Leibniz’s Art of Infallibility, Watson, and

the Philosophy, Theory, & Future of AI ∗

Selmer Bringsjord • Naveen Sundar G.Rensselaer AI & Reasoning (RAIR) Lab

Department of Computer ScienceDepartment of Cognitive Science

Rensselaer Polytechnic Institute (RPI)Troy NY 12180 USA

June 14, 2014

∗The authors are profoundly grateful to IBM for a grant through which the landmark Watson system was providedto RPI, and for an additional gift enabling us to theorize about this system in the broader context of logic, mathe-matics, AI, and the history of this intertwined trio. Bringsjord is also deeply grateful for the opportunity to speakat PT-AI 2013 in Oxford; for the leadership, vision, and initiative of Vincent Muller; and for the spirited objectionsand comments received at PT-AI 2013, which served to hone his thinking about the nature and future of (Weak) AI.Thanks are do as well to certain researchers working in the cognitive-computing space for helpful interaction: viz.,Dave Ferrucci, Chris Welty, Jim Hendler, Deb McGuiness, and John Licato.

Contents

1 Introduction 1

2 Leibniz, Logic, and “The Art of Infallibility” 22.1 Disclaimer: Leibniz’s Mill & Strong vs. Weak AI . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Central Aspects of Leibniz’s Dream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 The Leibnizian “Three-Ray” Conception of AI 4

4 An “Engineeringish” Leibnizian Theory of AI 54.1 Criteria for F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.1.1 Degree of Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.1.2 Degree of Quantification/Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.1.3 Degree of Homoiconicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.1.4 Degree of Possibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.1.5 Degree of Intensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.2 Criteria for SA and SF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3 Criteria for M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Watson Abstractly 105.1 Locating Watson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6 The Future of AI 12

List of Figures

1 Locating DCEC∗ in “Three-Ray” Leibnizian Universe . . . . . . . . . . . . . . . . . . . . . . . 52 DCEC∗ Syntax and Rules of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Representing Intensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Initial Stages Watson Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Final Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Assigning Confidence Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

List of Tables

1 Degree of Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Degree of Quantification/Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Degree of Homoiconicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Degree of Possibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Degree of Intensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Criteria SA and SF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Criteria M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1 Introduction

When IBM’s Deep Blue beat Kasparov in 1997, Bringsjord (1998) complained that despite theimpressive engineering that made this victory possible, chess is simply too easy a challenge forAI, given the full range of what the rational side of the human mind can muster.1 Systematichuman cognition leverages languages and logics that allow progress from set theory to cutting-edgeformal physics, and isn’t merely focused on an austere micro-language sufficient only to expressboard positions on an 8×8 grid, a small set of simple rules expressible in a simple set of first-orderformulae, and strategies expressible in off-the-shelf techniques and tools in AI.2 Even techniques forplaying invincible chess can be expressed in logical systems well short of those invented to captureaspects of human cognition, and the specific techniques used by Deep Blue were standard textbookones (e.g., alpha-beta “boosted” minimax search; again, see Russell & Norvig 2009).

However, arguably everything changed in 2011. For in that year, playing not a simple boardgame, but rather an open-ended game based in natural language, IBM’s Watson trounced the besthuman Jeopardy! players on the planet. What is Watson, formally, that is, logico-mathematically,speaking? And what does Watson’s prowess tell us about the philosophy, theory, and future ofAI? We present and defend snyoptic answers to these questions, ones based upon Leibniz’s seminalwritings on a universal one, on a Leibnizian “three-ray” space of computational formal logicsthat, inspired by those writings, we have invented, and on a “scorecard” approach to assessingreal AI systems based on turn on that three-ray space. It’s this scorecard approach that enablesa logico-mathematical understanding of Watson’s mind. And the scores that Watson earns inturn enables one to predict in broad terms the future of the interaction between homo sapienssapiens and increasingly intelligent computing machines. This future was probably anticipatedand called for in no small part by the founder of modern logic: Leibniz. He thought that God, ingiving us formal logic for capturing mathematics, sent thereby a hint to humans that they shouldsearch for a comprehensive formal logic able to capture and guide cognition across the full spanof rational human thought. This — as he put it — “true method” would constitute the “art ofinfallibility,” and would “furnish us with an Ariadne’s thread, that is to say, with a certain sensibleand palpable medium, which will guide the mind as do the lines drawn in geometry and the formulaefor operations which are laid down for the learner in arithmetic.”3 In modern terms, Leibniz wouldsay that God’s hint consists in this: The success of purely extensional first-order logic in modelingclassical mathematics indicates that more expressive logics can be developed in order to modelmuch more of human cognition.

Our plan for the remainder is this: Next (§2), after recording our agreement with Leibnizthat mechanical processing will never replicate human consciousness, we briefly summarize, in fourmain points, Leibniz’s original conception of the art of infallibility, and encapsulate his dream

1Note that we say: “rational side.” This is because in the present essay we focus exclusively upon what Leibnizaimed to systematize via his “art of infallibility.” Accordingly, in short, we target systematic human thought. (Inmodern terms, this sphere could be defined ostensively, by enumerating what is sometimes referred to as the formalsciences: logic and mathematics, formal philosophy, decision theory, game theory, much of modern analysis-basedeconomics (to which Leibniz himself paved the way), much of high-end engineering, and mathematical physics. Wearen’t concerned herein with such endeavors as poetry, music, drama, etc. Our focus shouldn’t be interpreted soas to rule out, or even minimize the value of, the modeling of, using formal logic, human cognition in these realms(indeed, e.g. see Bringsjord & Ferrucci 2000, Bringsjord & Arkoudas 2006); it’s simply that in the present essay weadopt Leibniz’s focus.

2Such as those nicely presented in (Russell & Norvig 2009).3From “Letter to Galois,” in the year 1677, included in (Leibniz & Gerhardt 1890).

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of a comprehensive logico-mathematical system able to render rigorous and infallible all humanthinking. In section 3, we present our “three-ray” Leibnizian logicist framework for locating agiven AI system. Inspired by this three-ray framework, we then (§4) provide a more fine-grained,engineeringish theory of AI systems (the aforementioned scorecard approach), and in section 5 usethis account to explain what Watson specifically is. We end with some brief remarks about thefuture of AI.

2 Leibniz, Logic, and “The Art of Infallibility”

2.1 Disclaimer: Leibniz’s Mill & Strong vs. Weak AI

It may be important to inform the reader that at least one of us believes with Leibniz not onlythat physicalism is false, but believes this proposition for specifically Leibnizian reasons. Leibnizfamously argued that because materialism is committed to the view that consciousness consistsin mere mechanical processing, materialism can’t explain consciousness, and moreover conscious-ness can’t consist merely in the movement of physical stuff. The argument hinges on a thought-experiment in which Leibniz enters what has become, courtesy of this very experiment, a ratherfamous mill.4 (The experiment and the anti-physicalist conclusion drawn from it, are defended in(Bringsjord 1992).) And he also argued elsewhere, more specifically, that self -consciousness is theinsurmountable problem for materialism. Note that Leibniz in all this argumentation takes aim atmechanical cognition: he argues that no computing machine can be self-conscious.5 In other words,Leibniz holds that Strong AI, the view that it’s possible to replicate even self-consciousness andphenomenal consciousness in a suitably programmed computer or robot, is false. Note that in ourmodern terminology, that which is computationally solvable (at least at the level of a standard Tur-ing machine or below) is the same as that which is — as it’s sometimes said in our recursion-theorytextbooks — mechanically solvable.

The previous paragraph implies that the Leibnizian “three-ray” conception of AI that we soonpresent, as well as our remarks about the future of AI following on that — all of this con-tent we view to be about Weak AI (the view that suitably programmed computers/robots cansimulate any human-level behavior whatsoever), not Strong AI.6 Put another way, the outwardbehavior of human persons, when confirmed by a well-defined test, can be matched by a suit-ably programmed computing machine or robot thereby able to pass the test in question. This

4In Leibniz’s words:

One is obliged to admit that perception and what depends upon it is inexplicable on mechanical prin-ciples, that is, by figures and motions. In imagining that there is a machine whose construction wouldenable it to think, to sense, and to have perception, one could conceive it enlarged while retaining thesame proportions, so that one could enter into it, just like into a windmill. Supposing this, one should,when visiting within it, find only parts pushing one another, and never anything by which to explaina perception. Thus it is in the simple substance, and not in the composite or in the machine, that onemust look for perception. [§17 of Monadology in (Leibniz 1991)]

5Leibniz came strikingly close to grasping universal or programmable computation. But nothing we say hererequires that he directly anticipated Post and Turing. This is true for the simple reason that Leibniz fully understoodde novo computation in the modern sense, and also was the first to see that a binary alphabet could be used toencode a good deal of knowledge; and the properties he saw as incompatible with consciousness are those inherent inde novo computation over a binary alphabet {0, 1}, and in modern Turing-level computation.

6(Bringsjord 1992) is a book-length defense of Weak AI, and a refutation of Strong AI.

2

specific “test-concretized” form of Weak AI has been dubbed Psychometric AI (Bringsjord &Schimanski 2003, Bringsjord 2011, Bringsjord & Licato 2012).

2.2 Central Aspects of Leibniz’s Dream

While Leibniz is without question the inventor of modern logic, the full details of his originalinventions in this regard aren’t particularly relevant to our purposes herein. What is relevant arefour aspects of this invention — aspects that we use as a stepping stone to create and present ouraforementioned “three-ray” conception.7 These four aspects are key parts of Leibniz’s dream of asystem for infallible reasoning in rational thought. Here, without further ado, is the quartet:

I. Universal Language (UL) for Rational Thought Starting with his dissertation, Leib-niz sought a “universal language” or “rational language” or “universal characteristic” inwhich to express all of science, and indeed all of rational thought. (We shall use ‘UL’to refer to this language.) Time and time again he struggled with particular alphabetsand grammars in order to render his dream of UL rigorous, and while the specifics arefascinating, he never succeeded — but modern logic is gradually moving closer and closerto his dream. An important point needs to be made about the universal language of whichLeibniz dreamed:

Pictographic Symbols & Constructs Allowed Leibniz knew that UL would have topermit pictographic symbols and constructs based on them. When most people thinkabout modern formal logic today, they think only of formulae that are symbolic andlinguistic in nature. This thinking reflects an ignorance of the fact that plenty of workin the rational sciences makes use of diagrams.8

II. Universal Calculus (UC) of Reasoning for Rational Thought Leibniz envisioned asystem for reasoning over content expressed in UL, and it is this system which, as Lewis(1960) explains, is the true “precursor of symbolic logic (p. 9).” Indeed, Lewis (1960)explains that there are seven “principles of [Leibniz’s] calculus” (= of — for us — UC),and some of them are strikingly modern. For instance, according to Principle 1, that“[W]hatever is concluded in terms of certain variable letters may be concluded in terms ofany other letters which satisfy the same conditions,” we are allowed to make deductive in-ferences by instantiating schemas, a technique at the very heart of Principia Mathematica,and still very much alive today.9

III. Distinction Between Extensional and Intensional Logic Part of Leibniz’s seminalwork in logic is in his taking account of not only ordinary objects in the extensional realm,but “possible” objects “in the region of ideas.” Leibniz even managed to make some of thekey distinctions that drive the modal logic of possibility and necessity. For example, he

7With apologies in advance for the pontification, Bringsjord maintains that there is really only one right routefor diving into Leibniz on formal logic: Start with the seminal (Lewis 1960), which provides a portal to the turningpoint in the history of logic that bears a fascinating connection to Leibniz (since Lewis took the first thoroughlysystematic move to intensional logic, anticipated by Leibniz). From there, move into and through that which Lewishimself mined, viz. (Leibniz & Gerhardt 1890). At this point, move to contemporary overviews of Leibniz on logic,and into direct sources, now much more accessible in translated forms than in Lewis’s day.

8Lewis (1960) asserts that the universal language, if true to Leibniz, would be ideographic pure and simple. Wedisagree. Ideograms can be pictograms, and that possibility is, as we note here, welcome — but Leibniz also explicitlywanted to allow UL to allow for traditional symbolic constructions, built out of non-ideographic symbols. We knowthis because of Leibniz’s seminal work in connection with giving us (the differential and integral) calculus, which isafter all routinely taught today using Leibniz’s symbolic notation (e.g., dx

dy, where y = f(x)).

9Axiomatic treatments of arithmetic, e.g., make use of this rule of inference, and then need only add modus ponensfor (first-order & finitary) proof-theoretic completeness.

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distinguished between things that don’t exist but could, versus things that don’t exist andcan’t possibly exist; and between those things that exist necessarily versus those thingsthat exist contingently. Clearly the influence of Leibniz on the thinker who gave us thefirst systems of modal logic, C.I. Lewis, was significant.10

IV. Welcoming Infinitary Objects and Reasoning Leibniz spent quite a bit time think-ing about infinitary objects and reasoning over them. This is of course a massive under-statement, in light of his invention of the calculus (see note 8; and note that

∫was also

given to us by Leibniz), which for him was based on the concept of an infinitesimal, noton the concept that serves today as a portal for those introduced to the differential andintegral calculus: the concept of a limit.11

To conclude this section, we point out that while Hobbes without question advanced the notionthat reasoning is computation (e.g., see Hobbes 1981), he had a very limited conception of reasoningas compared to Leibniz, and also had a relatively limited conception of computation as well.

3 The Leibnizian “Three-Ray” Conception of AI

Inspired by Leibniz’s vision of the “art of infallibility,” in the form of both UL and UC, a het-erogenous logic powerful enough to express and rigorize all of systematic human thought, we cannearly always position some particular AI work we are undertaking within a view of logic thatallows a particular logical system to be positioned relative to three color-coded dimensions, whichcorrespond to the three arrows shown in Figure 1. The blue ray corresponds to purely extensionallogical systems, the green the intensional logical systems, the orange to infinitary logical systems(and the red to diagrammatic logical systems). (In the interests of space, we leave aside the factthat each of the three main rays technically has a red sub-ray, which holds those logics that havepictographic elements, in addition to standard symbolic ones. We only show one red logic in Figure1.)

We have positioned our own logical system DCEC∗, which we use to formalize a good deal ofhuman communication and cognition (for a recent example, see e.g. Bringsjord, Govindarajulu,Ellis, McCarty & Licato 2014) within Figure 1; it’s location is indicated by the black dot therein,which the reader will note is quite far down the dimension of increasing expressivity that rangesfrom expressive extensional logics (e.g., FOL and SOL), to logics with intensional operators forknowledge, belief, and obligation (so-called philosophical logics; for an overview, see Goble 2001).Intensional operators like these are first-class elements of the language for DCEC∗. This language isshown in Figure 2. The reader will see that the language is at least generally aligned and inspiredby Leibniz’s UL and UC.

What does the three-ray framework described above imply about the theory and future of AI?Assuming that every AI system or AI agent can be placed within our three-ray conceptualization,which is directly inspired by Leibniz’s art of infallibility, every AI system or AI agent can be placedat a point along one of the three color-coded axes in this conceptualization. (Watson, for instance,has been placed on the blue axis or ray.)

10The systems of modal logic for which C.I. Lewis is rightly famous are given in the landmark (Lewis & Langford1932).

11An interesting way to see the ultimate consequences, for formal logic, of Leibniz’s infinitary reasoning in con-nection with infinitesimals and calculus, is to consider in some detail, from the perspective of formal logic, the“vindication” of Leibniz’s infinitesimal-based provided by Robinson (1996). Space limitations make the taking of thisway herein beyond scope.

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MiniMax

ularit

y

Logic

propo

stional

logic

semant

ic-web

logic

s

descr

iption l

ogics

fragm

ents

of FOL ...

temporal

temporal+epistemic

temporal+epistemic+deontic

Lw1,w

Art of Infallibility 1

UIMA outp

ut

+planning+arg semantics

epistemic

...

FOL

heterogeneous/visual

Infinitary (AoI 2)

SOL

...

DCEC ⇤Deontic Cognitive Event Calculus

(with Castañeda’s *)

1. natural language semantics (non-Montagovian)

2. higher-cognition tests (for Psychometric AI) (false-belief test, deliberative mind-reading mirror test for self-consciousness ...)

4. biz & econ simulation 3. ethically correct robots

Figure 1: Locating DCEC∗ in “Three-Ray” Leibnizian Universe

We can quickly note that contemporary AI is most a “blue” affair, with some “green” activity.As promised, we turn now to a framework that is more fine-grained than the three-ray one.

4 An “Engineeringish” Leibnizian Theory of AI

We shall be extremely lattitudinarian in our suppositions about the composition of a logicalsystem L , guided to a significant degree by the earlier framework presented in (Bringsjord 2008),and by the points I–IV we have just made about Leibniz’s work: Each such system will be aquintuple 〈F ,A,SA,SF ,M〉:

Logical System L

F A formal language defined by an alphabet and a grammar that generates a space of well-formed formulae.We here use ‘formula’ and ‘formulae’ in an aggressively ecumenical way that takes account of the foundingconceptions and dreams of Leibniz, so that formulas can for instance even be diagrammatic in nature.For example, in keeping with (Arkoudas & Bringsjord 2009), our relaxed concept of formula will allowdiagrammatic depictions of seating puzzles to count as formulae. Another example pointing to how widea scope we envisage for F is that any systematic kind of probabilistic or strength-factor parameters arepermitted to be included in, or attached to, formulae.

A An “argument theory” that regiments the concept of a structured, linked case in support of some formulaas conclusion. Notice that we don’t say ‘proof theory.’ Saying this would make the second element of alogical system much too narrow. We take proofs to be a special case of arguments.

SA A systematic scheme for assessing the validity of a given argument α in A.

SF A systematic scheme for assessing the semantic value of a given formula φ in F .

M Finally, a metatheory consisting of informative theorems, expressed and established in classical, deductive

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Syntax

S ::=Object | Agent | Self @ Agent | ActionType | Action v Event |Moment | Boolean | Fluent | Numeric

f ::=

action : Agent⇥ActionType ! Action

initially : Fluent ! Boolean

holds : Fluent⇥Moment ! Boolean

happens : Event⇥Moment ! Boolean

clipped : Moment⇥Fluent⇥Moment ! Boolean

initiates : Event⇥Fluent⇥Moment ! Boolean

terminates : Event⇥Fluent⇥Moment ! Boolean

prior : Moment⇥Moment ! Boolean

interval : Moment⇥Boolean

⇤ : Agent ! Self

payoff : Agent⇥ActionType⇥Moment ! Numeric

t ::= x : S | c : S | f (t1 , . . . , tn)

f ::=

t : Boolean | ¬f | f^y | f_y |P(a, t,f) | K(a, t,f) | C(t,f) | S(a,b, t,f) | S(a, t,f)

B(a, t,f) | D(a, t,holds( f , t0)) | I(a, t,happens(action(a⇤ ,a), t0))

O(a, t,f,happens(action(a⇤ ,a), t0))

Rules of Inference

C(t,P(a, t,f) ! K(a, t,f))[R1 ]

C(t,K(a, t,f) ! B(a, t,f))[R2 ]

C(t,f) t t1 . . . t tn

K(a1 , t1 , . . .K(an , tn ,f) . . .)[R3 ]

K(a, t,f)

f[R4 ]

C(t,K(a, t1 ,f1 ! f2)) ! K(a, t2 ,f1) ! K(a, t3 ,f2)[R5 ]

C(t,B(a, t1 ,f1 ! f2)) ! B(a, t2 ,f1) ! B(a, t3 ,f2)[R6 ]

C(t,C(t1 ,f1 ! f2)) ! C(t2 ,f1) ! C(t3 ,f2)[R7 ]

C(t,8x. f ! f[x 7! t])[R8 ]

C(t,f1 $ f2 ! ¬f2 ! ¬f1)[R9 ]

C(t, [f1 ^ . . .^fn ! f] ! [f1 ! . . . ! fn ! y])[R10 ]

B(a, t,f) f ! y

B(a, t,y)[R11a ]

B(a, t,f) B(a, t,y)

B(a, t,y^f)[R11b ]

S(s,h, t,f)

B(h, t,B(s, t,f))[R12 ]

I(a, t,happens(action(a⇤ ,a), t0))

P(a, t0 ,happens(action(a⇤ ,a), t0))[R13 ]

B(a, t,f) B(a, t,O(a⇤ , t,f,happens(action(a⇤ ,a), t0)))O(a, t,f,happens(action(a⇤ ,a), t0))

K(a, t,I(a⇤ , t,happens(action(a⇤ ,a), t0)))[R14 ]

f $ y

O(a, t,f,g) $ O(a, t,y,g)[R15 ]

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Figure 2: DCEC∗ Syntax and Rules of Inference

ways, regarding the other four members of L , their inter-relationships, and connections to other parts ofthe formal (theorem-driven) sciences.

There are well-known theorems in foundational computability theory that let us assert thefollowing general claim. In the case of computer programs written for Turing machines, the logicalsystem would be first-order logic.

Simulation in Logic

Given any computer program W that maps inputs from I to outputs in O, its processing can be captured bysearch for arguments in a logic system WL.

• Inputs would be be formulae from IF whose validity would have to be established.

• Outputs OF will be computed by filling in slots in the inputs if a successful argument has been found forthe input.

Given that we can simulate an AI system in a logic system, we could then evaluate the AI systemagainst Leibniz’s goal for a universal logic by looking for the minimal logic needed to simulate theAI system. There are several criteria for ranking logics, but we feel that they are not suited forevaluating AI systems from the standpoint of Leibniz’s dream. We propose the following Leibniziancriteria to evaluate logics when they are used to simulate AI. The criteria can be broken up intorepresentation and reasoning (in keeping with aspects I./UL and II./UC from §2.2). An earlierversion of these criteria can be found in (Govindarajulu, Bringsjord & Licato 2013). Very broadly,

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the representation criteria let us evaluate how good a system is at representing various mentaland worldly objects. The reasoning criteria let us evaluate how good the system is at representingvarious kinds of reasoning processes. We assign numeric scores for different criteria; the higher ascore, the more desirable it is.

4.1 Criteria for FWe propose the following criteria to establish how strong a representation system is.

4.1.1 Degree of Coverage

A representation system should be broad enough to cover all possible types of signs and structures.Pierce’s Theory of Signs provides a broad account of the types of signs that one could expect auniversal logic to possess. Briefly, an icon has a morphological mapping to the object it denotes.A symbol has no such mapping to the object it denotes.12 Mathematical logic used in AI haslargely focused on symbolic formulae. There has been some work in using iconic or visual logics(e.g., see the Vivid system introduced in Arkoudas & Bringsjord 2009).) A large part of humanreasoning would be difficult to reduce to a purely symbolic form. In (Govindarajalulu, Bringsjord& Taylor 2014), we show some examples of visual/geometric reasoning used to prove theorems inspecial relativity theory.

Table 1: Degree of Coverage

Coverage

Score Interpretation Example

1 Only symbolic Propositional logic, probability calculus, etc.1 Only iconic Image processing2 Both iconic and symbolic Vivid

4.1.2 Degree of Quantification/Size

This captures how many individual objects and concepts a formula can capture. Terms referring tosets or classes of objects would be counted as referring to only a single object. At the lowest levelin this scale, we have simple propositional systems. At the highest levels, we have infinitary logicscapable of expressing infinitely long statements. Such statements would be needed to formallycapture some statements in mathematics which cannot be expressed in a finite fashion (Barwise1980).13

12The distinction is not always perfect. E.g., {} for the empty set is both iconic and symbolic.13Infinitary logics are “measured” in terms of length not only of formulae, but e.g. number of quantifiers permitted

in formulae. In the present paper, we leave aside the specifics. But we do point out to the motivated reader thatFigure 1, on the infinitary “ray,” refers to the infinitary logic Lω1ω, which, in keeping with the standard notation,says that disjunctions/conjunctions can be of a countably infinite length, whereas the number of quantifiers allowedin given formulae must be finite.

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Table 2: Degree of Quantification/Size

Quantification/Size

Score Interpretation Example

0 Propositional SAT solvers0.5 First-order with limited quantification OWL (Baader, Calvanese & McGuinness 2007)1 First-order Mizar (Naumowicz & Kornilowicz 2009),2 Second-order, higher-order logics, Reverse mathematics project (Simpson 2009)3 Infinitary logics etc ω-rule (Baker, Ireland & Smaill 1992)

4.1.3 Degree of Homoiconicity

Ideally, the representation system should be able to represent its own formulae and structures initself easily. The term “homoiconicity” is usually used to refer to the Lisp class of languages whichare capable of natively representing, to various degrees, their own programs. First-order logic doesnot have this capability natively, but one can achieve this by using schemes like Godel numbering.In (Bringsjord & Govindarajulu 2012), we go through several such schemes for first-order logic.

The earliest such scheme, due to Godel, assigns a natural number nφ for every formula φ. Thenevery such number nφ is assigned a term nφ in the language. Thus one could write down sentencesin the fashion χ(nφ), in which we assert χ about φ. A more recent scheme termed reification assignsterms in a first-order language Lmain to formulae and terms in another first-order language Lobj .The event calculus (nicely covered and put to AI-use in Mueller 2006) formalism in AI employsthis scheme. In this scheme, states of the world are formulae in Lobj and are called fluents. Forexample, formulae in Lmain stating that the sentence raining in Lobj never holds would be:

¬∀t : holds(raining , t)

Ideally, we should be able to use F to state things not just about objects in F , but aboutSA, SF , and L itself in general. For example, first-order logic cannot directly represent first-orderproofs, but first-order denotational proof languages can do so (Arkoudas 2000).

Table 3: Degree of Homoiconicity

Homoiconicity

Score Interpretation Example

0 No native capability First-order logic1 Can talk about formulae First-order modal logic (Bringsjord & Govindarajulu 2013)1 Can talk about arguments First-order denotational proof languages2 Can talk about arguments and formulae

4.1.4 Degree of Possibility

Can the formulae represent alternate states-of-affairs or yet unrealized states-of-affairs? The systemshould be expressive enough to represent uncertain information, information that is false but couldbe true under alternate states of affairs, information that could hold in the future, etc.; notice

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the connection here to our point III about Leibniz’s art of infallibility. Note that by includingpossibility, we also include probablility and uncertainty.

Table 4: Degree of Possibility

Possibility

Score Interpretation Example

0 No possibility Propositional logic1 One mode of possibility Markov logic (Richardson & Domingos 2006)2 Multiple modes of possibility First-order logic (Halpern 1990)

4.1.5 Degree of Intensionality

How good is the system at representing other agents’ mental states? The system should be ableto represent and reason over knowledge, belief, and other intensionalities of other agents and itself.While one could refurbish a purely extensional system to represent intensionalities by various“squeezing” schemes, such squeezing leads to problems. In (Bringsjord & Govindarajulu 2012),we look at various schemes for squeezing knowledge into first-order logic and show how each ofthe schemes is either incoherent or leads to outright contradictions. These results suggest that auniversal logic should have the facility to directly represent intensionalities. What would such ascheme look like? A complex intensional statement and its possible representation in a semi-formalfirst-order modal notation is shown in Figure 3.

“Jones intends to convince Smith to believe that Jones believes that were the cat, lying in the foyer now, to be let out, it would settle, dozing, on the mat.”

subjunctive conditional

intensional operators

scoped term

I(j,C(s,B(s,B(j,◆[c : in(c, ◆(f : Foyer(f)), m : mat(m)]

out(c) !subj doze(c, m))))

Figure 3: Representing Intensionality

Table 5: Degree of Intensionality

Intensionality

Score Interpretation Example

0 No intensionality First-order logic1 Restricted intensionality Modal logic with no iterated beliefs2 Full intensionality Unlimited iterated beliefs (Bringsjord & Govindarajulu 2013)

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4.2 Criteria for SA and SFSA and SF capture reasoning in L. We would want them to be as general as possible. Formeasuring generality, we borrow a yardstick from computability theory. Since SA and SF arecomputer programs, we can measure how general they are by looking at where they are placedin the computability heirarchy. In this heirarchy, the lower levels would be populated by thecomputable schemes. Within the computable class, we would have the programs ordered by theirasymptotic complexity. After the computable class, we would have the semi-computable class ofprograms. The justification is that the higher a scheme is placed, the greater the number of kindsof reasoning processes it would be able to simulate.

Table 6: Criteria SA and SFSA and SFScore Interpretation Example

0 Computable and tractable Nearest neighbors1 Computable and intractable Bayesian networks (Chickering 1996)2 Uncomputable First-order theoremhood (Boolos, Burgess & Jeffrey 2007)

4.3 Criteria for MLogical systems are accompanied by a slew of meta-theorems. These frequently are soundness andcompleteness results, to start. Logical systems usually have other meta-theorems that might not be,at least on the surface, that useful for evaluating AI systems; for example, the compactness theoremfor first-order logic. We require that the logic system contain, at the least, two theorems we termthe boundary theorems. Boundary theorems correspond roughly to soundness and completeness.

Table 7: Criteria M

Boundary Theorems

Score Interpretation Example

0 No boundary theorems Adhoc learning systems1 Soundness or Completeness AIXI no notion of soundness (Hutter 2005)2 Soundness and Completeness First-order logic (Boolos et al. 2007)

5 Watson Abstractly

We now look at a highly abstract model of how Watson works.14 The abstract model is what anideal-observer logicist would see when examining Watson. Looking at Watson’s initial stages inFigure 4, we see that for a given question Q, multiple answers {A1,A2, . . . ,An} are generated. Each

14A more concrete overview of Watson can be found in (Ferrucci, Brown, Chu-Carroll, Fan, Gondek, Kalyanpur,Lally, Murdock, Nyberg, Prager, Schlaefer & Welty 2010). While (Ferrucci & Murdock 2012) delves much more intoWatson, for the material in this section one needs to primarily refer to (Gondek, Lally, Kalyanpur, Murdock, Dubou,Zhang, Pan, Qiu & Welty 2012).

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of these answers is paired with multiple pieces of data, the evidence, resulting in 〈eAi1 , . . . , e

Aim 〉.

Each such tuple 〈Q;Ai; eAi1 , . . . , e

Aim 〉 is then fed through answer and evidence scorers, resulting in a

feature vector v〈Q,Ai〉 for each 〈Q,Ai〉 pair.Each feature vector is then passed through multiple phases as shown in Figure 5. Within

each phase, there are classifiers for different broad categories of questions (e.g., Date, Number,Multiple Choice etc). Any given v〈Q,Ai〉 passes through at most one classifier in a phase. Thephases essentially serve to reduce the vast number of answers, so that a final ranking can focusonly on an elite set of answers. Each classifier performs logistic regression on the input, as shownin Figure 6. The classifier maps the feature vector into a confidence score c ∈ [0, 1].

While the details of the phases are not important for our discussion, we quickly sketch anabstract version of the phases. For any given question Q, all the v〈Q,Ai〉 are fed into one classifierin an initial filtering phase. This reduces the set of answers from n to a much smaller n′, the topn′ based on scores from the classifer. Then in a normalization phase, the feature values are firstnormalized with respect to the n′ instances. The answers are then re-ranked by one classifier inthis phase. The third transfer phase has specialized classifers to account for rare categories (e.g.Etymology). The final elite phase then outputs a final ranking of a smaller set of answers from theprevious phase.

Question

Q

A1

A2

An

answers

evidence answer scorers

evidence sco

rers

feature vector

...

...

eAij

vhQ,Aii

Figure 4: Initial Stages Watson Pipeline

5.1 Locating Watson

We confess that there is some “squeezing” that must be done in order to place Waston within thethree-ray space. We accomplish this squeezing by again invoking an “ideal-observer” perspectivewhen looking at Watson. This perspective is generated by imagining that an ideal observer, pos-sessed of great intellectual powers, and able to observe the entire pipeline that takes a question Qand returns an answer A, offers a classical argument in support of A being the correct answer. Ourideal observer is well-versed in all manner of statistical inference and machine learning, and has athis command not only what happens internally when Q as input yields A as output, but also fullycomprehends all the prior processing that went into the tuning of the pipeline that in the case athand gave A from Q. From this perspective, we obtain the following:

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C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

inst

ance

phase 1 phase 2 phase 3 phase 4

routes routesroutes routes

filtering transfer elite

a

Figure 5: Final Stages

Watson

Ray Criterion Score Reason

Ray 1

Coverage 1 Only symbolic at present.Quantification 0.5 First-order (due to the presence of a Prolog-based parser)Homoiconicity 1 First-orderPossibility 1 Support for probabilityIntensionality 0 No support for reasoning about intensionalities

Ray 2 SA and SF 1 Sub-first-order reasoning

Ray 3 Boundary Theorems 0 No formal study of Watson yet

6 The Future of AI

We end with some brief remarks about the future of AI in light of the foregoing.These days there’s much talk about the so-called “Singularity,” the point in time at which

AI systems, having reached the level of human intelligence, create AI+ systems that exceed thislevel of intelligence — which then leads to an explosion in which AI+ systems build even smartersystems, and the process (AI++, AI+++, . . .) iterates, leaving the poor humans in the dust.15 Givenour Leibnizian analysis above, anything like the Singularity looks, well, a bit . . . unreasonablysanguine. In terms of the above three-ray space, the reason is that, one, AI is currently definedas a “blue” enterprise, with just a tinge of “green” now in play. Can machines in the blue rangecreate machines that range across the green, red, and orange axes? No. And this negative is amatter of mathematical fact. For just as no finite-state automaton can create a full-fledged Turingmachine, no blue logic can create, say, an orange one. So humans must carry AI from blue intothe the other colors. And the fact is, there just is nothing out there about how that lifting is going

15The primogenitor of the case for this series of events is (Good 1965). A modern defense of the this original case isprovided by Chalmers (2010). A formal refutation of the original and modernized argument is supplied by Bringsjord(2012). An explanation of why belief in the Singularity is fideistic is provided by Bringsjord, Bringsjord & Bello(2013).

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Q A1

instance

C [0, 1]classifier

boosting, single and multilayer neural nets, decision trees,and locally weighted learning [11]; however, we haveconsistently found better performance using regularizedlogistic regression, which is the technique used in the finalWatson system. Logistic regression produces a scorebetween 0 and 1 according to the formula

f !x" # 1

1$ e%!0%

PMm#1

!mxm

;

where m ranges over the M features for instance x and !0 isthe Bintercept[ or Bbias[ term. (For details, the interestedreader is referred to [19].)An instance x is a vector of numerical feature values,

corresponding to one single occurrence of whatever thelogistic regression is intended to classify. Output f !x" may beused like a probability, and learned parameters !m may beinterpreted as Bweights[ gauging the contribution of eachfeature. For example, a logistic regression to classify carrotsas edible or inedible would have one instance per carrot,and each instance would list numerical features such as thethickness and age of that carrot. The training data consistsof many such instances along with labels indicating thecorrect f !x" value for each (e.g., 1 for edible carrots and 0 forinedible carrots). The learning system computes the model(the ! vector) that provides the best fit between f !x" andthe labels in the training data. That model is then usedon test data to classify instances. In Watson’s case, instancescorrespond to individual question/candidate-answer pairs,and the numerical values for the instance vector arefeatures computed by Watson’s candidate generation andanswer-scoring components. The labels on the training dataencode whether the answer is correct or incorrect. Thus,Watson learns the values for the ! vector that best distinguishcorrect answers from incorrect answers in the training data.Those ! values are then used on test data (such as a liveJeopardy! game) to compute f !x", our confidence that eachcandidate answer is correct. Candidate answers with higherconfidence scores are ranked ahead of candidate answerswith lower confidence scores.Logistic regression benefits from having no tuning

parameters beyond the regularization coefficient andoptimization parameters to set and therefore was robustwithout requiring parameter fitting in every experimental run.Answers must be both ranked and given confidence scores;thus, the framework does allow for separate models to betrained to perform ranking and confidence estimation. Inpractice, however, we found competitive performance usinga single logistic regression model where ranking was doneby sorting on the confidence scores.Ranking with classifiers is performed in the Classifier

Application step of each phase. It uses a model producedin the Classifier Training step for that phase.

StandardizationOne aspect of ranking that can be lost when converting aranking task to a classification task is the relative values offeatures within a set of instances to be ranked. It can beuseful to look at how the value of the feature for one instancecompares with the value of the same feature for otherinstances in the same class. For example, consider thefeatures that indicate whether the candidate answer is aninstance of the LAT that the question is asking for [12].In cases where a large proportion of the candidate answersthat the system has generated appear to have the desired type,the fact that any one of them does so should give the systemrelatively little confidence that the answer is correct. Incontrast, if our search and candidate generation strategieswere only able to find one answer of that appears to havethe desired type, that fact is very strong evidence thatthe answer is correct.To capture this signal in a system that ranks answers using

classification, we have augmented the existing features with aset of standardized features. Standardization is per query,where for each feature j and candidate answer i in theset Q of all candidate answers for a single question, thestandardized feature xstdij is calculated from feature xijby centering (subtracting the mean ") and scaling to unitvariance (dividing by the standard deviation #) over allcandidate answers for that question

xstdij #xij % "j#j

; "j #1

jQjXjQj

k#1xkj;

#j #

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1jQjXjQj

k#1!xkj % "j"2

vuut :

For each feature provided as input to the ranking component,we generate an additional new feature that is the standardizedversion of that feature. Both mean and standard deviationare computed within a single question since the purposeof standardization is to provide a contrast with the othercandidates that the answer is being ranked against. Wechose to retain the base features and augment with thestandardized features as that approach showed betterempirical performance than using standardized featuresor base features alone.Standardization is performed during the postprocessing

step of several different phases. It is performed in the HitlistNormalization phase because that is the first phase so nostandardized features exist. It is also performed in the Baseand Elite phases because some answers are removed atthe start of those phases; hence, the revised set of answerscan have different mean and standard deviation. Furthermore,it is performed in the Answer Merging and EvidenceDiffusion phases because the Feature Merging step for thesephases makes substantial changes to the feature values ofthe instances. It is not performed in other phases because

D. C. GONDEK ET AL. 14 : 5IBM J. RES. & DEV. VOL. 56 NO. 3/4 PAPER 14 MAY/JULY 2012

x f(x)

from training �i

Figure 6: Assigning Confidence Scores

to work, if it indeed can. On the other hand, Watson tells us something that may provide somebasis for the kind of optimism that “true believers” radiate these days. The something to whichwe allude is simply the stark fact that although the scorecard earned above by Watson, within thelarger Leibnizian context we have presented, shows that it’s fundamentally severely limited, thefact is that it’s behavior in rigidly controlled environments is surprisingly impressive. It remainsto be seen is how much of what humans do for a living would be replaceable by such mechanicalbehavior.

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