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Computational Hermeneutics:

An Integrated Approach for the Logical Analysis

of Natural-Language Arguments

David Fuenmayor and Christoph Benzmüller

Abstract We utilize higher-order automated deduction technologies for the logi-

cal analysis of natural language arguments. Our approach, termed computational

hermeneutics, is grounded on recent progress in the area of automated theorem

proving for classical and non-classical higher-order logics, and it integrates tech-

niques from argumentation theory. It has been inspired by ideas in the philosophy of

language, especially semantic holism and Donald Davidson’s radical interpretation;

a systematic approach to interpretation that does justice to the inherent circularity

of understanding: the whole is understood compositionally on the basis of its parts,

while each part is understood only in the context of the whole (hermeneutic circle).

Computational hermeneutics is a holistic, iterative approach where we evaluate the

adequacy of some candidate formalization of a sentence by computing the logical

validity of (i) the whole argument it appears in, and (ii) the dialectic role the argument

plays in some piece of discourse.

1 Motivation

While there have been major advances in the ﬁeld of automated theorem prov-

ing (ATP) during the last years, its main ﬁeld of application has mostly remained

bounded to mathematics and hardware/software veriﬁcation. We argue that the use

of ATP in argumentation (particularly in philosophy) can also be very fruitful,1not

only because of the obvious quantitative advantages of automated reasoning tools

(e.g. reducing by several orders of magnitude the time needed to test argument’s

David Fuenmayor

Freie Universität Berlin

e-mail: david.fuenmayor@fu-berlin.de

Christoph Benzmüller

Freie Universität Berlin and University of Luxembourg

(Funded by VolkswagenStiftung under grant CRAP: Consistent Rational Argumentation in Politics.)

e-mail: c.benzmueller@fu-berlin.de

1See e.g. the results reported in [5, 12–14, 32].

1

2 D. Fuenmayor and C. Benzmüller

validity), but also because it enables a novel approach to the logical analysis (aka.

formalization) of arguments, which we call computational hermeneutics.

As a result of reﬂecting upon previous work on the application of ATP for the

computer-supported evaluation of arguments in metaphysics [5,12–14, 30], we have

become interested in developing a methodology for formalising natural-language ar-

guments with regard to their assessment using automated tools. Unsurprisingly, the

problem of ﬁnding an/the adequate formalization of some piece of natural-language

discourse turns out to be far from trivial. In particular, concerning expressive higher-

order logical representations, this problem has already been tackled in the past

without much practical success.2In spite of the eﬀorts made in this area, carrying

out logical analysis of natural language (particularly of arguments) continues to

be considered as a kind of artistic skill that cannot be standardized or taught me-

thodically, aside from providing students with a handful of paradigmatic examples

supplemented with commentaries.

Our research aims at improving this situation. By putting ourselves in the shoes of

an interpreter aiming at ‘translating’ some natural-language argument into a formal

representation, we have had recourse to the philosophical theories of radical trans-

lation [46] and radical interpretation [22,25] (the latter being a further development

of the former), in which a so-called radical translator (Quine) or interpreter (David-

son), without any previous knowledge of the speaker’s language, is able to ﬁnd a

translation (in her own language) of the speaker’s utterances. The interpreter does

this by observing the speaker’s use of language in context and also by engaging

(when possible) in some basic dialectical exchange with him/her (e.g. by making

utterances while pointing to objects or asking yes/no questions). In our proposed

approach, this exchange takes place between a human (seeking to translate ‘unfamil-

iar’ natural-language discourse into a ‘familiar’ logical formalism) and interactive

proof assistants. The questions we ask concern the logical validity, invalidity and

consistency of formulas and proofs (our translation-candidates).

We also draw upon recent work aimed at providing adequacy criteria for logical

formalization of natural language discourse (e.g. [4,19, 43]), with a special emphasis

on the work of Peregrin and Svoboda [44], who, apart from providing syntactic

and pragmatic (inferential) adequacy criteria, also tackle the problem of providing a

systematic methodology for logical analysis. In this respect, they propose the method

of reﬂective equilibrium,3which is similar in spirit to the idealized scientiﬁc method

2See e.g. the research derived from Montague’s universal grammar program [39] and some of its

followers like Discourse Representation Theory (e.g. [37]) and Dynamic Predicate Logic (e.g. [36]).

3The notion of reﬂective equilibrium has been initially proposed by Nelson Goodman [35] as an

account for the justiﬁcation of the principles of (inductive) logic and has been popularized years

later in political philosophy and ethics by John Rawls [47] for the justiﬁcation of moral principles.

In Rawls’ account, reﬂective equilibrium refers to a state of balance or coherence between a set of

general principles and particular judgments (where the latter follow from the former). We arrive at

such a state through a deliberative give-and-take process of mutual adjustment between principles

Computational Hermeneutics 3

and, additionally, has the virtue of approaching this problem in a holistic way: the

adequacy of candidate formalizations for some argument’s sentences is assessed by

computing the argument’s validity as a whole (which depends itself on the way we

have so far formalized all of its constituent sentences).4As we see it, this circle

is a virtuous one: it does justice to holistic accounts of meaning drawing on the

inferential role of sentences. As Davidson has put it:

“[...] much of the interest in logical form comes from an interest in logical geography: to give

the logical form of a sentence is to give its logical location in the totality of sentences, to describe

it in a way that explicitly determines what sentences it entails and what sentences it is entailed by.

The location must be given relative to a speciﬁc deductive theory; so logical form itself is relative

to a theory." [23, p. 140]

2 Radical Interpretation and the Principle of Charity

What is the use of radical interpretation in argumentation? The answer is trivially

stated by Davidson himself, by arguing that “all understanding of the speech of

another involves radical interpretation" [22, p. 125]. Furthermore, the impoverished

evidential position we are faced with when interpreting some arguments (particu-

larly philosophical ones) corresponds very closely to the starting situation Davidson

contemplates in his thought experiments on radical interpretation, where he shows

how an interpreter could come to understand someone’s words and actions without

relying on any prior understanding of them. Davidson’s program builds on the idea

of taking the concept of truth as basic and extracting from it an account of inter-

pretation satisfying two general requirements: (i) it must reveal the compositional

structure of language, and (ii) it can be assessed using evidence available to the

interpreter [22, 24].

The ﬁrst requirement (i) is addressed by noting that a theory of truth in Tarski’s style

(modiﬁed to apply to natural language) can be used as a theory of interpretation. This

implies that, for every sentence sof some object language L, a sentence of the form:

«“s" is true in Liﬀ p» (aka. T-schema) can be derived, where pacts as a translation

of sinto a suﬃciently expressive language used for interpretation (note that in the

T-schema the sentence pis being used, while sis only being mentioned). Thus, by

virtue of the recursive nature of Tarski’s deﬁnition of truth [51], the compositional

structure of the object-language sentences becomes revealed. From the point of view

of computational hermeneutics, the sentence sis to be interpreted in the context of

a given argument (or a network of mutually attacking/supporting arguments). The

language Lthereby corresponds to the idiolect of the speaker (natural language), and

and judgments. More recent methodical accounts of reﬂective equilibrium have been proposed as

a justiﬁcation condition for scientiﬁc theories [28] and objectual understanding [3].

4In much the same spirit of Davidson’s theory of meaning [24] and Quine’s holism of theory

(dis-)conﬁrmation [45] in philosophy.

4 D. Fuenmayor and C. Benzmüller

the target language is constituted by formulas of our chosen logic of formalization

(some expressive logic XY) plus the turnstyle symbol `XY signifying that an infer-

ence (argument or argument step) is valid in logic XY. As an illustration, consider

the following instance of the T-schema:

«“Fishes are necessarily vertebrates" is true [in English, in the context of argu-

ment A] iﬀ A1, A2, ..., An`MLS4 “∀x. Fish(x) →Vertebrate(x)"»

where A1, A2, ..., Ancorrespond to the formalization of the premises of argument A

and the turnstyle `MLS4 corresponds to the standard logical consequence relation in

the chosen logic of formalization, e.g. a modal logic S4 (MLS4).5This toy example

aims at illustrating how the interpretation of a sentence relates to its logic of formal-

ization and to the inferential role it plays in a single argument. Moreover, the same

approach can be extended to argument networks. In such cases, instead of using the

notion of logical consequence (represented above as the parameterized logical turn-

style `XY), we can work with the notion of argument support. It is indeed possible

to parameterize the notions of support and attack common in argumentation theory

with the logic used for argument’s formalization (see example in section 5).

The second general requirement (ii) of Davidson’s account of radical interpretation

states that the interpreter has access to objective evidence in order to judge the appro-

priateness of her interpretations, i.e., access to the events and objects in the ‘external

world’ that cause sentences to be true (or, in our case, arguments to be valid). In our

approach, formal logic serves as a common ground for understanding. Computing

the logical validity of a formalized argument constitutes the kind of objective (or,

more appropriately, intersubjective) evidence needed to secure the adequacy of our

interpretations, under the charitable assumption that the speaker follows (or at least

accepts) similar logical rules as we do. In computational hermeneutics, the computer

acts as an (arguably unbiased) arbiter deciding on the truth of a sentence in the con-

text of an argument.

A central concept in Davidson’s account of radical interpretation is the principle of

charity, which he holds as a condition for the possibility of engaging in any kind of

interpretive endeavor. The principle of charity has been summarized by Davidson

by stating that “we make maximum sense of the words and thoughts of others when

we interpret in a way that optimizes agreement" [24, p. 197]. Hence the principle

builds on the possibility of intersubjective agreement about external facts among

speaker and interpreter. The principle of charity can be invoked to make sense of a

speaker’s ambiguous utterances and, in our case, to presume (and foster) the validity

of an argument. Consequently, in computational hermeneutics we assume from the

outset that the argument’s conclusions indeed follow from its premises and disregard

formalizations that do not do justice to this postulate.

5As described below, using the technique of semantical embeddings [10] (cf. also [6] and the

references therein) allows us to work with several diﬀerent non-classical logics (modal, temporal,

deontic, intuitionistic, etc.) while reusing existing higher-order reasoning infrastructure.

Computational Hermeneutics 5

3 Holistic Approach: Why Feasible Now?

Following a holistic approach for logical analysis was, until very recently, not feasible

in practice; since it involves an iterative process of trial-and-error, where the ade-

quacy of some candidate formalization for a sentence becomes tested by computing

the logical validity of the whole argument. In order to explore the vast combinatoric

of possible formalizations for even the simplest argument, we have to test its validity

at least several hundreds of times (also to account for logical pluralism). It is here

where the recent improvements and ongoing consolidation of modern automated

theorem proving technology (for propositional logic, ﬁrst-order logic and in partic-

ular also higher-order logic) become handy.

To get an idea of this, let us imagine the following scenario: A philosopher working

on a formal argument wants to test a variation on one of its premises or deﬁnitions

and ﬁnd out if the argument still holds. Since our philosopher is working with pen

and paper, she will have to follow some kind of proof procedure (e.g. tableaus or

natural-deduction calculus), which, depending on her calculation skills, may take

some minutes to be carried out. It seems clear that she cannot allow herself many of

such experiments on such conditions.

Now compare the above scenario to another one in which our working philosopher

can carry out such an experiment in just a few seconds and with no eﬀort, by em-

ploying an automated theorem prover. In a best-case scenario, the proof assistant

would automatically generate a proof (or the sketch of a countermodel), so she just

needs to interpret the results and use them to inform her new conjectures. In any

case, she would at least know if her speculations had the intended consequences, or

not. After some minutes of work, she will have tried plenty of diﬀerent variations of

the argument while getting real-time feedback regarding their suitability.6

We aim at showing how this radical quantitative increase in productivity does indeed

entail a qualitative change in the way we approach formal argumentation, since it

allows us to take things to a whole new level (note that we are talking here of many

hundreds of such trial-and-error ‘experiments’ that would take months or even years

if using pen and paper only). Most importantly, this qualitative leap opens the door

for the possibility of fully automating the process of argument formalization, as

it allows us to compute inferential (holistic) adequacy criteria of formalization in

real-time. Consider as an example Peregrin and Svoboda’s [43,44] proposed criteria:

(i) The principle of reliability: “φcounts as an adequate formalization of the sentence

Sin the logical system Lonly if the following holds: If an argument form in which

6The situation is obviously idealized, since, as is well known, most of theorem-proving problems

are computationally complex and even undecidable, so in many cases a solution will take several

minutes or just never be found. Nevertheless, as work in the emerging ﬁeld of computational

metaphysics [12–14, 29, 30, 48] suggests, the lucky situation depicted above is not rare and will

further improve in the future.

6 D. Fuenmayor and C. Benzmüller

φoccurs as a premise or as the conclusion is valid in L, then all its perspicuous

natural language instances in which Sappears as a natural language instance of φ

are intuitively correct arguments."

(ii) The principle of ambitiousness: “φis the more adequate formalization of the

sentence Sin the logical system Lthe more natural language arguments in which S

occurs as a premise or as the conclusion, which fall into the intended scope of Land

which are intuitively perspicuous and correct, are instances of valid argument forms

of Sin which φappears as the formalization of S." [44, pp. 70-71].

The evaluation of such inferential criteria clearly involves automatically comput-

ing the logical validity or consistency of formalized arguments (proofs). This is the

kind of work automated theorem provers are built for. Moreover, our focus on theo-

rem provers for higher-order logics is motivated by the notion of logical pluralism.

Computational hermeneutics targets the utilization of diﬀerent kinds of classical and

non-classical logics through the technique of semantical embeddings [10] (cf. also [6]

and the references therein), which allows us to take advantage of the expressive power

of classical higher-order logic (HOL) as a metalanguage in order to embed the syntax

and semantics of another logic as an object language. Using (shallow) semantical

embeddings we can, for instance, embed a modal logic by deﬁning the modal and

♦operators as meta-logical predicates in HOL and using quantiﬁcation over sets of

objects of a deﬁnite type w, representing the type of possible worlds or situations.

This gives us two important beneﬁts: (i) we can reuse existing automated theorem

proving technology for HOL and apply it for automated reasoning in non-classical

logics (e.g. free, modal, temporal or deontic logics); and (ii) the logic of formaliza-

tion becomes another degree of freedom and thus can be ﬁne-tuned dynamically by

adding/removing axioms in our metalanguage: HOL. A framework for automated

reasoning in diﬀerent logics by applying the technique of semantical embeddings

has been successfully implemented using automated theorem proving technology

(see e.g. [6, 30, 33]).

The following two sections illustrate some exemplary applications of the computa-

tional hermeneutics approach. They have been implemented using the Isabelle/HOL

[40] proof assistant for classical higher-order logic, aka. Church’s type theory [1].

4 Logical Analysis of Individual Structured Arguments

A ﬁrst application of computational hermeneutics for the logical analysis of ar-

guments has been presented in [32] (with its corresponding Isabelle/HOL sources

available in [31]). In that work, a modal variant of the ontological argument for the

existence of God, introduced in natural language by the philosopher E. J. Lowe [38],

has been iteratively analyzed using our computational hermeneutics approach and,

as a result, a ’most’ adequate formalization has been found. In a series of iterations

(seven in total) Lowe’s argument has been formally reconstructed using slightly dif-

Computational Hermeneutics 7

ferent sets of premises and logics, and the partial results have been compiled and

presented each time as a new variant of the original argument. We aimed at illus-

trating how Lowe’s argument, as well as our understanding of it, gradually evolves

as we experiment with diﬀerent combinations of (formalized) deﬁnitions, premises

and logics for formalization. We quote from [32] the following paragraph which best

summarizes the methodological approach taking us from a natural-language argu-

ment to its corresponding adequate logical formalization (and refer the interested

reader to [32] and [31] for further details):

“We start with formalizations of some simple statements (taking them as tentative) and use

them as stepping stones on the way to the formalization of other argument’s sentences, repeating

the procedure until arriving at a state of reﬂective equilibrium: A state where our beliefs and com-

mitments have the highest degree of coherence and acceptability. In computational hermeneutics,

we work iteratively on an argument by temporarily ﬁxing truth-values and inferential relations

among its sentences, and then, after choosing a logic for formalization, working back and forth

on the formalization of its premises and conclusions by making gradual adjustments while getting

automatic feedback about the suitability of our speculations. In this fashion, by engaging in a

dialectic questions-and-answers (‘trial-and-error’) interaction with the computer, we work our way

towards a proper understanding of an argument by circular movements between its parts and the

whole (hermeneutic circle)."

5 Logical Analysis of Arguments in their Extended

Argumentative Context

As mentioned in the previous section, an instance of the ontological argument by

E. J. Lowe has previously been employed to showcase the application of our compu-

tational hermeneutics approach to the problem of ﬁnding an adequate formalization

for a natural-language argument (but without considering the surrounding network

of arguments where it is embedded) [32]. In contrast, the example we discuss in

this section7additionally motivates and illustrates the fruitful combination of our

previous work with methods as typically used in abstract argumentation frameworks.

By doing so, we can now extend our holistic approach to logical analysis to include

the dialectic role an argument plays in some larger area of discourse represented

as a network of arguments. We see this as a novel contribution, which aligns our

work with other prominent structured approaches to argumentation in artiﬁcial in-

telligence [16, 17, 27].

Below we will show how our approach can extend methods from abstract argumenta-

tion [26,52] by adding a layer for deep semantical analysis to it and, vice versa, how

7The assessment presented here draws on previous work in [30] and particularly the more recent,

invited paper [8], which present an updated analysis of Gödel’s and Scott’s modal variants [34, 49]

of the ontological argument and illustrate how our method is able to formalize, assess and explain

those in full detail.

8 D. Fuenmayor and C. Benzmüller

our approach to deep semantical argument analysis becomes enriched by augmenting

it with methods as developed in argumentation theory. This way, argument analysis

becomes supported at both the abstract level and a concrete semantical level (i.e. with

fully formalized natural language content). We believe that such a combined, two-

level approach can provide a fruitful technological backbone for our computational

hermeneutics program. A particular advantage being the logical plurality we achieve

at both layers. At the abstract level, for instance, the support or attack relations8can

be replaced with little technical eﬀort with e.g. their intuitionistic logic or relevance

logic counterparts (exploiting the technique of shallow semantical embeddings).9At

a concrete level, we will demonstrate how the employed logics can be varied and

that it is, in fact, essential to do so, in order to achieve proper assessment results.

More precisely, in the example below we will switch between the higher-order modal

logics K, KB, S4 and S5.

5.1 Gödel’s Ontological Argument as a Showcase

Gödel’s and Scott’s variants of the ontological argument are direct descendants of

Leibniz’s, which in turn derives from Descartes’. These arguments have a two-part

structure: (i) prove that God’s existence is possible (see t3 in Figs. 1 and 2), and (ii)

prove that God’s existence is necessary, if possible (t5). The main conclusion (God’s

necessary existence, t6) then follows from (i) and (ii), either by modus ponens (in

non-modal contexts) or by invoking some axioms of modal logic (notably, but not

necessarily as we will see, the so-called modal logic system S5). Gödel’s ontological

argument, in its diﬀerent variants, is amongst the most discussed formal proofs in

modern literature, and so most of its premises and inferential steps have been subject

to criticism in some way or another (see e.g. [41], [42] and [50]). We can therefore

conceive of this argument as a network of (abstracted) nodes, some of them standing

for some argument supporting the respective premise and others standing for attack-

ing arguments (cf. bipolar argumentation frameworks [20, 21]).

The abstracted nodes of the natural language argument are introduced in Fig. 1

together with their associated formalizations in higher-order modal logic. The cor-

responding abstract argumentation network is displayed in Fig. 2. The network pre-

sented in Fig. 2 only comprises support relations, which is suﬃcient for the purpose

of this paper. Meaningful attack relations could of course be added. For example, it

is well known that Gödel’s and Scott’s versions of the ontological argument support

the modal collapse [50], which in turn can be interpreted as an attack to free will

(see the recent discussion of this aspect in [8]). Extending our work below to cover

8See lines 4-5 in Fig. 4, where their deﬁnitions are provided for classical logic

9The full ﬂexibility of our framework is not illustrated to its maximum in this paper due to

space restrictions. For example, for intuitionistic logic we would simply integrate the respective

embedding presented in earlier work [9] to model intuitionistic support/attack relations.

Computational Hermeneutics 9

d1 Being Godlike is equivalent to having all positive properties.

a1 Exactly one of a property or its negation is positive.

a2 Any property entailed by a positive property is positive.a

t1 Every positive property is possibly instantiated (if a property X is positive, then it is possible

that some being has property X).

t2bBeing Godlike is a positive property.

t3 Being Godlike is possibly instantiated.

a4 Positive (negative) properties are necessarily positive (negative).

d2 A property Y is the essence of an individual x iﬀ x has Y and all of x’s properties are entailed

by Y.c

t4 Being Godlike is an essential property of any Godlike individual (Eis standing for one of the

two notions above).

d3 Necessary existence of an individual is the necessary instantiation of all its essences.

a5 Necessary existence is a positive property.

t5 Being Godlike, if (possibly) instantiated, is necessarily instantiated.

t6 Being Godlike is actually instantiated.

aThe quantiﬁers ∃EXand ∀EXrepresent a kind of restricted (aka. actualist) quantiﬁcation over

a set of ‘existent’ objects. Its deﬁnition can be seen in lines 31-35 in Fig 3.

bGödel originally considered an additional assumption a3, which was used solely for deriving

theorem t2; Scott suggested to take t2 directly as an assumption instead, which is what we also

adopt here.

cThe underlined part in deﬁnition D2 has been added by Scott [49]. Gödel [34] originally omitted

this part; more on this in Section 2.3 below.

Fig. 1 The deﬁnitions (d1,d2,d3), assumptions (a1,a2,t2,a4,a5) and theorems/argumentation steps

(t1,t3,t4,t5,t6) of Gödel’s, respectively Scott’s, modal version of the ontological argument. Both the

natural language statements and the corresponding modal logic formalizations (in Isabelle/HOL)

are presented.

10 D. Fuenmayor and C. Benzmüller

a2a1a4

t2t1t4a5

t3t5

t6

d1,d2

d1d3

Fig. 2 Abstract argumentation network for Gödel’s ontological argument. The displayed arrows

indicate support relations. Arrow annotations (e.g., d1, d2in the support arrow from a4to t4)

indicate which deﬁnitions need to be unfolded for the respective support relations to apply.

the more elaborate analysis of the ontological argument as presented in [8] will be

addressed in future work.

5.2 Embedding a Higher-order Modal Logic in Isabelle/HOL

As previously mentioned, higher-order modal logic has been employed as a logic

for formalization of the natural-language content of the argument nodes. To turn

Isabelle/HOL into a ﬂexible modal logic reasoner we have adopted the shallow se-

mantical embedding approach [6, 10]. The respective embedding of higher-order

modal logic in Isabelle/HOL is the content of theory ﬁle «IHOML.thy», which is

displayed in Fig. 3. The base logic of Isabelle/HOL is classical higher-order logic

(HOL aka. Church’s type theory [1]). HOL is a logic of functions formulated on

top of the simply typed λ-calculus, which also provides a foundation for functional

programming. The semantics of HOL is well understood [7]. Relevant for our pur-

poses is that HOL supports the encoding of sets via their characteristic functions

represented as λ-terms. In this sense, HOL comes with a in-built notion of (typed)

sets that is exploited in our work for the explicit encoding of the truth-sets that

are associated with the formulas of higher-order modal logic. Since Isabelle/HOL-

speciﬁc extensions of HOL (except for preﬁx polymorphism) are not exploited in

our work, the technical framework we depict here can easily be transferred to other

HOL theorem proving environments.

Our semantical embedding in Isabelle/HOL encodes in lines 6-24 in Fig. 3 the

standard translation of propositional modal logic to ﬁrst-order logic in form of a

few (non-recursive) equations. Formula ϕ, for example, is modeled as an abbrevi-

ation (syntactic sugar) for the truth-set λwi.∀vi.wrv −→ ϕv, where rdenotes the

accessibility relation associated with the modal operator. All presented equations

Computational Hermeneutics 11

Fig. 3 Shallow semantical embedding of higher-order modal logic in Isabelle/HOL

exploit the idea that truth-sets in Kripke-style semantics can be directly encoded as

predicates (i.e. sets) in HOL. Possible worlds are thus explicitly represented in our

framework as terms of type iand modal formulas ϕare identiﬁed with their corre-

sponding truth sets ϕi→oof predicate type i→o. Note how validity and invalidity

is encoded in lines 21 and 24. A modal logic formula ϕis valid, denoted bϕc, if

its truth-set is the universal set, i.e., if ϕi→ois true in all words wi. Similarly, ϕ

is invalid, denoted bϕcinv, if ϕi→ois false in all worlds wi. In lines 26-35, further

equations are added to obtain actualist quantiﬁcation (here only for individuals) and

(polymorphic) possibilist quantiﬁcation for objects of arbitrary type (order). This

is where the shallow semantical embedding approach signiﬁcantly augments the

standard translation for propositional modal logics. For example, where ∀xα.φx is

12 D. Fuenmayor and C. Benzmüller

shorthand (binder-notation in HOL) for Πλxα.φx (the denotation of Πtest whether

its argument denotes the universal set of type α), then ∀xα.P x is now represented

as Π0λxα.λwi.P xw, where Π0stands for the lambda term λΦ.λwi.∀xα.Φxw and

the gets resolved as described above. In lines 37-42 some useful relations on

accessibility relations are stated, which are used to provide semantical deﬁnitions

for modal logics KB, S4 and S5. If none the latter abbreviations is postulated, the

content of Fig. 3 introduces higher-order modal logic K. Further details of the pre-

sented embedding, including proofs of faithfulness, have been presented elsewhere

(see e.g. [10], further references are given in [6]).

5.3 The Ontological Argument as an Abstract Argument Network

In lines 4-5 of the Isabelle/HOL ﬁle «ArgumentDeﬁnitions.thy», displayed in Fig. 4,

we import two central notions from argumentation theory: the binary relations “Sup-

ports” and “Attacks” [17, 21]. The former states that a valid modal formula Aand

another valid modal formula Btogether imply the validity of modal formula ψ. The

concretely employed implication and conjunction relations are those from classi-

cal logic (the meta-logic HOL). However, as mentioned before, our framework is

rich and expressive enough to replace classical consequence here by various other

notions of logical consequence. Alternatively, we could parameterize the three con-

tained validity statements for A,Band ψfor diﬀerent modal logics. In fact, the

diﬀerent notions of logics to be employed in these deﬁnitions could be modeled

as proper parameters (arguments) of the support and attack relations. This way we

would obtain a very expressive and powerful reasoning framework. In order to keep

things simple we will not further pursue this here, but leave it for further work.

Gödel’s argument is then speciﬁed in lines 13-14 of Fig. 4 as a network of ab-

stract nodes (recall Fig. 2). The modal validity of the assumptions “a1”, “a2”, “t2”,

“a4” and “a5” is assumed and the various support relations are stated as depicted

graphically in Fig. 3. The nodes itself have been introduced as uninterpreted con-

stant symbols in line 8 of Fig. 4 (and in line 10, we introduce further uninterpreted

constant symbols “kb”, “s4” and “s5” for characterizing the assumed modal logic

conditions). The “inner semantics” of these abstract nodes, and also the deﬁnitions

of the concepts «Godlike (G)», «Essence (E)» and «Necessary Existence (NE)», are

subsequently speciﬁed in lines 17-33. Since theory ﬁle «IHOML.thy» is imported,

we have access to the higher-order modal logic notions introduced there. At this

point we still leave it open whether the logic K, KB, S4 or S5 is considered, since we

will experiment with the diﬀerent settings later on (neither Gödel nor Scott explicitly

stated in their works which modal logic they actually assumed). We also do not ﬁx

the notion of essence here, but introduce the two alternative deﬁnitions proposed

by Gödel and Scott (see lines 26-27). In line 28, a respective uninterpreted constant

symbol for essence, E, is introduced and then used as a dummy in the formalization

of the subsequent argument nodes. In the experiments ahead we can now switch

Computational Hermeneutics 13

Fig. 4 Encoding of Gödel’s ontological argument as an abstract argument network, with a speciﬁ-

cation of the inner semantics of the argument nodes.

between Gödel’s and Scott’s notions of essence, by equating this dummy constant

symbol with the diﬀerent concrete deﬁnitions considered.

In lines 36-37, the abstract argument nodes are identiﬁed with their formalizations

as just introduced. Thus, when postulating the here deﬁned Boolean ﬂag “Instanti-

ateArgumentNodes”, the argument from lines 13-14 is no longer just abstract, but

in a sense instantiated through activation of the inner semantics of the argument

nodes. In line 39, the abstract logic conditions are analogously instantiated with their

concrete realizations by Boolean ﬂag “InstantiateLogics”. The “Instantiate” in line

41 then simply combines them into a single ﬂag.

14 D. Fuenmayor and C. Benzmüller

Fig. 5 Analysis of Gödel’s variant of the ontological argument.

5.4 Analysis of Gödel’s Variant of the Ontological Argument

In Fig. 5 we analyze Gödel’s variant of the ontological argument using the notions and

ideas as introduced in the imported theory ﬁles «IHOML.thy» and «ArgumentDeﬁ-

nitions.thy» from Figs. 3 and 4 respectively. To do so, the “essence" dummy constant

symbol is mapped to the deﬁnition as proposed by Gödel’s (see line 4). Then, in lines

7-9, we ask the model ﬁnder Nitpick [18], integrated with Isabelle/HOL, to compute

a model for the abstract Gödel argument. For the call in line 7, a model consisting of

one world (accessible from itself – not shown in the window) with one single indi-

vidual is presented in the lower window of Fig. 5. The duplicated calls in line 8 and 9

are of course redundant, since the Boolean logic ﬂags “kb”, “s4” and “s5” (and also

the argument nodes) are still uninterpreted. Hence, at the abstract level, the Gödel ar-

Computational Hermeneutics 15

gument has a model (we here even present the minimal model) and is thus satisﬁable.

For illustration purposes we show in line 12 that the abstract argument can eas-

ily become unsatisﬁable, for example, by adding an attack relation as displayed.

Automated theorem proving technology integrated with Isabelle/HOL can quickly

reveal such inconsistencies at the abstract argument level. A much more interesting

and relevant aspect is illustrated in lines 25-24 of Fig. 5: the satisﬁability of the

abstract Gödel argument provides no guarantee at all for its satisﬁability at instan-

tiated level, i.e. when the semantics of the argument nodes is added. In line 15, the

model ﬁnder Nitpick indeed fails to compute a satisfying model for the instantiated

argument (it terminates with a timeout).

However, if we now study the semantically instantiated (formalized) argument, which

is done by activating the link between the abstract nodes and their formalizations,

then we ﬁnd out that the Gödel argument is actually inconsistent (for all logic con-

ditions). And in lines 21-24, the inconsistency of the instantiated abstract argument

is then proven automatically by respective automation tools in Isabelle/HOL (here

the prover Metis is employed). The clue to the inconsistency is the “Empty Essence

Lemma (EEL)”, which is proven in line 18. The inconsistency of Gödel’s ontological

argument was unknown to philosophers until recently, when it was detected by the

automated higher-theorem prover LEO-II [11]; for more on this see [14,15].

5.5 Analysis of Scott’s Variant of the Ontological Argument

We analyze Scott’s version of the argument in Fig. 6. In line 4 of this ﬁle the notion of

essence according to Scott is activated. In lines 7-9, the model ﬁnder Nitpick again

conﬁrms the consistency of the argument at the abstract level, which was expected,

since the concretely employed notion of essence does not have an inﬂuence at this

level. It does so, however, at instantiated level, and this can be seen in lines 13-15 in

Fig. 6. In contrast to the instantiated Gödel argument from Fig. 5, where the model

ﬁnder Nitpick failed to conﬁrm satisﬁability, it now succeeds. And in fact it does so

for all logic conditions. The reported model (for logic S5, which in fact works for all

logic conditions) is displayed in the lower window of Fig. 6. This model is minimal,

it consists of one world (accessible to itself) and one object, and further details on

the interpretation of essence and the notion of positive properties are displayed.

However, as we illustrate next, the satisﬁability of Scott’s argument for logics KB, S4

and S5 does of course not imply the validity of the argument for these logic conditions.

In lines 19-29 in Fig. 7, we ﬁrst prove the validity of the Scott argument for logic

KB. This is done by automatically proving that all (instantiated) support relations

are validated. Note how concisely the exactly required dependencies are displayed in

the justiﬁcations of the proof steps. In lines 32-42 of Fig. 7, we analogously assess

the validity of the Scott argument for logic S4. However, the attempt to copy and

16 D. Fuenmayor and C. Benzmüller

Fig. 6 Analysis of Scott’s variant of the ontological argument – Part I

paste the previous proof does fail: in line 38, we obtain a countermodel, and this

countermodel comes with a non-symmetric accessibility relation between worlds.

Note that in line 25, in the proof from before, the logic condition KB (assms(1))

was indeed employed in the justiﬁcation. We here see that the symmetry condition

of logic KB indeed plays a role for the validity of Scott’s argument. In a logic such

as S4, where symmetric accessibility relations between worlds are not enforced, the

argument fails. This is conﬁrmed again in lines 44-45, where the countermodel is

computed directly for the stated validity conjecture (excluding the possibility that

there might be an alternative proof in S4 to the one attempted in lines 32-42). For

logic S5 the Scott argument is valid again, which is not a surprise given that logic

S5 entails logic KB. This is conﬁrmed in lines 48-58 in Fig. 7.

Computational Hermeneutics 17

Fig. 7 Analysis of Scott’s variant of the ontological argument – Part II

5.6 Section Summary

The analysis of non-trivial natural language arguments at the abstract argumentation

level is useful, but of limited explanatory power. Achieving such explanatory power

requires the extension of techniques from abstract argumentation frameworks with

means for deep semantical analysis as provided in our computational hermeneutics

approach. This has been illustrated in this section with the help of Gödel’s and

Scott’s versions of the ontological argument for the existence of God. Highly relevant

aspects, such as inconsistency of Gödel’s argument, invalidity of Scott’s argument

18 D. Fuenmayor and C. Benzmüller

for S4 and validity for KB and S5 could only be shown by the integration of the

abstract argumentation layer with our machinery.

6 Ongoing and Future Work

In previous work [32] we have illustrated how the computational hermeneutics ap-

proach can be carried out in a semi-automatic fashion for the logical analysis of an

isolated argument: We work iteratively on an argument by (i) tentatively choosing

a logic for formalization; (ii) ﬁxing truth-values and inferential relations among its

sentences; and (iii) working back and forth on the formalization of its axioms and

theorems, by making gradual adjustments while getting real-time feedback about the

suitability of our changes (e.g. validating the argument, avoiding inconsistency or

question-begging, etc.). This steps are to be repeated until arriving at a state of re-

ﬂective equilibrium: A state where our arguments and claims have the highest degree

of coherence and acceptability according to syntactic and, particularly, inferential

criteria of adequacy (see Peregrin and Svoboda’s criteria presented above [43,44]).

In this paper we have sketched another exemplary application of computational

hermeneutics to argumentation theory. We exploited the fact that our approach is

well suited for the utilization of diﬀerent kinds of classical and non-classical logics

through the technique of shallow semantical embeddings [6,10], which allows us to

take advantage of the expressive power of classical higher-order logic (as a metalan-

guage) in order to embed the syntax and semantics of another logic (object language).

This way it has become possible to parameterize the relations of argument support

and attack by adding the logic of formalization as a variable. This parameter can

then be varied by adding or removing premises (at a meta-level), which correspond

to the embedding of the logic in question. In future work, we aim at integrating our

approach with others in argumentation theory, which also take into account the logi-

cal structure of arguments (see e.g. [16, 17,27] and also [2] for a proof-theoretically

enriched approach).

Computational hermeneutics features the sort of (holistic) mutual adjustment be-

tween theory and observation, which is characteristic of scientiﬁc inquiry; we are

currently exploring the way to fully automate this process. The idea is to tackle the

problem of formalization as a combinatorial optimization problem, by using (among

others) inferential criteria of adequacy to deﬁne the ﬁtness/utility function of an

appropriate optimization algorithm. Davidson’s principle of charity would provide

our main selection criteria: an adequate formalization must (i) validate the argument

(among other qualitative requirements) and (ii) do justice to its intended dialectic

role in some discourse (i.e. it attacks/supports other arguments as intended). It is

worth noting that, for the kind of non-trivial arguments we are interested in (e.g.

from ethics, metaphysics and politics), such a selection criteria would aggressively

prune our search tree. Furthermore, the evaluation of our ﬁtness function is, with

Computational Hermeneutics 19

today’s technologies, not only completely automatizable, but also seems to be highly

parallelizable. The most challenging task remains how to systematically come up

with the candidate formalization hypotheses (an instance of abductive reasoning).

Here we see great potential in the combination of automated theorem proving with

other areas in artiﬁcial intelligence such as machine learning and, in particular, ar-

gumentation frameworks, by exploiting a layered, structured approach as illustrated

above.

Acknowledgements We thank the anonymous reviewers for their valuable comments which helped

to improve this paper.

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