Conditionals and Pseudo-Conditionals in Mathematical Texts Claus Zinn School of Informatics University of Edinburgh zinn@ inf. ed. ac. uk

Abstract Conditionals are the basic ingredient of a mathematical argument. We present an analysis of conditionals and pseudo-conditional in mathematical discourse, and then propose a computational framework for the (re-)construction and semantic representation of conditional dependencies.

1

Motivation

Informal, non-trivial mathematical discourse exhibits a complex net of conditional dependencies. A deep understanding of such discourse requires the disclosure of its logical structure. In stark contrast to the expression of a conditional in first order logic, mathematical writing has a large variety of linguistic constructions to express the logical consequence relation between statements. Constructions of the form “If . . ., then . . .” are complemented by many other (inter-sentential and intra-sentential) forms, some of which are expressed without the use of cue words. Moreover, statements that share the surface form of conditionals can be used for the expression of non-logical discourse relations. Unfortunately, the interpretation of conditionals in mathematical texts can exhibit a surprising degree of ambiguity. It is thus a complex if not impossible matter to distinguish the assumed parts from the derived statements of a mathematical argument on purely linguistic grounds. The complexity of processing conditional statements has already been recognised elsewhere. Kamp & Reyle note: “[Consequently,] no simple algorithm exists for deciding where the hypothetically asserted part ends and the text continues with assertions that are made categorically (i.e., not conditionally upon the assumptions expressed by the suppositional part).” As they point out, this question is subordinate to the general problem: “how the language user perceives the logical and rhetorical structure of a discourse or text is still poorly understood, and very far from being solved” [16, p. 145].

Our research work focuses on the development and implementation of a computational framework that uncovers, constructs and represents the logical structure of mathematical discourse. In this article, we will focus on conditionals, the basic ingredient of argument structures in mathematics. In the first part, we analyse four kinds of conditionals: classical conditionals that introduce assumptions and draw conclusions from them, categorical conditionals that express that a non-hypothetical statement implies another statement, counterfactuals, which are statements of the form “If A were the case, then B would be the case”, where A is supposed to be false, and pseudo-conditionals that share the surface form of classical conditionals but differ in their semantic and pragmatic function in yet another way. In the second part, we propose a framework that allows the computation of the logical structure of a mathematical discourse. It complements linguistic information with semantic (mathematical) and pragmatic (meta-mathematical) expertise.

2

Background

The literature on the use of connectives in general, and the use of conditionals in particular is vast. The English “or”, for instance, is treated by Zimmermann who finds that “or” is better modeled as marking an alternative than as expressing a logical disjunction [25]. Webber et al. argue for the position that “while adjacency and explicit conjunction (coordinating conjunctions such as “and”, “or”, and “but”; subordinating conjunction such as “although”, “whereas”, “when”, etc.) imply discourse relations between (the interpretation of) adjacent or conjoined discourse units, discourse adverbials such as “then”, “otherwise”, “nevertheless” and “instead” are anaphors, signalling a relation between the interpretation of their matrix clause and an entity in or derived from the discourse context” [24]. Statements of the form “If ..., then ...” have also been thoroughly discussed. The Ifs compendium [12] provides a good starting point and includes, for instance, Stalnaker’s and Lewis’ classic papers promoting a possible worlds approach to conditionals. Moreover, there are, among many others, a collection of papers on conditionals in the Journal of Philosophical Logic introduced by Donald Nute [19], the monographs of Adams and Jackson on conditionals [1,15] and, to mention a critical account on both Stalnaker’s and Lewis’ approach, Gabbay’s treatment of (subjunctive) conditionals using a ternary operation [10]. In [6], de Swart and Nederpelt give an interesting comparison between material implication, strict implication, relevant implication, counterfactuals, and intuitionistic implication. They argue that material implication, “although adequate for mathematics, is deficient in more than one respect” [6, p. 77]. For many uses, it is insufficient to give an interpretation of “If . . ., then . . . .” statements in terms of truth conditions. Often, Grice’s pragmatic principles governing discourse have to be taken into account as well. For the indicative conditional, de Swart and Nederpelt state [6, p. 82]: 2

“indicative conditional = material implication + conversational rules”, where such rules capture the concepts of necessity and relevance. In this paper, we argue that even mathematical textbook proofs may contain occurrences of “If . . ., then . . .” statements that cannot be captured by material implication.

3

Analysis of Conditionals in Mathematical Texts

We start with an introductory example that shows the necessity and complexity of a proper and fine-grained analysis of mathematical discourse. 3.1 Introductory Example Consider the mathematical argument (1) that proves Theorem 3 of Hardy & Wright’s textbook 1 [11, p. 21]. Theorem 3 (Euclid’s first theorem). If p is prime, and p | ab, then

p | a or p | b.

(1)

The theorem statement is followed by three proof sentences that build a quite complex net of argument interdependencies, in fact, a large multi-sentential conditional structure. A disclosure of its logical structure will need to include the identification of the quantification and scope of the variables it contains, its assumptions and their scope as well as the conclusions and the prior statement they depend on. Without the understanding of its conditional statements, none of these tasks can be achieved, as we shall see now. The theorem is an intra-sentential conditional of the classical if-then surface form. Its reading is given to the right-hand side of the initial proof state: ∆ ∪ {} ⊢ ∀p∈N ∀a∈N ∀b∈N . prime(p) ∧ p | ab → p | a ∨ p | b. Each of the theorem’s variables is universally quantified. Its proof will show that the conclusion p | a∨p | b solely depends on the premise prime(p)∧p | ab, given an underlying mathematical theory ∆ and an empty set of hypotheses. For the subsequent discussion, theory ∆ shall include at least these five lemmata: ∀a∈N ∀b∈N ∀c∈N . gcd(a, b) = c → ∃x∈Z ∃y∈Z .xa + yb = c ∀a∈N ∀b∈N . prime(a) ∧ ¬a|b → gcd(a, b) = 1 ∀a∈N ∀b∈N ∀d∈N . d|a → d|ab ∀a∈N ∀b∈N ∀d∈N ∀x∈N ∀y∈N . d|a ∧ d|b → d|xa + yb ∀a∈N ∀b∈N ∀x∈N . x|a ∧ a = b → x|b 1

All examples were taken from Hardy & Wright’s textbook on number theory [11].

3

(1) (2) (3) (4) (5)

The first important fact that a proof reader must realise is that the scope of the quantifiers for p, a and b does not extend beyond the theorem statement. However, the use of the names a, b and p in the theorem fore-shadows their respective use as free variables in the proof. 2 The first proof sentence is the beginning of an inter-sentential conditional. Its cue suppose must get wide scope, and thus, a complex assumption is added to the discourse (i.e., the proof state). Without the use of domain expertise, it is unclear whether the scope of the hypothetically asserted statement extends, say, beyond the next proof step. The first part of the second sentence shares the surface form of the theorem statement. Meta-mathematical knowledge, however, informs us that its semantic and pragmatic function is different. First, the symbols p and a that the conditional uses are not universally quantified but free variables. Second, the if-then statement does not assert a statement of the form A → B but adds an assumption A (p 6| a) to the argument’s hypotheses, and third, then concludes that B ((a, p) = 1) follows from the extended set of hypotheses. The textual proof so far can be summarised in more formal terms. The inference rule ∀-intro is applied to each of the quantifiers. Then, a proof by elimination method is performed. 3 Subsequently, the resulting conjunction prime(p) ∧ p | ab is split into its constituents. With these reasoning steps, we obtain a proof state that describes the underlying logical structure of the first sentence as well as the beginning of the second sentence: ∆ ∪ {p∈N , a∈N , b∈N , prime(p), p | ab, ¬p | a} ⊢ p | b. Then, a directed line of forward reasoning follows. Applying Lemma 2 to two hypotheses yields: {p∈N , a∈N , b∈N , prime(p), p|ab, ¬p|a, gcd(a, p) = 1} ⊢ p|b. The text continues with assertions that are made categorically: Once it is shown that the greatest common divisor of a and p equals 1, Theorem 24, encoded as Lemma 1, can be used to derive the existential statement that involves the two integers x and y. The formula gcd(a, p) = 1 can thus be rewritten as {p∈N , a∈N , b∈N , prime(p), p|ab, ¬p|a, ∃x∈Z ∃y∈Z xa + yp = 1} ⊢ p|b. Apparently, the textbook proof fragment xa+ yp = 1 or xab+ ypb = b does not constitute a disjunction, but an inference step of an algebraic nature (both sides of the first equation are multiplied by b to yield the second equation). For this reasoning step none of the prior assumptions is used. However, as 2

Hardy & Wright, following common practise in mathematical writing, omit to start their proof with a statement equivalent to “Now, let p, a and b be arbitrary natural numbers”. 3 To prove a statement of the form A → B ∨ C, assume A, assume ¬B, and show that C.

4

a prerequisite to this step, the specialisation method needs to be applied to each of the existential quantifiers: {p∈N , a∈N , b∈N , prime(p), p|ab, ¬p|a, X∈Z , Y∈Z , Xa + Y p = 1} ⊢ p|b. Following the informal proof, we multiply both sides of the equality by b: {p∈N , a∈N , b∈N , prime(p), p|ab, ¬p|a, X∈Z , Y∈Z , Xab + Y pb = b} ⊢ p|b. In the last sentence, the proof authors refer to the assumption p|ab, which suggests to the reader it is used to conclude p|b. Since this terminates the proof, the reader may infer that this is the last obligation that was to be shown. Formally, Lemma 3 is used twice: use p|ab to conclude p|xab, and use p|pb to conclude p|ypb. Lemma 4 is applied once, and then Lemma 5 (encoding a restricted form of equality) is applied to finish the proof. 3.2 A Brief Categorisation of Conditionals After this introductory analysis of a textbook proof, we now give a brief, more systematic classification of conditionals and pseudo-conditionals. 3.2.1 Classical Conditionals Intra-sentential conditionals of the form “If A, then B”, where the propositional content of B only depends on the propositional content of A, are exceptional. The following three examples are such classical conditionals or surface form variations thereof. (2) a. If 2n+1 − 1 is prime, then 2n (2n+1 −
1)
is perfect. ′ ′ p q b. If p and q are odd primes, then q p = (−1)p q , where p′ = 21 (p − 1), q ′ = 21 (q − 1).

c. If p and q are odd primes, then pq = pq , unless both p and q are of

the form 4n + 3, in which case pq = − pq . d. The square of an odd number 2m + 1 is 4m(m + 1) + 1. Given a mathematical theory, these sentences can be interpreted on the sentence level. Statement 2(a) is of the classical surface form “If A, then B”. Statement 2(b) is of the surface form “If A, then B, where C” while it reads as “If A and C, then B”. Sentence 2(c) has the surface form “If A, then B, unless C, in which case D”; logically it reads as the conjunction of two conditionals “If A and not C, then B” and “If C, then D”. As sentence 2(d) shows, a conditional can be expressed without cue words. However, the embedding of an indefinite noun phrase in a functional expression indicates a generic reading. 3.2.2 Categorical Conditionals Categorical conditionals constitute a relation between two statements of a different nature. They assert that A logically entails B, and they presuppose 5

that A has already been shown to hold in prior discourse or established valid in a theory. Typical surface forms include “A implies B” and “Since A, [we deduce that] B”. (3) a. Theorem 28 implies Theorem 29. b. Since log log x ≤ n, we deduce that π(x) ≥ log log x. Variations to 3(a)–3(b) include “A implies B because of C” and “A since B implies C”. The forms “Therefore B” or “Hence B” can be regarded as elliptic categorical conditionals. They do not explicitly verbalise or symbolise the statements that B depends on, either because the statements were mentioned earlier in the proof and are considered salient, or because the proof author considers them self-evident or trivial. 3.2.3 Counterfactuals A counterfactual is an expression of the form: “If A were the case, then B would be the case”, where A is supposed to be false. 4 Read these examples: (4) a. It will now be obvious why 1 should not be counted as a prime. If it were, Theorem 2 would be false, since we could insert any number of unit factors. b. If p were a prime of k(i), it would divide x + i or x − i, and this is false, since the numbers x i ± p p are not integers. Hence p is not a prime. In the first example, the premise “1 is prime” would invalidate a theorem, and thus, must be false. In the second example, the premise “p is a prime of k(i)” must be false as well, since assuming its truth, for the sake of argument, yields a contradiction. It is common practise to use counterfactuals within proof by contradiction, or reductio ad absurdum arguments. 3.2.4 Pseudo-Conditionals Pseudo-conditional statements share the syntactic form of classical conditionals but have a different semantic and pragmatic function. (5) a. If c is the remainder when n is divided by d and n = zd + c, then c ∈ S and 0 ≤ c < d. b. If pn is the nth prime then π(pn ) = n. c. If m is the least of these divisors, then m is prime. Example 5(a) is clearly a pseudo-conditional: instead of introducing an assumption, it introduces the name c for an object that is created by an algebraic operation (dividing n by d). Sentence 5(b) shows another pseudo-conditional. 4

As de Swart and Nederpelt point out, a counterfactual is a subjunctive conditional, but not all subjunctive conditionals are counterfactuals [6].

6

Here, the nth. prime number is given the shorter reference name pn . There are two interpretations for 5(c): In the first interpretation, it is assumed that m co-refers with “the least to these divisors” to the same object. In the second interpretation, the object that is being referred to by “the least of these divisors” is given the second, shorter reference m. This new reference is then used in the conclusion “m is prime”. Without the introduction of the name m, we could say “The least of these divisors is prime.” The choice among the two interpretations depends on whether the occurrence of variable m is old (prefer first reading) or new (second reading). Summary. The conditional has various uses in mathematical discourse. Its classical form states a generic statement of the form A → B (theorem contexts). A subtle but important difference is the conditional used in proof contexts (cf. example § 3.1). Here, A is added to the list of hypotheses, and then B follows from this extended list. A third use is in the context of proof by contraction or reductio ad absurdum arguments. A fourth use is in the context of algebraic manipulations, where A describes an operation and B its result. Moreover, a conditional’s sole purpose can be to introduce names, although naming can also be the byproduct of the other uses.

4

A Computational Framework

We now propose a computational framework for the interpretation of conditionals. It combines the proof planning technology from the field of automated reasoning with an extension of discourse representation theory from semantics. 4.1 Mathematical Reasoning: Proof Planning Proof planning is an enabling technology for the construction and representation of high-level proofs. To capture and operationalise informal mathematical reasoning, it requires the analysis of families of related proofs, the identification of common patterns in them, the representation of these pattern as proof plan schemata, and the use of these schemata to guide future proofs from the same family. Instead of generating an inference level proof from scratch, a global proof outline or proof plan of the conjecture is developed first. This is then used to guide the construction of a detailed proof, thus enabling us to fill in the details that mathematicians find trivial to write down [4]. Using proof planning it has been possible to automate more of the proof discovery process than is usually possible. Proof planning has been implemented in the systems Clam [5], λ-Clam [20] and Ωmega[3]. 4.2 Representation of Discourse: from DRSs to PRSs Discourse Representation Theory (DRT) provides a systematic approach to representing and updating discourse context [16,22]. However, in its pure form, it seems inadequate for the representation of mathematical discourse. 7

4.2.1 Inadequacy of DRT for Mathematical Discourse DRT’s rule set and language lacks coverage and expressiveness for the representation of mathematical discourse. Although Kamp & Reyle acknowledge the variety of expressions for conditionals in general forms of discourse, they only give the DRT construction rule CR.COND, depicted as (6), for such statements [16, p. 156]. 5 CR.COND Triggering configuration γ ∈ ConK :

(6)

Replace γ by:

mm S ESSS mm|m||| EEESESSSSSS m m m SSSS EE mmm || EE SSSS mmm 1 }|| E 1 m SS) m E m 1

" m vm

111

11 [{“,”} then] [if]

s s 11 21



1

111

s1 1

1

111

s2 1

→

While the rule CR.COND covers sentence 2(a), it fails on the examples 2(b)– 2(d). In addition, CR.COND is also triggered by all of the pseudo-conditionals 5(a)–5(c). Moreover, standard DRT does not provide propositional discourse entities, a representational means which we will need to compute and formulate the consequence relation between conditionals. Propositional discourse entities will be necessary to properly handle anaphora that refer to statements (e.g., “the theorem”, “the induction hypothesis”, “the first part of Theorem 20”), a set of statements (e.g., “Therefore A”, “It follows that A”) or larger portions of the text (e.g., “the argument”, “the first alternative”). Moreover, mathematical discourse is a highly structured form of discourse (e.g., in a proof by cases, the initiating assumption of the first case cannot be used for the derivation of statements in the second case), and its adequate representation requires a language that allows the expression of such structures. As a result, we have extended discourse representation structures to proof representation structures, which we will discuss next. 4.2.2 Proof Representation Structures (PRSs) Fig. 1 shows the first few PRS lines for Hardy & Wright’s textbook proof of Theorem 3, (cf. discourse 1 on page 2). A PRS line is a quintuple [α, β, γ, µ, ν] such that α is the proof line’s number, β is a discourse marker of the set {theorem, ind base case, ind hyp, . . .}, γ is a modality of the set {let, assp, then, goal}, µ is its content, and ν is its justification. The accessibility of discourse referents and conditions is imposed by a numbering scheme that is defined over a partially ordered set of abstract discourse referents. For example, take the condition labelled 1.5.6, which contains the referents a and p. Their quantification is determined by the closest accessible let construc5

However, there are accounts that treat modal subordination [21,9].

8

— — — thm 1 1.1 1.2 1.3 1.4 1.5 1.5.1 1.5.2 1.5.3 1.5.4 1.5.5 1.5.6 1.5.7 ...

— — — thm(3) — — — — — — — — — — — — — ...

let let let goal — let let let goal — assp assp goal then then then then ...

p ∈ N be universally quantified a ∈ N be universally quantified b ∈ N be universally quantified prime(p) ∧ div(p, times(a, b)) → div(p, a) ∨ div(p, b) — p ∈ N be arbitrary a ∈ N be arbitrary b ∈ N be arbitrary prime(p) ∧ div(p, times(a, b)) → div(p, a) ∨ div(p, b) — prime(p) ∧ div(p, times(a, b)) ¬div(p, a) div(p, b) not divides(p, a) prime(p) divides(p, times(a, b)) equal(gcd(a, p), 1) ...

— — — — (choose/3, thm(3)) — — — — (elim 1, 1.4) — — — (notation, 1.5.2) (and, 1.5.1) (and, 1.5.1) (lem, gcd, 1.5.5, 1.5.4) ...

Fig. 1. The first few lines of a PRS for Theorem 3

tions that introduce these names: p gets its binding from PRS line 1.1, a from PRS line 1.2 and b from PRS line 1.3. For justifying the propositional part of 1.5.6, all formulae of all prior PRS lines, except the PRS header, are accessible. Discourse markers provide a non-numeric way to refer to parts of the PRS. For example, the marker thm, which indices a line in the PRS header, is used to resolve references of the form “the theorem”. 4.3 A Discourse Update Engine for the Construction of PRSs The construction of semantic representations on the sentence level is based upon an adapted version of discourse representation theory. The result of the linguistic analysis (performed by a DCG-based parser) is an intermediate and underspecified semantic representation. Each intermediate representation that is passed to the discourse update engine is then interpreted and incorporated into a proof representation structure that captures the current proof context. The discourse update engine is supported by a proof planner that is informed by two sources: a set of proof plan schemata that captures meta-mathematical expertise, and knowledge specific to the domain of number theory. Fig. 2 depicts a high-level view of the discourse update algorithm. The alprs_update(+TRS, +PRS, +DRS, -NewPRS) :add_drs_to_prs(TRS, PRS, DRS, NewPRS). prs_update(+TRS, +PRS, +DRS, -NewPRS) :accommodate_proof_context(TRS, PRS, PRS1), prs_update(TRS, PRS1, DRS, NewPRS). accommodate_proof_context(+TRS, +PRS, -NewPRS) :do_backward_reasoning(TRS, PRS, NewPRS) ; do_forward_reasoning(TRS, PRS, NewPRS).

Fig. 2. The Discourse Update Algorithm

gorithm’s main predicate is prs update/4; it has three input parameters, pre9

fixed with “+”, and one output parameter, prefixed with “–”. Two cases are covered. The first clause of prs update/4 succeeds if its subgoal add drs to prs/4 succeeds; otherwise the second clause of prs update/4 is tried. The interpretation and integration of a DRS is non-trivial. The predicate add drs to prs/4 interprets the discourse representation structure DRS within the proof context PRS and the theory TRS. The integration of a DRS into a proof context consists of linking its discourse referents and its discourse conditions to the corresponding ones in the PRS. That is, the discourse update engine regards each DRS referent and condition as anaphoric to its place in the proof context. If these links cannot be established, then the proof context has to be accommodated to provide such links. This behaviour is ensured by Prolog’s backtracking behaviour; if the first clause of the predicate prs update/4 fails, then its second clause is activated. A call to the predicate accommodate proof context/3 extends a proof context PRS into a context PRS1 in an attempt to allow the integration of the given DRS in the subsequent, recursive call to prs update/4. The discourse update engine’s accommodation clause makes use of a self-built, simple pragmatic component, the proof planner. The proof context can be extended by reasoning forwards from accessible PRS proof lines of modality assp and then, as well as by reasoning backwards from an open goal obligation PRS line. Fig. 3 depicts three proof plan schemata encoding argumentation schemes for induction, choose and elimination. Proof plan schemata are represented as X X.1 X.2 X.3 X.4

∀n∈N .n > n0 → P (n) proof by induction ind base case — ind hyp —

— goal let assp goal

— P (n0 ) n ∈ N be arbitrary P (n) P (n + 1)

(ind 1, Line) — — — —

X X.1 X.2 X.3 X.4

∀x∈N ∀y∈N ∀z∈N .P (x, y, z) — — — — —

— let let let goal

— x ∈ N be arbitrary y ∈ N be arbitrary z ∈ N be arbitrary P (x, y, z)

(choose/3,Line) — — — —

X X.1 X.2 X.3

A → (B1 ∨ B2) — — — —

— assp assp goal

— A ¬B1 B2

(elim 1, Line) — — —

Fig. 3. Proof Plan Schemata as Underspecified Proof Representation Structures

underspecified proof representation structures. The schema for induction, for instance, introduces the five underspecified abstract discourse entities X, X.1, X.2, X.3, X.4, the discourse markers ind base case, proof by induction and ind hyp, the discourse referent n, the assumption P (n) as well as the proof obligations P (n0 ) and P (n + 1). The schema is applicable for theorems of the form ∀n∈N .P (n). When applied and introduced into the context, its underspecified content is fully instantiated. 10

4.4 Example Reconsider the two surface form conditionals of discourse (1). The discourse update engine needs to interpret the sentence-level representation of the first conditional in the context that has been created after the processing of the theorem announcement “Theorem 3 (Euclid’s first theorem)”. Given this theorem context, the discourse update engine interprets the first conditional as a classical conditional, and thus assigns a generic reading. As a result, we obtain the PRS header of Fig. 1. The sentence-level representation of the second if-then conditional is:

(7)

¬

v12∈N , v13∈N name(v12 , p), v12 name(v13 , a), v13 div(v12 , v13 )

. =? . =?

→

1∈N , v14∈N , v15∈N , r3∈N , gcd∈(N→N)→N . name(v14 , a), v14 =? . name(v15 , p), v15 =? r3 = gcd(v14 , v15 ), f un result(r3 ), equal(r3 , 1)

The semantic representation (7) must be interpreted in a much richer context. In fact, it is first interpreted in the PRS (8), which resulted from processing the first proof sentence.

(8)

— — — thm 1 1.1 1.2 1.3 1.4 1.5 1.5.1 1.5.2

— — — thm(3) — — — — — — — —

let let let goal — let let let goal — assp goal

p ∈ N be universally quantified a ∈ N be universally quantified b ∈ N be universally quantified prime(p) ∧ div(p, times(a, b)) → div(p, a) ∨ div(p, b) — p ∈ N be arbitrary a ∈ N be arbitrary b ∈ N be arbitrary prime(p) ∧ div(p, times(a, b)) → div(p, a) ∨ div(p, b) — prime(p) ∧ div(p, times(a, b)) div(p, a) ∨ div(p, b)

— — — — (choose/3, thm) — — — — (impl elim, 1.4) — —

While the discourse update succeeds wrt. the incorporation of the named discourse referents p, a and b, it fails linking the representation for p 6| a to its place in the proof context. Therefore, the accommodation predicate must be invoked. However, the proof planner also fails to return a continuation of (8) that allows the interpretation and integration of p 6| a. Consequently, the PRS (8) must be abandoned alltogether, and another PRS continuation from a previous PRS has to be sought. Backtracking within prs update/4 initiates the reprocessing of the DRS that lead to its construction, and eventually, a proper interpretation of the conditional (7) yields the PRS shown in Fig. 1.

5

Related Work, Discussion and Conclusion

5.1 Linguistic Phenomena Most of the phenomena remarked on this paper have been noticed before, of course. Reconsider, for example, the second proof sentence of Theorem 3, the conditional “If p 6| a then (a, p) = 1”. Our logical-mathematical analysis of 11

the argument shows that the premise p 6| a is added to the discourse context and made accessible to justify the conclusion (a, p) = 1 as well as subsequent proof statements. The fact that conditionals can introduce assumptions has been observed (and given a DRT treatment) by several authors (cf. Roberts [21] and Frank & Kamp [9], among others). Roberts gives the following two pairs of sentences: (9) a. b. c. d.

If Edna forgets to fill the birdfeeder, she will feel very bad. The birds will get hungry. If John bought a book, he’ll be home reading it by now. It’ll be a murder mystery.

In the first pair, intuitively, 9(b) is not a categorical assertion; it is only asserted as following from the antecedent of 9(a). The same is true for 9(d) wrt. 9(c). Moreover, sentence 9(d) contains an anaphoric expression that has its antecedent introduced by the premise of 9(c). Both phenomena occur in a combined manner when we have a referring expression that has a proposition, or a set thereof, as an antecedent. As we argued in § 3.2.2, sentences of the form “Therefore A” can be regarded as elliptic, and therefore, anaphoric in character, referring to a proposition (or set thereof) that allow to conclude A. This pragmatic use of “therefore” has been studied by Webber et al. who show that discourse adverbials (like “therefore”, “but”, “then”, “so”) “make an anaphoric, rather than a structural, connection with the previous discourse” [24]. They function anaphorically to pick up a salient proposition from the discourse context. Reconsider the surface conditional 5(a), which we identified as a pseudoconditional, or better, action conditional. The question is whether it is justified to treat the action in “If A then B is obtained”, or the cause in “If A then causally B” as a proposition. In dynamic logic, such conditionals are expressed by a formula [a]B, where a is an action, or a computer program, and B is the proposition expressing the result of executing the program. In our approach, we consider the consequent and the antecedent of a conditional as propositions, and therefore, our analysis of the conditional is reduced to the characterisation of a formal link between them. Proof planning is an enabling device to help establishing such formal links. While proof plans might capture the proof author’s intensional structure behind an argumentation, they also encode the logical relations between parts of the argument. 5.2 Discourse Understanding as Inferring Rhetorical Relations Work on generating human-readable, natural language proofs from machinegenerated formal proofs shows the necessity to abstract from “obvious” steps in the formal proof, being thus able to identify the underlying main argumentation of the formal proof (e.g., Fiedler’s doctoral dissertation [8], the generation module of the tactics-based prover Nuprl [14]). In addition, one might argue that proof verbalisation techniques yield more natural and read12

able proofs when they complement the main logical structure of the proof by a rhetorical structure. In the case of interpreting natural language proofs, one might see the reconstruction of a formal proof (in our approach: proof plan) underlying the informal mathematical discourse as analogous to inferring its rhetorical relations. However, it is unclear whether discourse relations other than logical consequence (for example, explanation, elaboration, narration or background ) do actually appear in textbook proofs. Usually, informal mathematical proofs do not contain temporal relations, and postulating the presence of events may be criticised as pushing it too far. Of course, proof authors make use of discourse markers that help structuring the argument. However, note that the use of “but” in our introductory example does not indicate contrast, but “remembers” the reader of a proposition that has been introduced as an assumption at an earlier proof stage. This is not to say that the cue “but” cannot be used to mark contrast, say, in a reductio ad absurdum argument. In our approach, however, this is captured in purely logical terms.

5.3 On Representation In our approach, we construct a proof object, represented as a proof representation structure, as the representation of the discourse itself. At the same time, we treat the proof object as a mathematical argument that the informal mathematical discourse describes and that parts of this discourse therefore refer to. As discussed in § 4.3, it is the discourse update engine that regards each DRS referent and condition of a DRS-based representation of each proof sentence as anaphoric to its place in the proof object. Proof representation structures are richer than the proof plans used by the proof planning systems [3,5,20]. The added features (e.g., numbering scheme, discourse markers) were necessary to facilitate the processing of (some of the) linguistic phenomena that occur in informal mathematical arguments. In this respect, the work of Ferguson and Allen about plan representation in mixed-initiative planning (several participants cooperate to develop plans) is particularly interesting [7]. They found, through the collection and analysis of empirical data, that “in no case did one agent simply describe the plan by describing a sequence of actions”. Rather, they found that users “identified the overall goals, identified subgoals to focus on, identified important actions in the plan, stated relevant facts that would help the development of the plan, identified problems with what the other agent proposed, confirmed what the other agents suggested, and requested clarification were not fully understood”. Moreover, Ferguson and Allen point out that “many of the details of the plan are never mentioned at all, and yet are implicitly agreed upon by the agents”. Thus, it is clear that a STRIPS representation of plans is inadequate for supporting the communication about plans. A planning system that engages in a mixed-initiative dialogue with human users must thus use a richer data structure, extending the representation of a plan as a sequence of actions. 13

5.4 Conclusion and Future Work Building a text understander for mathematical discourse is more feasible than building one for other, less scientific domains. The reasons are three-fold: (i) the universe of discourse is mathematics, a science more formally complete and precise than others; (ii) textbook proofs are a highly structured form of discourse, and mathematicians agree on a variety of different methods for how to prove theorems as well as how to present them; and (iii) the major discourse relation in mathematical discourse is logical consequence, and proof engines that operationalise mathematical reasoning are ready for deployment. Nevertheless, as we have seen in the first part of the paper, the interpretation of the basic ingredient of mathematical writing, the conditional statement, is complex. This is true despite of the good style of mathematical writing that is propagated by many guidelines that are directed at teaching a clear exposition of mathematical ideas (e.g., [13], [17], [23]). But, as we have demonstrated in the second part of the paper, an extension of discourse representation structures combined with an operationalisation of mathematical reasoning can provide a computational framework for the automatic reconstruction of the logical structure of mathematical discourse. The relationship between our approach of reconstructing the logical structure of an informal mathematical proof and more general kinds of inference in other discourse genres clearly deserves more discussion. Future work will need to further investigate and compare our approach with alternative treatments of conditionals.

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