.(λw.ϕw ∧ψw)
and λϕ.(λw.∀v.r(w, v)→ϕv, where constant symbol rde-
notes an accessibility relation between possible worlds). The
mapping of constant symbols and variables of Lis then triv-
ial, since only a type-lifting is required. Most importantly,
the mapping of L to HOL can be given in form of a finite set
of quite simple equations (in fact, abbreviations); no explicit
recursive definitions are required. Generally note the way in
which the dependency of logic L on possible worlds is made
explicit while other aspects and parameters of its semantic
interpretation, such as the underlying semantic domains, re-
main (implicitly) shared between both logics.
An interesting aspect is that the approach scales well
even for first-order and higher-order quantifiers. Thus, we
can identify a fragment of HOL which, modulo the above
sketched world-type-lifting, corresponds to HOML. This may
seem astonishing, since HOML may appear more expres-
sive than HOL at first sight. Figure 1 presents such an ex-
emplary equational theory encoded in the proof assistant Is-
abelle/HOL. Formalisation tasks in challenging application
areas (such as metaphysics) requiring HOML can now be
carried out within Isabelle/HOL by using the HOML syntax
as introduced. The HOL meta-logic guarantees global co-
herence and e.g. also enables for global consistency checks.
And, modulo the embeddings in HOL, the automated reason-
ing tools available in Isabelle/HOL can now be reused.
Similar equational theories can be given for a wide range
of non-classical logics (see e.g. the logics mentioned in §5
many of which have prominent applications in artificial intel-
ligence, computer science, philosophy, maths and computa-
tional linguistics. Note that there is currently no other prac-
tically available approach in which a comparative range of
logic embeddings has been established in practice. Moreover,
soundness and completeness of the approach has already been
Figure 1: A lean and elegant equational theory encoded in Is-
abelle/HOL which semantically embeds source logic HOML
in target logic HOL; the equations are stated in meta-logic
HOL. Here, a modal logic S5 with universal accessibility
relation is exemplarily embedded. Moreover, type poly-
morphism is employed in the equations for the quantifiers.
This way the otherwise required enumeration of quantifier-
equations for all types can be avoided.
established for a wide range of logics; thereby Henkin se-
mantics is typically assumed for both HOL and the embedded
source logics L (in case L goes beyond first-order).
5 Some Exemplary Applications
Obviously, the range of possible applications of the approach
is very wide. In fact, due to its generality, very few concep-
tual limitations are known at this point.12 Some exemplary
application directions, which have already been addressed in
pilot studies, are outlined. From a practical perspective a rel-
evant question clearly is whether the theorem provers perfor-
mance scales beyond small proof of concept examples. This
question has to be assessed individually for each application
12Eventually the use of HOL at the meta-level, as opposed to an
even more expressive meta-logic, can be seen as conceptual limita-
tion. However, there is no reason why HOL could not be exchanged
by an even more expressive meta-logic, provided that practical rea-
soning tools are available for it.
domain. However, the experience from the pilot studies men-
tioned below is that the approach indeed matches and may
even outperform human reasoning capabilities in individual
application domains (e.g. flaws in human refereed research
papers and textbooks have been revealed). Another reassur-
ing fact is that in particular in the area of metaphysics the ar-
gumentation granularity (size of single argumentation steps)
that was supported in full automatic mode by the approach
well matched the typical argumentation granularity in hu-
man generated, rational (masterpiece) arguments. Moreover,
note that only the propositional fragments (and in a few cases
the first-order fragments) of the logics mentioned below have
been automated in practice before. The semantical embed-
ding approach, however, scales for their propositional, first-
order and even higher-order logic fragments. Future work
includes the widening of the range of application pilot stud-
ies, in particular, towards the modeling and assessment of ra-
tional arguments between intelligent machines and humans.
It can be expected that, in the long-run, the combination of
expressive quantified non-classical logics will become highly
relevant in this context.
5.1 Philosophy
Masterpiece Rational Arguments in Metaphysics. Nu-
merous modern variants of the Ontological Argument for the
existence of God, one of the still vividly debated masterpiece
arguments13 in metaphysics, have been rigorously analysed
on the computer. In the course of these experiments, the
higher-order ATP LEO-II (Benzmüller et al., 2015) detected
an (previously unknown!) inconsistency in Kurt Gödel’s
(1970) prominent, higher-order modal logic variant of the
argument, while Dana Scott’s (1972) slightly different vari-
ant of the argument was completely verified in the interactive
proof assistants Isabelle/HOL and Coq. Further relevant in-
sights contributed or confirmed by ATPs e.g. include the sep-
aration of relevant from irrelevant axioms, the determination
of mandatory properties of modalities, and undesired side-
implications of the axioms such as the “modal collapse”14.
The main results about Gödel’s and Scott’s proofs have been
presented at ECAI and IJCAI conferences (Benzmüller and
Woltzenlogel Paleo, 2014, 2016a).
Further variants of Gödel’s axioms were proposed by An-
derson, Bjordal and Hájek (Anderson, 1990; Anderson and
Gettings, 1996; Hájek, 1996, 2001; Hájek, 2002; Bjørdal,
1999). These variants have also been formally analysed,
and, in the course of this work, theorem provers have even
contributed to the clarification of an unsettled philosophi-
cal dispute between Anderson and Hájek (Benzmüller et al.,
2017). Moreover, the modal collapse, whose avoidance has
been the key motivation for the contributions of Anderson,
13See e.g. Sobel (2004) for more details on the ontological argu-
ment.
14The modal collapse 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 Gödel’s axioms as the ATPs quickly confirm. In other words,
Gödel’s axioms, as a side-effect, imply that everything is determined
(we may even say: that there is no free will).
Bjordal and Hájek (and many others), has been further inves-
tigated (Benzmüller and Woltzenlogel-Paleo, 2016b). Several
further contributions complete these initial experiments on
the formal assessment of rational arguments in metaphysics
(Benzmüller and Woltzenlogel Paleo, 2013a; Benzmüller
and Woltzenlogel Paleo, 2015c; Benzmüller and Woltzen-
logel Paleo, 2015a; Benzmüller, 2015b; Benzmüller and
Woltzenlogel Paleo, 2015b; Benzmüller, 2015a; Benzmüller
and Woltzenlogel Paleo, 2015d; Benzmüller and Woltzenlo-
gel Paleo, 2013b).
Principia Metaphysica. Analyzing masterpiece rational
arguments in philosophy with the semantical embedding ap-
proach on the computer is not trivial. However, it still leads to
comparably small corpora of axioms, lemmata and theorems,
and it does thus not yet provide feedback on the scalability
of the approach for larger and more ambitious projects. For
that reason another challenge has recently been tackled: the
Principia Logico-Metaphysica (PLM) by Zalta (2016). The
PLM is intended to provide a rigorous formal basis for all of
metaphysics and the sciences; this includes a (flexible) foun-
dation for mathematics and in this sense it is more ambitious
than Russel’s Principia Mathematica. Since Zalta has cho-
sen a hyperintensional (relational) higher-order modal logic
S5 as the logical foundation of his PLM, it has hence been
an open challenge question whether this very specific logical
setting can still be suitably encoded in the semantical em-
bedding approach. Besides hyperintensionality, a particular
challenge concerns the conceptional gap between the rela-
tional and functional bases of the logic of the PLM and HOL,
which imply different strengths of comprehension principles,
which in turn are of significant impact to the entire theory
(full comprehension in the PLM causes paradoxes and incon-
sistencies, Oppenheimer and Zalta (2011)).
Despite these challenges, the ongoing work on the PLM
has progressed very promisingly. In fact, most of the PLM
has meanwhile been represented and partially automated in
Isabelle/HOL by using the semantical embedding approach.15
Other Logics in Philosophy. The approach has recently
been successfully applied to other prominent logics in phi-
losophy, including quantified conditional logics (Benzmüller,
2016, 2013), multi-valued logics (Steen and Benzmüller,
2016) and paraconsistent Logics (Benzmüller and Woltzen-
logel Paleo, 2015b, Sec. 5.4).
Award Winning Lecture Course The successes presented
above and below have inspired the design of a worldwide new
lecture course on Computational Metaphysics at FU Berlin
(Wisniewski et al., 2016).16 In this course the above research
on the formalisation of ontological arguments and the foun-
dations of metaphysics led into a range of further formali-
sation projects in philosophy, maths and computer science.
15See https://github.com/ekpyron/TAO, respectively https://
github.com/ekpyron/TAO/blob/master/output/document.pdf
16The lecture course, held in summer 2016, has received
FU Berlin’s central teaching award; see http://www.fu-berlin.de/
campusleben/lernen-und-lehren/2016/160428-lehrpreis/index.html.
Some of the student projects conducted in this course have
resulted in impressive new contributions. For example, a
computer-assisted reconstruction of an ontological argument
by Leibniz will appear as a chapter in a book dedicated to the
300th anniversary of Leibniz’s death (Bentert et al., 2017).
Also core parts of the textbooks by Fitting (2002) and Boo-
los (1993) have meanwhile been formalised.17 A key factor
in the successful implementation of the course has been, that
a single methodology and overall technique (the semantical
embedding approach) was used throughout, enabling the stu-
dents to quickly adopt a wide range of different logic vari-
ants in short time within a single proof assistant framework
(Isabelle/HOL). The course concept is potentially suited to
significantly improve interdisciplinary, university level logic
education.
5.2 Mathematics
Free Logics. Prominent, open challenges for formalisation
in mathematics (and beyond) include the handling of par-
tiality and definite descriptions. Free logic (Lambert, 2012;
Scott, 1987) adapts classical logic in a way particularly suited
for handling such challenges. Free logics have interesting ap-
plications, e.g. in natural language processing and as a logic
of fiction. In mathematics, free logics are particularly suited
in applications domains such as category theory or projective
geometry (e.g. morphism composition in category theory is
a partial operation). Similar to the other non-classical logics
mentioned before, free logics can be elegantly embedded in
HOL (Benzmüller and Scott, 2016).
Category Theory. Utilizing this embedding of Scott’s
(1987) approach to free logic in HOL, a systematic theory de-
velopment in category theory has recently been contributed.
In this exemplarily study six different but closely related ax-
iom systems for category theory have been formalized in Is-
abelle/HOL and proven mutually equivalent with automated
theorem provers via Sledgehammer. In the course of these ex-
periments, the provers revealed a technical flaw (constricted
inconsistency or missing axioms) in the well known category
theory textbook by Freyd and Scedrov (1990).
5.3 Artificial Intelligence and Computer Science
Most of the above mentioned logics (and respective experi-
ments) are obviously relevant also for applications in artifi-
cial intelligence and computer science. Further relevant ex-
periments include:
Epistemic and Doxastic Logics. Epistemic logic supports
e.g. the modeling of knowledge of rational agents. Doxastic
logic is about the modeling of agent beliefs. Both are just par-
ticular multi-modal logics and thus amenable to the semanti-
cal embedding approach. Respective experiments show that
the approach indeed works well for elegantly solving promi-
nent puzzles about knowledge and belief in artificial intelli-
gence (Benzmüller, 2011; Steen et al., 2016), including the
well known wise men puzzle resp. muddy children puzzle.
17The sources of the formalisation of Fitting’s work are available
at https://github.com/cbenzmueller/TypesTableauxAndGoedelsGod.
Time and Space. The reasoning about time and space has
been a long standing challenge in artificial intelligence, in
particular, when combined reasoning about time, space and
eventually further modal concepts is required. Again, the se-
mantical embedding approach can provide a possible solu-
tion, see e.g. the combination of spatial and epistemic rea-
soning outlined in (Benzmüller, 2011, Sec. 6).
Description Logics. Description logics are prominent e.g.
in the semantic web community. However, description are
basically just a reinvention multi-modal logics (the base de-
scription logic ALC corresponds to a basic multi-modal logic
K), and thus the semantical embedding approach elegantly
applies. Hence, the shallow embedding approach applies also
to a range of prominent description logics, and the mentioned
logic correspondences can even be verified in it.
Many-valued Logics. Many-valued logics have applica-
tions, for example, in philosophy, mathematics and com-
puter science. Theorem provers for various propositional,
first-order and higher-order many-valued logics can easily
be obtained by utilising the semantical embedding approach.
An exemplary semantical embedding of the multi-valued
logic SIXTEEN has been provided in (Steen and Benzmüller,
2016).
Access Control Logics (Security). The semantical embed-
ding approach also applies to security logics, and respec-
tive experiments for access control logics have been reported
(Benzmüller, 2009).
6 Summary and Outlook
The semantical embedding approach, which utilises classi-
cal higher-order logic at meta-level to encode (combinations
of) a wide range of non-classical logics, has many applica-
tions e.g. in artificial intelligence, computer science, philos-
ophy, mathematics and (deep) natural language processing.
Automation of reasoning in these logics (and their combina-
tions) is achieved indirectly with off-the-shelf reasoning tools
as currently developed, integrated and deployed in modern
higher-order proof assistants. The range of possible applica-
tions of this universal reasoning approach is far reaching and,
as has been demonstrated, even scales for non-trivial rational
arguments, including masterpiece arguments in philosophy.
A relevant and challenging future application direction
concerns the application of the semantical embedding ap-
proach for the modeling of ethical, legal, social and cultural
norms in intelligent machines, ideally in combination with
the realisation of human-intuitive forms of rational arguments
in machines complementing internal decision making means
at the level of statistical information and subsymbolic rep-
resentations. To enable such applications, the author is cur-
rently adapting the semantic embedding to cover also recent
works in the area of deontic logics (such as Makinson and
van der Torre (2000) and Carmo and Jones (2013)).
Acknowledgements:
This work has been supported by the following research
grants of the German Research Foundation DFG: BE 2501/9-
2 (Towards Computational Metaphysics) and BE 2501/11-
1 (Leo-III). I cordially thank all my collaborators of these
and other related projects. This includes (in alphabetical or-
der): Larry Paulson, Dana Scott, Geoff Sutcliffe, Alexander
Steen, Max Wisniewski, Bruno Woltzenlogel-Paleo and Ed-
ward Zalta.
References
P. Abate and R. Goré. The tableaux work bench. In M. C.
Mayer and F. Pirri, editors, Automated Reasoning with Ana-
lytic Tableaux and Related Methods, International Conference,
TABLEAUX 2003, Rome, Italy, September 9-12, 2003. Proceed-
ings, volume 2796 of Lecture Notes in Computer Science, pages
230–236. Springer, 2003.
S. Abramsky, D. Gabbay, and T. Maibaum, editors. Handbook of
Logic in Computer Science, volume 1-5. Oxford University Press,
1992-2001.
S. F. Allen, M. Bickford, R. L. Constable, R. Eaton, C. Kreitz,
L. Lorigo, and E. Moran. Innovations in computational type the-
ory using nuprl. J. Applied Logic, 4(4):428–469, 2006.
A. Anderson and M. Gettings. Gödel ontological proof revisited. In
Gödel’96: Logical Foundations of Mathematics, Computer Sci-
ence, and Physics: Lecture Notes in Logic 6, pages 167–172.
Springer, 1996.
C. Anderson. Some emendations of Gödel’s ontological proof. Faith
and Philosophy, 7(3), 1990.
H. Andreka, I. Németi, and I. Sain. Universal Algebraic Logic. Stud-
ies in Universal Logic. Birkhäuser Basel, 2017.
M. Bentert, C. Benzmüller, D. Streit, and B. Woltzenlogel-
Paleo. Analysis of an ontological proof proposed by Leib-
niz. In C. Tandy, editor, Death and Anti-Death, Volume 14:
Four Decades after Michael Polanyi, Three Centuries after
G.W. Leibniz. Ria University Press, 2017. To appear (http:
//christoph-benzmueller.de/papers/B16.pdf).
C. Benzmüller and D. Miller. Automation of higher-order logic. In
D. M. Gabbay, J. H. Siekmann, and J. Woods, editors, Handbook
of the History of Logic, Volume 9 — Computational Logic, pages
215–254. North Holland, Elsevier, 2014.
C. Benzmüller and L. Paulson. Exploring properties of nor-
mal multimodal logics in simple type theory with LEO-II. In
C. Benzmüller, C. Brown, J. Siekmann, and R. Statman, editors,
Reasoning in Simple Type Theory — Festschrift in Honor of Peter
B. Andrews on His 70th Birthday, Studies in Logic, Mathemati-
cal Logic and Foundations, pages 386–406. College Publications,
2008. (Superseded by 2013 article in Logica Universalis).
C. Benzmüller and L. Paulson. Multimodal and intuitionistic logics
in simple type theory. The Logic Journal of the IGPL, 18(6):881–
892, 2010.
C. Benzmüller and D. Scott. Automating free logic in Isabelle/HOL.
In G.-M. Greuel, T. Koch, P. Paule, and A. Sommese, edi-
tors, Mathematical Software – ICMS 2016, 5th International
Congress, Proceedings, volume 9725 of LNCS, pages 43–50,
Berlin, Germany, 2016. Springer.
C. Benzmüller and B. Woltzenlogel Paleo. Gödel’s God in Is-
abelle/HOL. Archive of Formal Proofs, 2013. (Formally verified).
C. Benzmüller and B. Woltzenlogel Paleo. Gödel’s God on the com-
puter. In S. Schulz, G. Sutcliffe, and B. Konev, editors, Proceed-
ings of the 10th International Workshop on the Implementation of
Logics, 2013. (Invited paper).
C. Benzmüller and B. Woltzenlogel Paleo. Automating Gödel’s on-
tological proof of God’s existence with higher-order automated
theorem provers. In T. Schaub, G. Friedrich, and B. O’Sullivan,
editors, ECAI 2014, volume 263 of Frontiers in Artificial Intelli-
gence and Applications, pages 93 – 98. IOS Press, 2014.
C. Benzmüller and B. Woltzenlogel Paleo. Experiments in compu-
tational metaphysics: Gödel’s proof of god’s existence. In S. C.
Mishram, R. Uppaluri, and V. Agarwal, editors, Science & Spir-
itual Quest, Proceedings of the 9th All India Students’ Confer-
ence, 30th October – 1 November, 2015, IIT Kharagpur, India,
pages 23–40. Bhaktivedanta Institute, Kolkata, www.binstitute.
org, 2015. (Invited paper).
C. Benzmüller and B. Woltzenlogel Paleo. Higher-order modal log-
ics: Automation and applications. In A. Paschke and W. Faber,
editors, Reasoning Web 2015, number 9203 in LNCS, pages 32–
74, Berlin, Germany, 2015. Springer. (Invited paper).
C. Benzmüller and B. Woltzenlogel Paleo. Interacting with modal
logics in the Coq proof assistant. In L. D. Beklemishev and D. V.
Musatov, editors, Computer Science - Theory and Applications -
10th International Computer Science Symposium in Russia, CSR
2015, Listvyanka, Russia, July 13-17, 2015, Proceedings, volume
9139 of LNCS, pages 398–411. Springer, 2015.
C. Benzmüller and B. Woltzenlogel Paleo. On logic embeddings
and Gödel’s God. In M. Codescu, R. Diaconescu, and I. Tutu, ed-
itors, Recent Trends in Algebraic Development Techniques: 22nd
International Workshop, WADT 2014, Sinaia, Romania, Septem-
ber 4-7, 2014, Revised Selected Papers, number 9563 in LNCS,
pages 3–6, Sinaia, Romania, 2015. Springer. (Invited paper).
C. Benzmüller and B. Woltzenlogel Paleo. The inconsistency in
Gödel’s ontological argument: A success story for AI in meta-
physics. In S. Kambhampati, editor, IJCAI 2016, volume 1-3,
pages 936–942. AAAI Press, 2016.
C. Benzmüller and B. Woltzenlogel-Paleo. The modal collapse as
a collapse of the modal square of opposition. In J.-Y. Béziau
and G. Basti, editors, The Square of Opposition: A Corner-
stone of Thought (Collection of papers related to the World
Congress on the Square of Opposition IV, Vatican, 2014), http://
www.springer.com/ us/book/9783319450612, Studies in Univer-
sal Logic. Springer International Publishing Switzerland, 2016.
C. Benzmüller, F. Theiss, L. Paulson, and A. Fietzke. LEO-II - a
cooperative automatic theorem prover for higher-order logic. In
IJCAR 2008, number 5195 in LNAI, pages 162–170. Springer,
2008.
C. Benzmüller, J. Otten, and T. Raths. Implementing and evaluating
provers for first-order modal logics. In L. D. Raedt, C. Bessiere,
D. Dubois, P. Doherty, P. Frasconi, F. Heintz, and P. Lucas, edi-
tors, ECAI 2012, volume 242 of Frontiers in Artificial Intelligence
and Applications, pages 163–168, Montpellier, France, 2012. IOS
Press.
C. Benzmüller, L. C. Paulson, N. Sultana, and F. Theiß. The
higher-order prover LEO-II. Journal of Automated Reasoning,
55(4):389–404, 2015.
C. Benzmüller, L. Weber, and B. Woltzenlogel-Paleo. Computer-
assisted analysis of the Anderson-Hájek controversy. Logica Uni-
versalis, 2017. Accepted for publication; to appear presumably
in volume 11 (2017) issue 1.
C. Benzmüller. Automating access control logic in simple type the-
ory with LEO-II. In D. Gritzalis and J. López, editors, Emerging
Challenges for Security, Privacy and Trust, 24th IFIP TC 11 In-
ternational Information Security Conference, SEC 2009, Pafos,
Cyprus, May 18-20, 2009. Proceedings, volume 297 of IFIP,
pages 387–398. Springer, 2009.
C. Benzmüller. Simple type theory as framework for combining
logics. In Contest paper at the World Congress and School on
Universal Logic III (UNILOG), Lisbon, Portugal, 2010. The con-
ference had no published proceedings; the paper is available as
arXiv:1004.5500v1.
C. Benzmüller. Combining and automating classical and non-
classical logics in classical higher-order logic. Annals of Mathe-
matics and Artificial Intelligence (Special issue Computational
logics in Multi-agent Systems (CLIMA XI)), 62(1-2):103–128,
2011.
C. Benzmüller. Automating quantified conditional logics in HOL. In
F. Rossi, editor, 23rd International Joint Conference on Artificial
Intelligence (IJCAI-13), pages 746–753, Beijing, China, 2013.
AAAI Press.
C. Benzmüller. Gödel’s ontological argument revisited – findings
from a computer-supported analysis (invited). In R. S. Silvestre
and J.-Y. Béziau, editors, Handbook of the 1st World Congress on
Logic and Religion, João Pessoa, Brazil, page 13, 2015. (Invited
abstract).
C. Benzmüller. Invited talk: On a (quite) universal theorem proving
approach and its application in metaphysics. In H. D. Nivelle,
editor, TABLEAUX 2015, volume 9323 of LNAI, pages 213–220,
Wroclaw, Poland, 2015. Springer. (Invited paper).
C. Benzmüller. Cut-elimination for quantified conditional logic.
Journal of Philosophical Logic, 2016.
Y. Bertot and P. Casteran. Interactive Theorem Proving and Program
Development. Springer, 2004.
F. Bjørdal. Understanding Gödel’s ontological argument. In
T. Childers, editor, The Logica Yearbook 1998. Filosofia, 1999.
P. Blackburn, J. van Benthem, and F. Wolter, editors. Handbook of
Modal Logic. Elsevier, 2006.
J. Blanchette and T. Nipkow. Nitpick: A counterexample generator
for higher-order logic based on a relational model finder. In ITP
2010, number 6172 in LNCS, pages 131–146. Springer, 2010.
J. C. Blanchette, S. Böhme, and L. C. Paulson. Extending sledge-
hammer with SMT solvers. J. Autom. Reasoning, 51(1):109–128,
2013.
J. C. Blanchette, C. Kaliszyk, L. C. Paulson, and J. Urban. Ham-
mering towards QED. J. Formalized Reasoning, 9(1):101–148,
2016.
G. Boolos. The Logic of Provability. Cambridge University Press,
1993.
C. E. Brown. Satallax: An automatic higher-order prover. In
B. Gramlich, D. Miller, and U. Sattler, editors, Automated Rea-
soning - 6th International Joint Conference, IJCAR 2012, Manch-
ester, UK, June 26-29, 2012. Proceedings, volume 7364 of Lec-
ture Notes in Computer Science, pages 111–117. Springer, 2012.
J. Carmo and A. J. I. Jones. Completeness and decidability results
for a logic of contrary-to-duty conditionals. J. Log. Comput.,
23(3):585–626, 2013.
A. Church. A formulation of the simple theory of types. Journal of
Symbolic Logic, 5:56–68, 1940.
L. M. de Moura, S. Kong, J. Avigad, F. van Doorn, and J. von
Raumer. The lean theorem prover (system description). In A. P.
Felty and A. Middeldorp, editors, Automated Deduction - CADE-
25 - 25th International Conference on Automated Deduction,
Berlin, Germany, August 1-7, 2015, Proceedings, volume 9195
of Lecture Notes in Computer Science, pages 378–388. Springer,
2015.
M. Fitting. Types, Tableaus, and Gödel’s God. Kluwer, 2002.
S. Foster and G. Struth. On the fine-structure of regular algebra.
Journal of Automated Reasoning, 54(2):165–197, 2015.
G. Frege. Begriffsschrift. Eine der arithmetischen nachgebildete
Formelsprache des reinen Denkens. Halle, 1879.
P. J. Freyd and A. Scedrov. Categories, Allegories. North Holland,
1990.
D. Gabbay and F. Guenthner, editors. Handbook of Philosophical
Logic, volume 1-17. Springer, 2001-2014.
D. Gabbay, C. Hogger, and J. Robinson, editors. Handbook of Logic
in Artificial Intelligence and Logic Programming. Oxford Uni-
versity Press, 1993-1998.
D. Gabbay, J. Woods, and EtAl., editors. Handbook of the History
of Logic, volume 1-11. Elsevier, 2004-2014.
O. Gasquet, A. Herzig, D. Longin, and M. Sahade. Lotrec: Logical
tableaux research engineering companion. In B. Beckert, editor,
Automated Reasoning with Analytic Tableaux and Related Meth-
ods, International Conference, TABLEAUX 2005, Koblenz, Ger-
many, September 14-17, 2005, Proceedings, volume 3702 of Lec-
ture Notes in Computer Science, pages 318–322. Springer, 2005.
C. Gerhardt, editor. Die philosophischen Schriften von G. W. Leib-
niz, volume 7. Weidmannsche Buchhandlung, 1890.
K. Gödel. Appx. A: Notes in Kurt Gödel’s Hand, pages 144–145. In
Sobel Sobel (2004), 1970.
M. J. C. Gordon and T. F. Melham, editors. Introduction to HOL. A
theorem proving environment for higher order logic. Cambridge
University Press, 1993.
W. Guttmann, G. Struth, and T. Weber. Automating algebraic meth-
ods in isabelle. In S. Qin and Z. Qiu, editors, Formal Methods and
Software Engineering - 13th International Conference on Formal
Engineering Methods, ICFEM 2011, Durham, UK, October 26-
28, 2011. Proceedings, volume 6991 of Lecture Notes in Com-
puter Science, pages 617–632. Springer, 2011.
P. Hájek. Magari and others on Gödel’s ontological proof. In
A. Ursini and P. Agliano, editors, Logic and algebra, pages 125–
135. Dekker, New York etc., 1996.
P. Hájek. Der Mathematiker und die Frage der Existenz Gottes. In
B. Buldt et al., editor, Kurt Gödel. Wahrheit und Beweisbarkeit,
pages 325–336. öbv & hpt, Wien, 2001. ISBN 3-209-03835-X.
P. Hájek. A new small emendation of Gödel’s ontological proof.
Studia Logica, 71(2):149–164, 2002.
T. Hales et al. A formal proof of the kepler conjecture. CoRR,
abs/1501.02155, 2015.
J. Harrison. HOL light: An overview. In S. Berghofer, T. Nip-
kow, C. Urban, and M. Wenzel, editors, Theorem Proving in
Higher Order Logics, 22nd International Conference, TPHOLs
2009, Munich, Germany, August 17-20, 2009. Proceedings, vol-
ume 5674 of Lecture Notes in Computer Science, pages 60–66.
Springer, 2009.
U. Hustadt and R. A. Schmidt. MSPASS: modal reasoning by trans-
lation and first-order resolution. In R. Dyckhoff, editor, Auto-
mated Reasoning with Analytic Tableaux and Related Methods,
International Conference, TABLEAUX 2000, St Andrews, Scot-
land, UK, July 3-7, 2000, Proceedings, volume 1847 of Lecture
Notes in Computer Science, pages 67–71. Springer, 2000.
B. Jacobs. Categorical Logic and Type Theory, volume 141 of Stud-
ies in Logic and the Foundations of Mathematics. North Holland,
Elsevier, 1999.
C. Kaliszyk and J. Urban. Hol(y)hammer: Online ATP service for
HOL light. Mathematics in Computer Science, 9(1):5–22, 2015.
J. Lambek and P. Scott. Introduction to Higher Order Categorical
Logic. Cambridge University Press, 1986.
K. Lambert. Free Logic. Selected Essays. Cambridge University
Press, 2012.
W. Lenzen. Leibniz’s logic. In D. Gabbay and J. Woods, editors,
The Rise of Modern Logic: From Leibniz to Frege, volume 3 of
Handbook of the History of Logic, pages 1–83. Elsevier, 2004.
D. Makinson and L. W. N. van der Torre. Input/output logics. J.
Philosophical Logic, 29(4):383–408, 2000.
L. Moss. Coalgebraic logic. Annals of Pure and Applied Logic,
96(1-3):277–317, 1999.
T. Nipkow, L. Paulson, and M. Wenzel. Isabelle/HOL: A Proof As-
sistant for Higher-Order Logic. Number 2283 in LNCS. Springer,
2002.
H. Ohlbach, A. Nonnengart, M. de Rijke, and D. Gabbay. Encoding
two-valued nonclassical logics in classical logic. In J. Robinson
and A. Voronkov, editors, Handbook of Automated Reasoning (in
2 volumes), pages 1403–1486. Elsevier and MIT Press, 2001.
P. E. Oppenheimer and E. N. Zalta. Relations versus functions at
the foundations of logic: Type-theoretic considerations. J. Log.
Comput., 21(2):351–374, 2011.
S. Owre and N. Shankar. A brief overview of PVS. In O. A. Mo-
hamed, C. A. Muñoz, and S. Tahar, editors, Theorem Proving
in Higher Order Logics, 21st International Conference, TPHOLs
2008, Montreal, Canada, August 18-21, 2008. Proceedings, vol-
ume 5170 of Lecture Notes in Computer Science, pages 22–27.
Springer, 2008.
V. Peckhaus. Calculus ratiocinator versus characteristica univer-
salis? The two traditions in logic, revisited. History and Phi-
losophy of Logic, 25(1):3–14, 2004.
J. Rutten. Universal coalgebra: a theory of systems. Theoretical
Computer Science, 249(1):3–80, 2000.
R. Schmidt and U. Hustadt. First-order resolution methods for
modal logics. In A. Voronkov and C. Weidenbach, editors, Pro-
gramming Logics - Essays in Memory of Harald Ganzinger, vol-
ume 7797 of Lecture Notes in Computer Science, pages 345–391.
Springer, 2013.
D. Scott. Appx. B: Notes in Dana Scott’s Hand, pages 145–146. In
Sobel Sobel (2004), 1972.
D. Scott. Gödel’s ontological proof. In On Being and Saying. Essays
for Richard Cartwright, pages 257–258. MIT Press, 1987.
J. Sobel. Logic and Theism: Arguments for and Against Beliefs in
God. Cambridge U. Press, 2004.
A. Steen and C. Benzmüller. Sweet SIXTEEN: Automation via em-
bedding into classical higher-order logic. Logic and Logical Phi-
losophy, 25:535–554, 2016.
A. Steen, M. Wisniewski, and C. Benzmüller. Tutorial on reasoning
in expressive non-classical logics with Isabelle/HOL. In C. Ben-
züller, R. Rojas, and G. Sutcliffe, editors, GCAI 2016, 2nd Global
Conference on Artificial Intelligence, volume 41 of EPiC Series
in Computing, pages 1–10, Berlin, Germany, 2016. EasyChair.
D. Tishkovsky, R. A. Schmidt, and M. Khodadadi. Mettel2: To-
wards a tableau prover generation platform. In P. Fontaine, R. A.
Schmidt, and S. Schulz, editors, Third Workshop on Practical
Aspects of Automated Reasoning, PAAR-2012, Manchester, UK,
June 30 - July 1, 2012, volume 21 of EPiC Series in Computing,
pages 149–162. EasyChair, 2013.
J. van Benthem and A. ter Meulen, editors. Handbook of Logic and
Language (2nd ed.). Elsevier, 2011.
M. Wisniewski, A. Steen, and C. Benzmüller. Leopard - A generic
platform for the implementation of higher-order reasoners. In
M. Kerber, J. Carette, C. Kaliszyk, F. Rabe, and V. Sorge, ed-
itors, Intelligent Computer Mathematics - International Confer-
ence, CICM 2015, Washington, DC, USA, July 13-17, 2015, Pro-
ceedings, volume 9150 of LNCS, pages 325–330. Springer, 2015.
M. Wisniewski, A. Steen, and C. Benzmüller. Einsatz von The-
orembeweisern in der Lehre. In A. Schwill and U. Lucke,
editors, Hochschuldidaktik der Informatik: 7. Fachtagung des
GI-Fachbereichs Informatik und Ausbildung/Didaktik der Infor-
matik; 13.-14. September 2016 an der Universität Potsdam,
Commentarii informaticae didacticae (CID), Potsdam, Germany,
2016. Universitätsverlag Potsdam.
E. N. Zalta. Principia logico-metaphysica. Draft version, preprint
available at https://mally.stanford.edu/principia.pdf, 2016.
