Towards Representing What Readers of Fiction Believe

Abstract

Despite the extensive literature on the problem of story understanding, there has been little focus on formally representing some forms of knowledge that are specific to stories, such as how the reader expects information to be presented over the course of reading. To illustrate, the reader of a mystery story may expect to eventually find out who is guilty, and also that the author may first try to mislead them about who is guilty. We propose literary logic, a formalism based on work by Friedman and Halpern for reasoning about dynamic systems, and apply it in representing this sort of knowledge. We also consider issues relating to carrying over world knowledge into fiction, and knowledge of genre conventions.

1 Introduction

Story understanding is a long-standing problem in artificial intelligence, with notable early work from the 1970s (Charniak 1972; Schank and Abelson 1977) . McCarthy (1990) , in a memo originally from 1976, pointed out that stories raise problems for commonsense reasoning. Research has continued, and recent years have seen a proliferation of corpora of stories in various mediums with accompanying questions for machine learning purposes, including MCTest (Richardson, Burges, and Renshaw 2013) , ROCStories (Mostafazadeh et al. 2016) , MovieQA (Tapaswi et al. 2016) , and COMICS (Iyyer et al. 2017) . In this paper, we are concerned with the task of a reader answering questions about a story after reading (some prefix of) it.1 We propose a logic for the purpose of determining how the reader would answer such questions based on their various types of background knowledge.

Much of the work on story understanding has focused on the “world knowledge” needed to understand stories. For example, Charniak (1972) devoted a chapter to how knowledge about piggy banks can be used in understanding passages about them. In representing stereotypical events, scripts (Schank and Abelson 1977) also encode world knowledge, like that tips are given at restaurants after eating.

However, there are other forms of knowledge that are also relevant. Diakidoy et al. (2014) suggested that readers have “story knowledge” such as expectations that characters’ efforts would meet with complications, but they did not try 1We will not be considering summarization or further tasks that have been suggested as part of story understanding (Michael 2013) . to represent that in their argumentation-based approach to story understanding. Some information of that sort could be represented in a story grammar (Rumelhart 1975) . Charniak and Goldman (1989) pointed out the significance of readers assuming that mentioned objects are going to be relevant. In work on using abduction to interpret text (Hobbs, Stickel, and Martin 1993) , it’s been suggested that the abductive explanations might refer to such things as authors’ plans. Despite interest in interpreting literature (Hobbs 1990) , this has not been much focused on in the context of stories. We may note that if scripts are learned from corpora, as by Chambers and Jurafsky (2009) , they probably end up also capturing information about what events authors find noteworthy. Chaturvedi, Peng, and Roth (2017 ) consider several forms of knowledge in trying to predict the correct ending of a story, including knowledge of patterns of sentiment in stories.

Forms of knowledge which have not been explored in much depth include what the reader believes that they will come to learn from reading (parts of) the story, and what the reader thinks the author will try to make them believe over time. For example, the reader may believe that they will learn from reading a mystery who was guilty, but that the author will try to make them believe at some time that an innocent character is guilty. Or the reader may believe that if they haven’t been told the main character’s eye color by halfway through a book, they’ll never find it out. In this paper, we apply an approach to modelling belief and time to this sort of representational problem. We also focus specifically on fiction, unlike most AI story understanding research.

Applying world knowledge to fiction is more complicated than to non-fiction. In the philosophical literature, there has been substantial work on defining “truth in fiction”. Lewis (1978) noted the phenomenon of carry-over, that “factual premisses [...] may carry over into the fiction, not because there is anything explicit in the fiction to make them true, but rather because there is nothing to make them false” (p. 42). To use an example of his, we may assume that Sherlock Holmes does not have a third nostril. Lewis offered multiple definitions of truth in fiction; his “Analysis I” said that A sentence of the form “In the fiction f , ” is nonvacuously true iff some world where f is told as known fact and is true differs less from our actual world, on balance, than does any world where f is told as known fact and is not true.

His “Analysis II” was similar but instead of considering differences from the actual world, considered differences from the worlds where the common beliefs of the fiction’s community of origin were true. Others have used similar ideas, e.g. Walton (1990) had his “Reality Principle” and “Mutual Belief Principle” which roughly correspond to Lewis’s analyses, though Walton regarded them only as rules of thumb. Genre information – e.g., about time travel (Morgenstern 2014) , or that dragons breathe fire – is another sort of knowledge, which is unclear how to incorporate into these sorts of definitions; perhaps the most detailed approach attempting to do so was given by Bonomi and Zucchi (2003) .

These philosophical approaches were not fully formalized and expressed in logic. The logics designed by philosophers for dealing with fiction (Woods 1974; Heintz 1979) have usually focused on other issues, like handling inconsistent stories (which we will not be addressing in this paper).

The formal logic we present in this paper, which we call literary logic (LL), is a variant of the logic used by Friedman and Halpern (1999) to model belief revision in dynamical systems. We argue that LL can be used to represent various forms of knowledge relevant to story understanding. We focus on two main issues: representing reader’s expectations about stories (which may take into account genrespecific information), and the carry-over of world knowledge and its interaction with genre knowledge (e.g. about dragons). Literary logic provides temporal features that we apply to the first issue (though they may also have a role to play with respect to the second), and non-monotonic aspects that are useful for both. The outline of this paper is as follows. Section 2 describes the syntax and semantics of literary logic, and notes some of its properties. Section 3 shows how the question-answering task can be formalized, describing how we can make use of abnormality predicates (McCarthy 1986) in specifying the reader’s initial epistemic state. Section 4 formalizes some examples of reader knowledge: we consider carry-over (and incorporating genre knowledge) in section 4.1, and then expectations about mystery stories in section 4.2. Section 5 discusses related work, and section 6 concludes with a discussion of future work. 2

Literary logic

This section describes the language LL, which is closely based on the logic of Friedman and Halpern, which provided for modelling the accessibility and plausibility of possible worlds over time. The major differences include that LL is first-order, is evaluated with respect to finite rather than infinite timelines (because stories are finite and are read in finite time), and includes the complete set of past and future temporal operators from Lichtenstein, Pnueli, and Zuck (1985 ). LL describes the beliefs of a reader over time as they read a discourse, a sequence of logical sentences representing a story, one sentence per time step.

A very visible feature of LL (that is mostly just for clarity of presentation) is that we have two sorts of predicates, “real” and “imaginary” ones, and have a special “in imagination” operator I. The idea is that real predicates describe properties in the real world (including “literary” properties, like the genre of the story being read) while imaginary predicates (that hold only “in imagination”) describe properties that apply within the world of the story being read. The reader’s beliefs about the extensions of both sorts of predicates can change over the course of reading. 2.1

Syntax

The syntax of LL involves both terms and predicates.

A term is either a standard name or a variable. There is a countably infinite set N = f#1; #2; #3; : : : g of standard names. Intuitively, these stand for all the objects that we may want to refer to, including not just real-life things like piggy banks, but also theoretical literary concepts like what Van Inwagen (1977 ) called “creatures of fiction”, like the character Sherlock Holmes or his pipe. There also is a countably infinite set of variables. Note that the logic does not have constants or function symbols, though the standard names can be thought of as constant symbols that satisfy the unique name assumption and an infinitary version of domain closure. For a discussion of why standard names are useful, see Levesque and Lakemeyer (2000, section 2.2).

As previously indicated, there are two (non-empty) sets of predicate symbols, the real r and the imaginary f (which do not have to be disjoint). Each predicate P from either set has an arity, ar(P ), which is the number of terms that it takes as arguments. So, to give a typical example, there could be a real unary predicate Rabbit that indicates its argument is a rabbit in reality, and an imaginary unary predicate also called Rabbit that indicates its argument is a rabbit in the world of the story under consideration. The set of real predicates would also typically include predicates to express literary propositions; for example, there could be a 0-ary real predicate FantasyGenre which would indicate that the story was in the fantasy genre.

A real atom is a string of the form P (t1; : : : ; tk), where P 2 r, k = ar(P ), and t1; : : : ; tk are terms. Similarly, an imaginary atom is a string of the form Q(t1; : : : ; tk), where Q 2 f . We will say that an atom is ground if no variables appear in it. We will assume that there is a unary predicate Mentioned 2 r, which we will later give the special meaning of picking out those standard names that appear within the discourse.

The formulas of LL are the expressions of the form generated by the grammar below, where P is a real atom, Q is an imaginary atom, x is a variable, and t1 and t2 are terms. := Q j : j ( ^ ) j (t1 = t2) j 9x( ) := P j : j ( ^ ) j (t1 = t2) j 9x( ) j I j

D j # j j U j S j € We will also be talking about -type formulas, which are expressions of the form generated by the grammar (though the -type formulas are what we will mean when we refer to LL formulas). A variable x appearing in a ( - or -type) formula is said to be free if it does not appear within a subformula of the form 9x( ), and a formula with no free variables is called a sentence. The use to which we will put type sentences is that the discourse being read is a sequence of -type sentences, which describe the world of the story. An LL sentence (that is, a -type sentence) describes the real world, and can include modal operators to describe the reader’s beliefs over time.

The operators :; ^; 9, and = are familiar from first-order logic, and we can use them to define abbreviations like _, , , and 8 in the usual ways. It’s convenient to have a symbol > that always takes a true truth value; let > := 8x(x = x).

We will read I as “ is imagined” or “ is true in imagination” (though the I operator serves a technical function and is not intended to formalize a commonsense notion of imagination). We will read D as “The last sentence read of the discourse was ”.

The operators # (“next”), (“previous”), U (“until”), and S (“since”) are standard temporal logic operators.2 They describe time for the reader, who reads one sentence of the discourse each time step. We can define further temporal operators in the usual ways: (“eventually”) by := > U , (“always in the future”) by := : : , (“sometime in the past”) by := > S , and (“always in the past”) by := : : . We can also define an operator ¸ (“after reading”) by ¸ := ( ^ :#>), so ¸ means that is true at the final time (i.e., when the entire story has been read), and µ (“initially”) by µ := ( ^ : >), so that µ ' means is true at time 0 (the initial time).

The formula 1 € 2 means that 2 is true in all the most plausible accessible worlds in which 1 is true. We could follow Friedman and Halpern in defining a belief operator B with the abbreviation B := > € (that is, is believed if it is true in all the most plausible accessible worlds), but instead let us give a more general definition: if is any sentence, let We can think of B as indicating that is believed by an agent who initially considers it impossible that is false. We will call the knowledge base (or KB) of the agent (though is not necessarily true). Note that cannot include the B operator, for then B would not expand to a finite sentence, but can contain B 0 for a suitable different sentence 0. We also define a “knowledge” operator K by

:= ((µ ) ^ : ) € :>: The result is that K is true if the agent with knowledge base considers it impossible that is false.

We define (a subjective version of) fictional truth to be what the reader, after reading the entire story, believes is imagined:

F := ¸ B I : (3) We can read F as saying that is (subjectively) fictionally true. The reason why we want to consider the final time (and so use the ¸ operator) is that fictional truth is determined by the story as a whole, which has only been fully consumed at the final time.

Furthermore, we define [ ] by [ ] := (#D # ). So [ ] says that is true after reading (provided that the next sentence is actually ). We will abbreviate sequences of such operators with [ 1; : : : ; k] := [ 1] [ k] . So, e.g., [ 1; 2] abbreviates (#D 1 #(#D 2 # )). 2Our “next” and “previous” operators are the “strong” versions. (1) (2) 1. 2. 3. 4. 5. 2.2

Semantics

The grammar provides two types of formulas, denoted by and . While our goal in this section is to define satisfaction and validity with respect to -type sentences, let us first define a satisfaction relation j= that specifies when an -type sentence is satisfied by an interpretation (which we take to be a set of imaginary ground atoms). We will use the notation [x=c] to indicate the formula obtained by replacing all free occurrences of the variable x in by c 2 N . j= Q(c1; : : : ; ck) iff Q(c1; : : : ; ck) 2 j= :

iff 6j= j= ( 1 ^ 2) iff j= 1 and j= 2 j= (c1 = c2) iff c1 and c2 are identical names j= 9x( ) iff j= [x=c] for some c 2 N This is just an established way of giving the semantics of a first-order logic with substitutional quantification (Levesque and Lakemeyer 2000) . Now, let us make some definitions. Definition 1 (discourse). A discourse is a finite sequence of -type sentences, ending with End, a special sentence not appearing earlier (we do not have to introduce a new symbol for this; we can just take End = >).

Definition 2 (complex world). A (complex) world is a tuple w = hwr; wf ; wdi where wr is a set of real ground atoms, wf is a set of imaginary ground atoms, and wd is a discourse s.t. (1) if wd(i) = for any i then wf j= , and (2) iff c 2 N appears in a sentence of wd, then Mentioned (c) 2 wr.

Intuitively, wr is the set of all real ground atoms that are true in the world w, wf is the set of all imaginary ground atoms that are true in w, and wd is a formal representation of the story that is told in w. Note that wf represents one way of “completing” the fictional world in a way compatible with the story being told. What wf makes true is not the same as what is fictionally true (as determined by a F operator).

The sentences of a discourse, unlike those of a natural language story, are not indexical relative to the “current” time within the story. So, for example, the rather trivial story “John picked up a block. Then he put it back down.” could get encoded (in a style based after Maslan, Roemmele, and Gordon (2015)) as the following discourse: hJohn(#1) ^ PBulotcdko(w#n2()#^1; #2; #4); Endi. The last arguments to Pickup

Pickup(#1; #2; #3); Precedes(#3; #4) ^ and Putdown are meant to be the names of event instances, so e.g. Pickup(#1; #2; #3) says that #3 is an event in which #1 picked up #2, and Precedes expresses the events’ temporal ordering (with respect to time within the story, not time for the reader). The point here however is not the specific way these sentences represent time, but that they are like what Quine (1968) called “eternal sentences” in that their truth does not depend on their time of evaluation. We will also expect a discourse to usually provide standard names for relevant objects and events, as our example did.

In order to provide semantics for the € operator, we need a way to represent plausibility. Friedman and Halpern did so using the very general notion of a plausibility space; we will use what can be considered a special case of that, a version of the popular “system of spheres” representation (Lewis 1973; Grove 1988; Bonomi and Zucchi 2003) . Below we will use W to denote the set of all complex worlds. Definition 3 (system of spheres). A system of spheres is a set S of subsets (“spheres”) of W such that (1) for any two spheres U 2 S and V 2 S, either U V or V U , (2) for any non-empty set V W, there is a -minimal sphere C such that C \ V 6= ;, and (3) W 2 S.

A system of spheres can also be thought of as a total preorder on worlds, where w v (“w is at least as plausible as v”) if every sphere containing v also contains w. Every system of spheres has a “central” sphere (the -minimal sphere C such that C \ W 6= ;) containing the -minimal (most plausible) worlds.

The € operator depends not just on the plausibility of worlds, but on which worlds are (currently) accessible. Definition 4. For b a non-negative integer, the accessibility relation at time b, b W W, is given by w b v iff jwdj b, jvdj b, and wd(i) = vd(i) for 1 i b.

Intuitively, at time b the reader will not consider possible any world with a discourse not starting with the same b sentences they have read so far. For a world w with jwdj b we may use the notation [w] b := fv 2 W : w b vg. That is, [w] b is the set of worlds accessible from w at time b.

The satisfaction of a literary logic sentence is given relative to a system of spheres , a world w = hwr; wf ; wdi 2 W, and a time b 2 f0; 1; : : : ; ng, where n = jwdj, the length of the discourse wd. The recursive rules for when ; w; b satisfy , written ; w; b k , are given below: ; w; b k

P (c1; : : : ; ck) iff P (c1; : : : ; ck) 2 wr ; w; b k 1 € 2 iff ; v; b k 2 for every v 2 min fv 2 [w] b : ; v; b k 1g We will write ; w k if ; w; 0 k . We will write k (“ is valid”) if ; w k for every system of spheres and world w. 2.3

Properties

To understand the B

operator, it is helpful to introduce another accessibility relation, and a sentence.

b

W

W, where b is a time b v iff w Definition 5. Given a system of spheres , a time b, and sentence , define w . b v and ; v; 0 k ; w; b k : iff ; w; b 6k ; w; b k ( 1 ^ 2) iff ; w; b k ; w; b k 9x( ) iff ; w; b k ; w; b k I iff wf j=

D iff b > 0 and wd(b) = ; w; b k # iff b < n and ; w; b + 1 k iff b > 0 and ; w; b 1 k ; w; b k (c1 = c2) iff c1 and c2 are identical names 1 and ; w; b k

2 [x=c] for some c 2 N 1 U 2 iff ; w; j k 2 for some j such that n and ; w; k k 1 for all k s.t. b k < j

1 S 2 iff ; w; j k 2 for some j such that b and ; w; k k 1 for all k s.t. j < k b 1. 2. 3. 4. 5.

Intuitively, w b v if at world w and time b, the reader with knowledge base considers world v possible.

Observation 1.

v 2 min ([w] b ).

; w; b k

B iff ; v; b k for every

B can be shown to be a K45 operator (supporting positive and negative introspection). There is also remembrance of past beliefs, e.g. we have k B B B . Observation 2. While (I( 1 _ 2) (I 1 _ I 2)) and (I(9x ) 9xI ) are valid for any 1 and 2, F ( 1 _ 2) (F 1 _ F 2) and F (9x 1) 9xF 1 are not (assuming for the last that some Q 2 f has nonzero arity).

Note that if Observation 2 did not hold the behavior of the F operator would contradict the generally accepted idea that fiction is incomplete ( Dolezˇel 1995 ) and so there is no answer to the question of, for example, exactly how many children Lady Macbeth had in Shakespeare’s Macbeth (Wolterstorff 1976) . As McCarthy (1990) wrote, “In a made-up story, questions about middle names or what year the story occurred in do not necessarily have an answer”. Observation 3. k (D F ). That is, whatever the discourse includes is fictionally true for any reader.

Note this means we cannot encode metaphorical language. Also, Walton (1990, x4.5) raised philosophical questions on how literally some other aspects of stories should be taken, such as whether Shakespearean characters are really fictionally uttering the poetic speeches attributed to them (or, more simply, whether characters are really speaking English in English-language stories). Our formalism does not offer a choice in how to answer that.

3

Applying LL to question-answering We want to formalize how a reader would answer questions after reading (a prefix of) a story. As part of this formalization, we want to specify (within the language) not just what the reader initially believes, but what things the reader initially considers more plausible than others (so as to determine exactly how the reader’s beliefs evolve in response to reading). In LL, following some previous work on belief revision (Friedman and Halpern 1999; Shapiro et al. 2011) , the plausibility of worlds does not actually change over time, but only the accessibility relation. That nonetheless suffices to allow whether a proposition is believed to change back and forth over time (see (Shapiro et al. 2011, section 6)). This suggests we can fix one system of spheres to always use, and just set the initial accessibility relation appropriately (which the in the B operator has the effect of doing). In this section, we will define the ‘k ’ relation, the analogue of k for a particular fixed system of spheres. Then question-answering can be done by determining which expressions of the form k [ 1; : : : ; k]B hold, i.e. what a reader with a KB of background knowledge (of possibly various types) believes after reading the first k sentences of the discourse.

To define the specific system of spheres, we will apply the idea of circumscription (McCarthy 1986) and have the plausibility of worlds be inversely related to the sizes of the extensions of distinguished “abnormality” predicates. Suppose that we have a finite set of abnormality predicates, each with an associated priority (a positive integer). If Ab is a k-ary abnormality predicate of priority i, we will say that Ab(c1; : : : ; ck) is a priority i ground atom. For a world w, let Ci(w) 2 f0; 1; 2; : : : g [ f1g be the sum of numbers of priority i ground atoms from wr and wf . Let the partial order

CIRC W W be defined by w CIRC v if there is some i for which Ci(w) < Ci(v) and Cj (w) Cj (v) for all j i. Note that CIRC is a prioritized version of the preference relation from cardinality-based circumscription (Liberatore and Schaerf 1997; Moinard 2000) . The associated preorder

CIRC can be seen to satisfy the system of spheres definition. Definition 6 (k ). For w a world, b a time, and an LL sentence, we define k by w; b k if CIRC; w; b k , and we define k if for every world w.

CIRC; w k

Using the fixed system of spheres CIRC, essentially the reader represented using the B operator “only knows” the knowledge base (see Levesque (1990) ) but also applies circumscription to determine their beliefs. So we have, e.g., k B(P ab):P . Note that Observations 2 and 3 still apply if the use of ‘k ’ in them is replaced by ‘k ’, and Observation 1 works for any system of spheres, including CIRC. 4

Examples of formalizing reader knowledge As a reader reads, they draw conclusions about the imaginary world of the story, and also about real-world literary truths, like the genre of a story. Consider the knowledge base = I(8x(Knight (x) ^ :Ab(x) Man(x))) ^ (I(9xDragon(x )) FantasyGenre) which states that (in imagination) knights are normally men, and that the existence of (imaginary) dragons is a sign of the story belonging to the fantasy genre. If a reader with this knowledge reads as the first sentence of discourse (Dragon(#1) ^ Knight (#2)), which is a formal version of “There was a knight and a dragon”, we would want them to believe that the story is a fantasy and that there is in imagination a man, and that is what we have: k [Dragon(#1) ^ Knight (#2)](B FantasyGenre ^ B IMan(#2)).

Expectations about the story’s development can also be represented. The expectation that a story will literally follow a version of the rule of “Chekhov’s gun” – if a gun is shown hanging on the wall in one scene, it should be fired by the end of the story – can be written as 8x8e19e2 Mentioned (e1) ^ I(HangingOnWall (x; e1) ^ Gun(x)) ^ :Ab (Mentioned (e2) ^ I(Firing (x; e2))) . That is, if the eventuality e1 of a gun hanging on the wall is mentioned, then normally a firing event e2 is also mentioned. (The reader would also need further knowledge about events to prevent considering that e1 = e2.) How to encode the general underlying pragmatic principle is less clear.

4.1 Carry-over and genre conventions

In the example with the knight, to make the belief that knights are normally men applicable to fiction, we enclosed it in an I operator. However, we would prefer to write beliefs about the real world, and have them automatically get carried over to fiction. A first, syntactic, approximation to that is the following: Suppose the knowledge base is a conjunction including a conjunct 8~x( (~x)) (~x abbreviates the sequence of all leading universally quantified variables). If (~x) uses only operators from first-order logic and does not include real atoms for which there are not imaginary counterparts, then I( (~x)) is also a formula. Then you could automatically generate the sentence 8~x(Ab(~x) _ I( (~x))), where Ab is some abnormality predicate (of appropriate arity) not used in . This new sentence, roughly a defeasible imaginary copy of 8~x( (~x)), could be conjoined with .

Carry-over by humans is probably more complicated than that. Ryan (1991 , ch. 3) proposed restrictions on what should get carried over, including that the existence of real people or geographic locations should only be carried over into fictions that name at least one real person or location. A psychological experiment of Weisberg and Goldstein (2009) suggested that people are less likely to carry over facts into fictions differing from reality in other ways.

Below we consider the interaction of carry-over with fictional conventions in two examples of philosophical origin. Scrulch the dragon Lewis (1978, p. 45) gave a case where fictional truth depends on more than world knowledge: Suppose I write a story about the dragon Scrulch, a beautiful princess, a bold knight, and what not. It is a perfectly typical instance of its stylized genre, except that I never say that Scrulch breathes fire. Does he nevertheless breathe fire in my story? Perhaps so, because dragons in that sort of story do breathe fire. But the explicit content does not make him breathe fire. Neither does background, since in actuality and according to our beliefs there are no animals that breathe fire. For us there is no difficulty in writing additional sentences that describe how things in imagination are different from in reality, such as Ab _I(8x(Dragon(x ) BreathesFire(x)). Here Ab would represent the abnormality of a story about dragons which didn’t breathe fire. We would have to give Ab sufficiently high priority so that this sentence would overrule any carried over beliefs about animals not breathing fire in general. Note that the sentence does not do anything to specify the fire-breathing abilities of real dragons; despite believing that fictional dragons normally breathe fire, the reader could still regard real dragons that breathe fire as (even) less plausible than real dragons that do not breathe fire. Recognizing a witch Walton (1990, x4.3) gave a number of examples of tricky cases about fictional truth, including one about what information is needed to recognize a fictional character as a witch. He wrote (p. 161, 164) the following (about drawing, but clearly also relevant to other media): Any child can draw a witch. Depicting a woman with a black cape, conical hat, and long nose will usually do the trick. [...] The fact that fictionally there is a witch is implied by the fact that fictionally there is a woman with a black cape, conical hat, and long nose. But it is not the case that were there (in the real world) a longnosed woman decked out in black cape and conical hat [...], there would be a witch. [...] Although it is fictional in a mutually recognized legend that there are witches and that they have long noses and wear conical hats, it is much less clearly fictional in it that, were there a woman of that description, she would be a witch. Is it part of the legend that there are no Halloween parties, or that nonwitches never dress thus [...]? This idea, that someone described in a stereotypically witchlike way is a witch while not necessarily everyone in the world of the story with witch-like characteristics is a witch, can be expressed in literary logic. We can do so by writing 8x(I(WitchLike(x)) ^ Mentioned (x) ^ :Ab(x) IWitch(x)). Then the IWitch(x) conclusion is not drawn for every x having witch-like characteristics, but only for those also mentioned in the discourse (recall the special Mentioned predicate). This is an example of scoped nonmonotonic reasoning (Etherington, Kraus, and Perlis 1991) . 4.2

Expectations about mystery stories

A reader may expect when reading a mystery story to eventually find out who is guilty. In this section, we will use the unary imaginary predicate symbol G (x) to mean that x is guilty (in imagination).

Suppose that is the KB of a reader, and we want to inform this reader that they should expect to find out who is guilty. The proposition below shows how we can extend into a KB 0 so a reader knowing only 0 believes they will find out who is guilty (assuming that a reader knowing only the original KB, , does not believe that they won’t find out who’s guilty – in other words, that k :B :9x(F G (x))). Proposition 1. Let G be a unary imaginary predicate. Suppose is an LL sentence s.t. k :B :9x(F G (x)). Let 0 = ^ 9x(F G (x)). Then k B 0 9x(F 0 G (x)). Proof. We want to prove that for every world w, we have w; 0 k B 0 9x(F 0 G (x)). To do that, we want to show that v; 0 k 9x(F 0 G (x)) for every v 2 min([w] 0 0 ). Fix an arbitrary such v (if there are none, we are done), and let n = jvdj. We have that v; 0 k 0 and so (by the definition of 0) v; 0 k and v; 0 k 9x(F G (x)). Let c 2 N be such that v; 0 k F G (c). Then v; n k B IG (c). Therefore, for each v0 2 min([v] n ), we have v0; n k IG (c). It can be shown that min([v] n0 ) min([v] n ), which means that for each v 2 min([v] n0 ), we have v ; n k IG (c). Hence v; n k B 0 IG (c), and v; 0 k 9x(F 0 G (x)).

We could also consider representing the knowledge that the reader won’t find out who’s guilty until very near the end, which Brewer and Lichtenstein (1982) considered to be an example of a “curiosity discourse organization”. The non-trivial part would be formalizing the vague “very near” (to say the reader won’t have a belief about who’s guilty at a precise time – say, three sentences before the end – we could simply write something like ¸ :9xB IG (x)). Brewer and Lichtenstein suggested that the purpose of stories is to entertain, and that three ways that authors accomplish this is by creating suspense, surprise, and curiosity by manipulating when information gets revealed to the reader. Our next example might be considered a case of surprise.

A genre-savvy reader might think that in a mystery story, it’s true that “The first character the author tries to make you suspect of being guilty is innocent.” Under an assumption of authorial competence we can roughly paraphrase that as “The first character a na¨ıve reader would suspect of being guilty is innocent.” Consider a reader with KB ; let us suppose that they are the reader the author would be able to trick into suspecting the wrong character. How could we extend their KB to make them genre-savvy?

As a prelude to that, consider the following formula: (x) = B IG (x) ^ 8y(y 6= x : B IG (y)) (4) Recalling that “ ” is the “previously” operator, we can read (x) as “x is believed to be guilty (in imagination) and for all y not equal to x, y was not previously believed to be guilty (in imagination)”, where the beliefs are understood to be those of the reader with KB . So, for a standard name c, (c) is true at a time iff c is the (unique) first character believed to be guilty (in imagination).

Below, proposition 2 shows a sentence (incorporating (x) as a subformula) we can conjoin to to produce the knowledge base 0 of a savvy reader, and establishes that if (c) is ever true (i.e., that c is the first character the reader with KB believes is guilty) then the reader with KB 0 will believe that c is not guilty (unless that reader knows that c is guilty, e.g. because the discourse includes G (c) explicitly). Proposition 2. Let be an LL sentence, let Ab be a 0ary real abnormality predicate of higher priority than any abnormality predicate appearing in , let G be a unary imaginary predicate, and (as in Equation 4) let (x) = B IG (x) ^ 8y(y 6= x : B IG (y)). Then define 0 = ^ Ab _ 8x( (x) I:G (x)) : Then

8x ( (x) ^ :K 0 IG (x)) k Proof. Suppose for a world w, time b, and name c, we have w; b k (c) ^ :K 0 IG (c). We want to show that w; b k B 0 I:G (c). It can be shown that w; b k :K 0 Ab, and therefore (because all worlds in which Ab is false are more plausible than all others) w; b k B 0 :Ab. So w; b k B 0 8x( (x) I:G (x)), and so w; b k B 0 ( (c) I:G (c)). So we will be done if we can show that w; b k B 0 (c), i.e. that v; b k (c) for every v 2 min([w] b 0 ). This follows because for any such v, the discourse vd must agree with wd on the first b entries, which is enough to give (c) the same truth value there.

B 0 I:G (x) : 5

Discussion and related work

5.1

Regarding non-monotonic reasoning

We have space only to make a couple remarks in this section.

Many forms of circumscription, including prioritized circumscription, have been considered in the literature (see e.g. (Lifschitz 1994) ). Cardinality-based circumscription (Liberatore and Schaerf 1997; Moinard 2000) has the advantage of being simple to work with because there will always be a set of most plausible worlds in which any sentence is true, if that sentence is true in any worlds.

The way that we can use the in the B operator to refer to what is and isn’t believed by an agent with knowledge 0 recalls hierarchic autoepistemic logic (Konolige 1988) . In common with that system, and unlike standard autoepistemic logic (Levesque 1990) , the issue of there being multiple “stable expansions” of an agent’s beliefs does not arise. Wilensky (1983) briefly discussed “dynamic points” in stories, involving violations of the expectations of a character or the reader. He wrote (p. 616) that “Only with recourse to events that are supposed to transpire in the reader during the course of understanding a text can the discourse structure of a theory of stories be stated.” LL, of course, is expressly designed to allow referring to changing beliefs of the reader.

Michael (2013) also considered encoding reader expectations in a formal system. He gave an example of encoding the expectation “that the story clarifies at each instance whether it is day or night” (which is not unreasonable for a story told in a visual medium), though the brief outline given of the semantics for his system does not cover how disjunctive expectations like that should be handled. Michael also considers the case of the reader being told additional information about what the author expects them to infer.

We may note that there is also work in the AI subfield of narrative generation that concerns itself with reader expectations. For example, the “Prevoyant” system (Bae and Young 2014) is supposed to generate narratives that are surprising.

Rapaport and Shapiro (1995) presented a computational approach to handling carry-over in story understanding in their SNePS system. Beliefs about reality are copied into a “story world context” and belief revision is used to deal with any conflicts that may arise as the story is read (they also discuss an alternative approach using a “story operator”). Genre conventions are not discussed, and since time for the reader does not play an explicit role, it’s not clear how reader beliefs about the future could be represented or queried.

The ISAAC story understanding system (Moorman and Ram 1994) has a so-called “creative understanding” process that allows for modifying pre-existing concepts in an attempt to understand a story. Moorman and Ram did not relate this to the philosophical literature on carry-over, and further investigation of that would be interesting.

Bonomi and Zucchi (2003) gave an approach to combining carry-over with genre conventions. They consider having two systems of spheres (the worlds in these, unlike our complex worlds, are not split into real and imaginary parts), one centered on Bx, the set of worlds conforming to the “overt beliefs” of the author of x (the fiction in question), and one centered on Rx, the set of worlds following the conventions for x. Fictional truth is determined by what is true in all the closest worlds to Bx from among those worlds that are closest to Rx in which the “directly generated content”3 of x is true. Note this requires which conventions x follows to be already known, while LL allows for reasoning about that.

We have not much considered interaction between expectations and carry-over. However, Mart´ınez-Bonati (1983) suggested that the reader expects to quickly find out how realistic the world of the story is (p. 188):

If I read a few narrative sentences implying a system of reality not different from ordinary life, I will rapidly tend to solidify my expectations into a “realistic” fictional horizon. [...] A similar promptness will be an at3This is not the same as the literal content, as they also consider (in an unformalized way) the narrator and their reliability. tribute of the projection of fictional horizons that are traditional and well-known (for example, the fabulous world of speaking animals).

At an intermediate point in reading, the reader’s beliefs about what carries over may be influenced not just by what the author has said about the fictional world, but by what the reader believes the author will say in the rest of the story.

Our formal discourses require temporal information be explicitly encoded, like some other approaches (Diakidoy et al. 2014; Maslan, Roemmele, and Gordon 2015) . While this is not like the ordinary use of natural language, it is much less complicated. For logic-based approaches that do try to deal with those kinds of issues, see Episodic Logic (Schubert and Hwang 2000) and Segmented Discourse Representation Theory (Asher and Lascarides 2005) .

The form of fictional truth we have formalized is readerdependent. Whether what a text means should depend on the reader is controversial (Hirst 2008) . An objective form might be implemented in a multi-agent version of literary logic, by defining objective fictional truth in terms of the knowledge of an ideal reader which other readers have beliefs about. (LL in this paper is not truly multi-agent, despite the parameterized B operators, since one reader cannot reason about another’s beliefs without specifying what the latter’s KB is.) 6

Conclusion

We have argued that our logic LL, following on the work of Friedman and Halpern, can represent various forms of knowledge relevant to story understanding, and so be used to determine how a reader with such knowledge would answer questions about a story. We encourage investigating how other formal approaches developed for modelling belief over time could similarly be useful in story understanding.

In future work, we plan to further investigate carry-over and to construct fully worked out examples with complete stories. Also, it should be possible to replace the Mentioned predicate with epistemic constructions. Another point is that LL models the reader as logically omniscient (Hintikka 1975) , seeing all consequences of its own beliefs, but it would be interesting to consider resource-bounded readers, as that is what real authors write for.

Acknowledgments

This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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