Universal Reasoning, Rational Argumentation and Human-Machine Interaction
UUniversal Reasoning, Rational Argumentation and Human-Machine Interaction
Christoph Benzmüller
University of Luxemburg & Freie Universität [email protected] | [email protected]
Abstract
Classical higher-order logic, when utilized as ameta-logic in which various other (classical andnon-classical) logics can be shallowly embedded, iswell suited for realising a universal logic reasoningapproach. 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 anda range of exemplary applications are discussed. Inthe future, universal logic reasoning in combinationwith appropriate, controlled forms of rational argu-mentation may serve as a communication layer be-tween humans and intelligent machines.
The ambition to understand, model and implement rationalargumentation and universal logical reasoning independentof the human brain has a long tradition in the history of hu-mankind. It reaches back at least to the prominent study ofsyllogistic arguments by Aristoteles. Today, with the event ofincreasingly intelligent computer technology, the question ismore topical than ever: if humans and intelligent machinesare 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 ina form that is accessible to human understanding and judge-ment. This will be crucial for achieving a reconcilable andsocially 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, puta strong focus on statistical information, machine learningand subsymbolic representations, all of which are rather de-tached from human-level rational explanation, understandingand 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, thatis, rational arguments.
Via exchange of rational argumentsat human-intuitive level the much needed mutual understand-ing and acceptance between humans and intelligent machinescan eventually be guaranteed.
This is particularly relevant forthe assessment of machine actions in terms of legal, ethical,moral, social and cultural norms purported by humans. Butwhat formalisms are available that could serve as a most gen-eral basis for the modeling of human-level rational argumentsin machines?
The quest for a most general framework supporting univer-sal reasoning and rational argumentation is very prominentlyrepresented 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 theworld and the sciences can be encoded. This universal logicframework 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. Leibniz’ envisioned, for example, that disputes be-tween philosophers could be resolved by formalisation andcomputation: “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 totake the pens into the hand and to sit down by the abacus,saying to each other (and if they wish also to a friend calledfor help): Let us calculate.” (Leibniz 1690, translation byLenzen (2004, p. 1)). Leibniz’ visionary proposal, which became famous underthe slogan
Calculemus!: “Let us Calculate.” , is very ambi-tious and far reaching: “If we had it [a characteristica uni- Leibniz characteristica universalis and calculus ratiocinator areprominently discussed in the numerous philosophy books and pa-pers. Recommended texts include Lenzen (2004) and Peckhaus(2004). Quo 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. (Leibniz1684; cf. Gerhardt (1890, p. 200)). a r X i v : . [ c s . A I] M a r ersalis], we should be able to reason in metaphysics andmorals in much the same way as in geometry and analysis.” (Leibniz 1677, Leter to Gallois; translation by Russell). From the perspective of the initially depicted challenge, anobvious 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 isa characteristica universalis and an associated calculus ratio-cinator? What has modern logic to offer?
A quick study of the survey literature on logical formalisms suggest that quite the opposite to Leibniz’ dream has becometodays reality. Instead of a characteristica universalis,a most general universal formalism supporting rigorousformalisations across all scientific disciplines, we are todayactually facing a very rich and heterogenous zoo of differentlogical systems . Their development is typically motivatedby e.g. different practical applications, different theoreticalproperties, different practical expressivity, or differentschools of origin. Some exemplary species in the logic zooare 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, wearrive at intuitionistic and constructive logics, where we mayagain distinguish propositional, first-order and higher-order vari-ants. Higher-order logics, classical or non-classical, are typicallytyped (to rule out paradoxes and inconsistencies) and differenttype systems have been developed. This brings us in the area oftype theories (some proof assistants may (additionally) apply thepropositions as types paradigm and encode theorems as typesand 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 explosionprinciple (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 forparticular applications. Many of the mentioned logic species, e.g.modal logics, have again a wide range of subspecies (e.g. logicsK, KB, KT, S4, S5 and different domain conditions for quantifiedmodal logics, etc.). And, to further complicate matters, certainpractical applications may even require flexible combinations oflogics.
Many of the above logic formalisms have their origin inphilosophy 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 Car si nous l’avions telle que je la concois, nous pourrionsraisonner 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)). See for example various handbooks on logical formalisms suchas 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 aconsequence, 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 logicalreasoning. This approach has a very pragmatic motivation,foremost reuse of tools, simplicity and elegance. It utilisesclassical higher-order logic (HOL) as a unifying meta-logicin which (the syntax and semantics) of varying other logicscan be explicitly modeled and flexibly combined. Off-the-shelf higher-order interactive and automated theorem proverscan then be employed to reason about and within the shal-lowly embedded logics. This way Leibniz vision can (at leastpartially) be realised.However, note the difference to Leibniz original idea: In-stead of a single, universal logic formalism, the semanticalembedding 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 visionis realized in an indirect way: universal logical reasoning isestablished at the meta-level in HOL . Translations between logic formalisms are not new. For ex-ample, by suitably encoding Kripke style semantics (possibleworld 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, withthe help of respective (external) logic translation tools im-plementing these mappings, off-the-shelf theorem proversfor FOL have been turned into a practical reasoning sys-tems for PMLs. A reasoning tool based on this idea isMSPASS (Hustadt and Schmidt, 2000). Related approaches Classical higher-order logic has its roots in the logic of Frege’sBegriffsschrift (Frege, 1879). However, the version of HOL as usedhere is a (simply) typed logic of functions, which has been proposedby Church (1940). It provides lambda-notation, as an elegant anduseful means to denote unnamed functions, predicates and sets (bytheir characteristic functions). Types in HOL eliminate paradoxesand inconsistencies: e.g. the well known Russel paradox (set ofsets which do not contains themselves), which can be formalizedin Frege’s logic, cannot be represented in HOL due to type con-straints. More information on HOL and its automation is providedby Benzmüller and Miller (2014). t 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 thetableau workbench (Abate and Goré, 2003). These systemsallow the syntax and proof rules of the logic of interest tobe explicitly specified in a respective interface from whichthey then generate a custom-tailored, tableau based theoremprover on the fly. However, they are typically restricted topropositional non-classical logics only, which significantlylimits their range of applications. In particular, non-trivialrational arguments in philosophy and metaphysics are clearlybeyond their scope. Fact is: There are numerous reasoningtools available for PMLs, but only a handful implementedsystems for first-order modal logics (FOML) (Benzmüller etal. , 2012). And, prior to the semantical embedding approach,there was not a single, practically available theorem proverfor higher-order modal logics (HOML).In the translation approach, which is generally not re-stricted to PMLs and FOL, the external transformation tooltypically 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 tooloperates at an (extra-logical) meta-level in which a semanti-cally justified bridge is established between the former andthe latter. But do we actually need to segregate all these com-ponents? Why not realising the very same basic idea withinone and the same logic framework, so that the source logic,the target logic and the meta-level are all “living” in the samespace, and so that the logic transformations can themselvesbe 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 modelthe source and target logic, and the mapping between them.Moreover, in contrast to related work, the approach does notstop at the level of propositional non-classical logics, butrather 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 therebynot 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 characteristicfunctions (e.g. λx.p ( x ) ), is well suited to elegantly encodemany such set theoretic notions of semantics explicitly inform of a simple equational theory. The fact that HOL issufficiently expressive is actually not so surprising when not-ing that an (informal) notion of classical higher-order logicis typically also the meta-logic of coice in most contemporary logic or maths textbooks . (B) Interactive theorem proving in HOL is already wellsupported in practice.
Powerful interactive provers havebeen 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 andShankar, 2008). They often come with comfortable user-interfaces and intuitive user interaction support. Relatedproof assistants, which can also be turned into reasoners forHOL, include Coq (Bertot and Casteran, 2004), Nuprl (Allen et al. , 2006) and Lean (de Moura et al. , 2015). Note thatproof assistants have recently attracted lots of attention inmathematics, for example, in the context of Hales’ success-ful verification of his proof of the Kepler conjecture . Whilehuman experts alone had previously failed to fully assess hisproof (this has happened for the first time in history) his for-mal verification attempt within the proof assistant HOL lightsucceeded (Hales and others, 2015). We will be facing an in-creasing number of analogous situations in the future: humanand machine interactions will generate increasingly complexartefacts across all sciences, which, due to their sheer com-plexity and/or reasoning depth, will be deprived of traditionalmeans of human assessment. We instead need new forms andmeans of scientific judgement, which again employ computertechnology to overcome these challenges. However, ideallythis computer technology is trusted (e.g. verified) and/or de-livers rational arguments back in a form amenable to humanunderstanding and judgement. (C)
Also automated theorem proving in HOL has recentlymade significant progress.
Theorem provers such as LEO-II (Benzmüller et al. , 2008), Satallax (Brown, 2012) and themodel finder Nitpick (Blanchette and Nipkow, 2010) havebeen 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 hasrecently 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 proofassistants such as Isabelle/HOL and HOL light to conve-niently call FOL and HOL reasoners in the background (evenin parallel and remotely over the internet). Suitable logictransformations are realized within these systems and resultsare appropriately mapped back into trusted proofs the hostingproof assistants. Further projects have recently been fundedin this area, including Matryoshka , AI4REASON andSMART . These projects, which (partly) integrate latest ma-chine learning techniques, will significantly further improveproof automation of routine tasks in interactive proof assis- Johannes Kepler (1571-1630) stated the conjecture that the mostdense way of stapling cannon balls (or oranges and alike) is the formof a pyramid; the conjecture can be generalized beyond 3 dimen-sional space. See also the following articles in New Scientist: http://tinyurl.com/gvxzx42 and http://tinyurl.com/jr8rdfq. http://matryoshka.gforge.inria.fr http://ai4reason.org/ http://cordis.europa.eu/project/rcn/206472_en.html ants, 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 asoften required in metaphysics.
The overall idea is to pro-vide a lean and elegant equational theory which interpretsthe syntactical constituents of logic L as terms of the target(and meta-)logic HOL.
Different to the traditional translationapproach, this connection, i.e. the equational theory, is itselfformalized in HOL. Moreover, in contrast to a deep logicalembedding, where (the syntax and) the semantics of L wouldbe formalized in full detail, only the crucial differences inthe semantics of both are addressed in the equational theoryand the commonalities, such as the notions of domains, areshared. Regarding the HOML L and HOL, for example, acrucial 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 ϕ o of Lwith world-predicates ( λ -abstractions) ϕ i → o in HOL (where i stands for a reserved type for worlds). To establish such amapping it essentially suffices to equate the logical connec-tives of L (e.g. ∧ and (cid:50) ) with corresponding world-liftedpredicates and relations in HOL (e.g. λϕ.λψ. ( λw.ϕw ∧ ψw ) and λϕ. ( λw. ∀ v.r ( w, v ) → ϕv , where constant symbol r de-notes an accessibility relation between possible worlds). Themapping of constant symbols and variables of L is 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 setof quite simple equations (in fact, abbreviations); no explicitrecursive definitions are required. Generally note the way inwhich the dependency of logic L on possible worlds is madeexplicit while other aspects and parameters of its semanticinterpretation, such as the underlying semantic domains, re-main (implicitly) shared between both logics.An interesting aspect is that the approach scales welleven for first-order and higher-order quantifiers . Thus, wecan identify a fragment of HOL which, modulo the abovesketched world-type-lifting, corresponds to HOML. This mayseem 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 applicationareas (such as metaphysics) requiring HOML can now becarried out within Isabelle/HOL by using the HOML syntaxas 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 rangeof non-classical logics (see e.g. the logics mentioned in §5many 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 oflogic 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 HOMLin target logic HOL; the equations are stated in meta-logicHOL. Here, a modal logic S5 with universal accessibilityrelation 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 embeddedsource logics L (in case L goes beyond first-order). Obviously, the range of possible applications of the approachis very wide . In fact, due to its generality, very few concep-tual limitations are known at this point. Some exemplaryapplication directions, which have already been addressed inpilot 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. Thisquestion has to be assessed individually for each application Eventually the use of HOL at the meta-level, as opposed to aneven more expressive meta-logic, can be seen as conceptual limita-tion. However, there is no reason why HOL could not be exchangedby an even more expressive meta-logic, provided that practical rea-soning tools are available for it. omain. However, the experience from the pilot studies men-tioned below is that the approach indeed matches and mayeven outperform human reasoning capabilities in individualapplication domains (e.g. flaws in human refereed researchpapers 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 approachwell matched the typical argumentation granularity in hu-man generated, rational (masterpiece) arguments . Moreover,note that only the propositional fragments (and in a few casesthe first-order fragments) of the logics mentioned below havebeen automated in practice before.
The semantical embed-ding approach, however, scales for their propositional, first-order and even higher-order logic fragments.
Future workincludes 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 ofexpressive quantified non-classical logics will become highlyrelevant in this context.
Masterpiece Rational Arguments in Metaphysics.
Nu-merous modern variants of the
Ontological Argument for theexistence of God, one of the still vividly debated masterpiecearguments in metaphysics, have been rigorously analysedon the computer. In the course of these experiments, thehigher-order ATP LEO-II (Benzmüller et al. , 2015) detectedan (previously unknown!) inconsistency in Kurt Gödel’s(1970) prominent, higher-order modal logic variant of theargument, while Dana Scott’s (1972) slightly different vari-ant of the argument was completely verified in the interactiveproof 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 determinationof mandatory properties of modalities, and undesired side-implications of the axioms such as the “modal collapse” .The main results about Gödel’s and Scott’s proofs have beenpresented at ECAI and IJCAI conferences (Benzmüller andWoltzenlogel Paleo, 2014, 2016a).Further variants of Gödel’s axioms were proposed by An-derson, Bjordal and Hájek (Anderson, 1990; Anderson andGettings, 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 evencontributed to the clarification of an unsettled philosophi-cal dispute between Anderson and Hájek (Benzmüller et al. ,2017). Moreover, the modal collapse, whose avoidance hasbeen the key motivation for the contributions of Anderson, See e.g. Sobel (2004) for more details on the ontological argu-ment. The modal collapse is a sort of constricted inconsistency at thelevel of possible world semantics. The assumption that there mayactually be more than one possible world is refuted; this followsfrom 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). Severalfurther contributions complete these initial experiments onthe formal assessment of rational arguments in metaphysics(Benzmüller and Woltzenlogel Paleo, 2013a; Benzmüllerand Woltzenlogel Paleo, 2015c; Benzmüller and Woltzen-logel Paleo, 2015a; Benzmüller, 2015b; Benzmüller andWoltzenlogel Paleo, 2015b; Benzmüller, 2015a; Benzmüllerand Woltzenlogel Paleo, 2015d; Benzmüller and Woltzenlo-gel Paleo, 2013b).
Principia Metaphysica.
Analyzing masterpiece rationalarguments in philosophy with the semantical embedding ap-proach on the computer is not trivial. However, it still leads tocomparably small corpora of axioms, lemmata and theorems,and it does thus not yet provide feedback on the scalabilityof the approach for larger and more ambitious projects. Forthat reason another challenge has recently been tackled: the
Principia Logico-Metaphysica (PLM) by Zalta (2016). ThePLM is intended to provide a rigorous formal basis for all ofmetaphysics and the sciences; this includes a (flexible) foun-dation for mathematics and in this sense it is more ambitiousthan Russel’s Principia Mathematica. Since Zalta has cho-sen a hyperintensional (relational) higher-order modal logicS5 as the logical foundation of his PLM, it has hence beenan open challenge question whether this very specific logicalsetting can still be suitably encoded in the semantical em-bedding approach. Besides hyperintensionality, a particularchallenge 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 PLMhas progressed very promisingly. In fact, most of the PLMhas meanwhile been represented and partially automated inIsabelle/HOL by using the semantical embedding approach. Other Logics in Philosophy.
The approach has recentlybeen 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 presentedabove and below have inspired the design of a worldwide newlecture course on
Computational Metaphysics at FU Berlin(Wisniewski et al. , 2016). In this course the above researchon 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. See https://github.com/ekpyron/TAO, respectively https://github.com/ekpyron/TAO/blob/master/output/document.pdf ome of the student projects conducted in this course haveresulted in impressive new contributions. For example, acomputer-assisted reconstruction of an ontological argumentby Leibniz will appear as a chapter in a book dedicated to the300th 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. A key factorin the successful implementation of the course has been, thata single methodology and overall technique (the semanticalembedding 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 tosignificantly improve interdisciplinary, university level logiceducation.
Free Logics.
Prominent, open challenges for formalisationin 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 suitedfor handling such challenges. Free logics have interesting ap-plications, e.g. in natural language processing and as a logicof fiction. In mathematics, free logics are particularly suitedin applications domains such as category theory or projectivegeometry (e.g. morphism composition in category theory isa partial operation). Similar to the other non-classical logicsmentioned before, free logics can be elegantly embedded inHOL (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 automatedtheorem provers via Sledgehammer. In the course of these ex-periments, the provers revealed a technical flaw (constrictedinconsistency or missing axioms) in the well known categorytheory textbook by Freyd and Scedrov (1990).
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 supportse.g. the modeling of knowledge of rational agents. Doxasticlogic 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 thatthe 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 thewell known wise men puzzle resp. muddy children puzzle. The sources of the formalisation of Fitting’s work are availableat https://github.com/cbenzmueller/TypesTableauxAndGoedelsGod.
Time and Space.
The reasoning about time and space hasbeen a long standing challenge in artificial intelligence, inparticular, when combined reasoning about time, space andeventually 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 arebasically just a reinvention multi-modal logics (the base de-scription logic ALC corresponds to a basic multi-modal logicK), and thus the semantical embedding approach elegantlyapplies. Hence, the shallow embedding approach applies alsoto a range of prominent description logics, and the mentionedlogic 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 easilybe obtained by utilising the semantical embedding approach.An exemplary semantical embedding of the multi-valuedlogic 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).
The semantical embedding approach, which utilises classi-cal higher-order logic at meta-level to encode (combinationsof) 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 toolsas currently developed, integrated and deployed in modernhigher-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 rationalarguments, including masterpiece arguments in philosophy.A relevant and challenging future application directionconcerns the application of the semantical embedding ap-proach for the modeling of ethical, legal, social and culturalnorms in intelligent machines, ideally in combination withthe realisation of human-intuitive forms of rational argumentsin machines complementing internal decision making meansat 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 recentworks in the area of deontic logics (such as Makinson andvan der Torre (2000) and Carmo and Jones (2013)). cknowledgements:
This work has been supported by the following researchgrants 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 theseand other related projects. This includes (in alphabetical or-der): Larry Paulson, Dana Scott, Geoff Sutcliffe, AlexanderSteen, 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 , pages230–236. Springer, 2003.S. Abramsky, D. Gabbay, and T. Maibaum, editors.
Handbook ofLogic 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.
Faithand 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 afterG.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. InD. M. Gabbay, J. H. Siekmann, and J. Woods, editors,
Handbookof the History of Logic, Volume 9 — Computational Logic , pages215–254. North Holland, Elsevier, 2014.C. Benzmüller and L. Paulson. Exploring properties of nor-mal multimodal logics in simple type theory with LEO-II. InC. Benzmüller, C. Brown, J. Siekmann, and R. Statman, editors,
Reasoning in Simple Type Theory — Festschrift in Honor of PeterB. 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 logicsin 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 InternationalCongress, 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 ofLogics , 2013. (Invited paper).C. Benzmüller and B. Woltzenlogel Paleo. Automating Gödel’s on-tological proof of God’s existence with higher-order automatedtheorem 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
Reasoning Web 2015 , number 9203 in LNCS, pages 32–74, Berlin, Germany, 2015. Springer. (Invited paper).C. Benzmüller and B. Woltzenlogel Paleo. Interacting with modallogics 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, CSR2015, Listvyanka, Russia, July 13-17, 2015, Proceedings , volume9139 of
LNCS , pages 398–411. Springer, 2015.C. Benzmüller and B. Woltzenlogel Paleo. On logic embeddingsand Gödel’s God. In M. Codescu, R. Diaconescu, and I. Tutu, ed-itors,
Recent Trends in Algebraic Development Techniques: 22ndInternational 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 inGödel’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 asa collapse of the modal square of opposition. In J.-Y. Béziauand G. Basti, editors, , Studies in Univer-sal Logic. Springer International Publishing Switzerland, 2016.C. Benzmüller, F. Theiss, L. Paulson, and A. Fietzke. LEO-II - acooperative 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 evaluatingprovers 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 Intelligenceand Applications , pages 163–168, Montpellier, France, 2012. IOSPress.C. Benzmüller, L. C. Paulson, N. Sultana, and F. Theiß. Thehigher-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 presumablyin volume 11 (2017) issue 1.. Benzmüller. Automating access control logic in simple type the-ory with LEO-II. In D. Gritzalis and J. López, editors,
EmergingChallenges 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 combininglogics. In
Contest paper at the World Congress and School onUniversal Logic III (UNILOG) , Lisbon, Portugal, 2010. The con-ference had no published proceedings; the paper is available asarXiv: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 Computationallogics in Multi-agent Systems (CLIMA XI)) , 62(1-2):103–128,2011.C. Benzmüller. Automating quantified conditional logics in HOL. InF. Rossi, editor, , pages 746–753, Beijing, China, 2013.AAAI Press.C. Benzmüller. Gödel’s ontological argument revisited – findingsfrom a computer-supported analysis (invited). In R. S. Silvestreand J.-Y. Béziau, editors,
Handbook of the 1st World Congress onLogic and Religion, João Pessoa, Brazil , page 13, 2015. (Invitedabstract).C. Benzmüller. Invited talk: On a (quite) universal theorem provingapproach 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 ProgramDevelopment . Springer, 2004.F. Bjørdal. Understanding Gödel’s ontological argument. InT. Childers, editor,
The Logica Yearbook 1998 . Filosofia, 1999.P. Blackburn, J. van Benthem, and F. Wolter, editors.
Handbook ofModal Logic . Elsevier, 2006.J. Blanchette and T. Nipkow. Nitpick: A counterexample generatorfor higher-order logic based on a relational model finder. In
ITP2010 , 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. InB. 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 resultsfor 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 ofSymbolic Logic , 5:56–68, 1940. L. M. de Moura, S. Kong, J. Avigad, F. van Doorn, and J. vonRaumer. 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 9195of
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 nachgebildeteFormelsprache des reinen Denkens . Halle, 1879.P. J. Freyd and A. Scedrov.
Categories, Allegories . North Holland,1990.D. Gabbay and F. Guenthner, editors.
Handbook of PhilosophicalLogic , volume 1-17. Springer, 2001-2014.D. Gabbay, C. Hogger, and J. Robinson, editors.
Handbook of Logicin Artificial Intelligence and Logic Programming . Oxford Uni-versity Press, 1993-1998.D. Gabbay, J. Woods, and EtAl., editors.
Handbook of the Historyof Logic , volume 1-11. Elsevier, 2004-2014.O. Gasquet, A. Herzig, D. Longin, and M. Sahade. Lotrec: Logicaltableaux 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. InSobel Sobel (2004), 1970.M. J. C. Gordon and T. F. Melham, editors.
Introduction to HOL. Atheorem proving environment for higher order logic . CambridgeUniversity Press, 1993.W. Guttmann, G. Struth, and T. Weber. Automating algebraic meth-ods in isabelle. In S. Qin and Z. Qiu, editors,
Formal Methods andSoftware Engineering - 13th International Conference on FormalEngineering 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. InA. 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. InB. 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 inHigher Order Logics, 22nd International Conference, TPHOLs2009, Munich, Germany, August 17-20, 2009. Proceedings , vol-ume 5674 of
Lecture Notes in Computer Science , pages 60–66.Springer, 2009.. 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
LectureNotes 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 forHOL light.
Mathematics in Computer Science , 9(1):5–22, 2015.J. Lambek and P. Scott.
Introduction to Higher Order CategoricalLogic . Cambridge University Press, 1986.K. Lambert.
Free Logic. Selected Essays . Cambridge UniversityPress, 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. Encodingtwo-valued nonclassical logics in classical logic. In J. Robinsonand A. Voronkov, editors,
Handbook of Automated Reasoning (in2 volumes) , pages 1403–1486. Elsevier and MIT Press, 2001.P. E. Oppenheimer and E. N. Zalta. Relations versus functions atthe 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 Provingin Higher Order Logics, 21st International Conference, TPHOLs2008, 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.
TheoreticalComputer Science , 249(1):3–80, 2000.R. Schmidt and U. Hustadt. First-order resolution methods formodal 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. InSobel Sobel (2004), 1972.D. Scott. Gödel’s ontological proof. In
On Being and Saying. Essaysfor Richard Cartwright , pages 257–258. MIT Press, 1987.J. Sobel.
Logic and Theism: Arguments for and Against Beliefs inGod . 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 reasoningin expressive non-classical logics with Isabelle/HOL. In C. Ben-züller, R. Rojas, and G. Sutcliffe, editors,
GCAI 2016, 2nd GlobalConference on Artificial Intelligence , volume 41 of
EPiC Seriesin 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 PracticalAspects 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 andLanguage (2nd ed.) . Elsevier, 2011.M. Wisniewski, A. Steen, and C. Benzmüller. Leopard - A genericplatform for the implementation of higher-order reasoners. InM. 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,