Studies in Logic, Vol. 1, No. 3 (2008): 19–31 PII: 1674-3202(2008)-03-0019-13
Sphere Semantics for Aspect∗ Ahti-Veikko Pietarinen Department of Philosophy, University of Helsinki
[email protected]
Abstract. This paper introduces sphere semantics, which derives from Tarski’s geometry of solids, and applies it to aspectual phenomena in natural language. Sphere semantics is particularly geared for the English progressive. The approach has the following virtues: (i) It extends interval-based semantics but omits its pitfalls, (ii) it solves the imperfective puzzle, and (iii) the proposed solution needs no appeal to the strategy of eventual outcomes. Logical approaches to natural-language aspect have proved elusive. Point-based structures of time and the classical Priorian tense logic [18] have turned out to be too weak. Intervalbased semantics lacks expressive power and suffers from poor interpretation of the progressive ([4, 8]). A new model and the semantics of time is in this paper based on the mereological notion of spheres. Sphere semantics builds upon Tarski’s geometry of solids introduced in 1927. Differing from both point-based and interval-based structures, many aspectual distinctions can be characterised in a unifying logical manner in sphere semantics. In a universe of closed spheres, an alternativeness relation is given in terms of tangentiality. The basic notions of external and internal tangents, external and internal diameters and concentricity may then be defined accordingly. Unlike in interval semantics, sphere semantics evaluates events not in segments of time but in spheres, which are primitives of the universe. Consequently, the problem of taking intervals as the primitive notion and at the same time dispensing with points of time at the extremities of an interval disappears. The English progressive is characterised as a continuous action on spheres. An action is non-terminating in so far as a sphere is not exited via external tangents. Accordingly, tangential exit characterises completion. The mereological ‘part of’ relation differentiates between those events holding in homogeneous and those holding in heterogeneous spheres. Non-duratives are null-diametric concentric spheres. Sphere semantics defines a notion of time at the locations of those moments in which an action or performance completes, while dispensing with the notion of time at an extremity. It suggests a qualitative notion of possible worlds, and takes a step towards the cyclic notion of time, while not departing too far from the interval-based models.
1.
Introduction
To get a good grip of aspectual distinctions of verbs by logical approaches to natural-language semantics has proved elusive. Point-based structures of time and the classical Priorian tense logic [18] have turned out to be too weak. Interval-based Received 2008-11-20 ∗ Supported by the Academy of Finland Project 1103130 (Logic and Game Theory) and the University of Helsinki Research Funds (2104027).
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semantics lacks expressive power and suffers from poor interpretation of the English progressive ([4, 8]). I will propose a new model and the semantics of time based on the mereological notion of spheres. Sphere semantics builds upon Tarski’s geometry of solids introduced in 1927. Differing from both point and interval-based structures, many aspectual distinctions can be characterised in a unifying logical manner in sphere semantics. I will confine the discussion to the English progressive. In a universe of closed spheres, an alternativeness relation is given in terms of tangentiality. The basic notions of external and internal tangents, external and internal diameters and concentricity may then be defined accordingly. Unlike in interval semantics, sphere semantics evaluates events, states, processes and episodes [2] not in segments of time but in closed spheres, which are primitives of the universe. Consequently, the problem of taking intervals as the primitive notion and at the same time dispensing with points of time at the extremities of an interval disappears. The English progressive is characterised as a continuous action on spheres. An action is non-terminating in so far as a sphere is not exited via external tangents. Accordingly, tangential exit characterises completion. The mereological ‘part of’ relation differentiates between those events holding in homogeneous and those holding in heterogeneous spheres. Non-duratives are null-diametric concentric spheres. Further, we obtain a qualitative notion of possible worlds. External diametricity defines a passage from one event to another being maximally protracted, while internal diametricity captures that two subevents are maximally distant. Two notions of action are thus produced and semantically modelled: the former for atelic and the latter for telic events. Moreover, that some subevent both begins and ends involves centric tangentiality.
2.
Semantics of the progressive
Tenses as sentential operators It is fairly commonplace to considered naturallanguage tenses in logical semantics as sentential operators: p = John builds a house. P(ast) p = John built a house. F(uture) p = John will build a house. Their semantics is routinely provided in terms of possible-worlds semantics. This may be fine, as far as it goes. However, such an approach does not differentiate between simple tenses and PROGRESSIVES: ? p = John is / was / will be / would be building a house.
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Ahti-Veikko Pietarinen / Sphere Semantics for Aspect
A number of semantic methods to cope with the English progressive have been proposed, with little consensus reached ([6–8, 11, 13, 20]).1 The Imperfective Puzzle At the heart of the semantics of the progressive lies the Imperfective Puzzle [5, 12].2 Suppose we have Scenario 1: John stepped onto the street and walked to the other side. Both (1) and (2) are true of this scenario: John crossed the street.
(1)
John was crossing the street.
(2)
One evident possibility for the definition of truth of the progressive would be the following. P1: PROG(p) is true at t if and only if the event e of PROG(p) is part of an event e0 of p. For instance, the event e is the progressive crossing the street in (2), and the event e0 is the imperfective crossed the street in (1). However, consider Scenario 2: John stepped onto the street, but twisted his ankle and quickly turned back. Now rival consequences are possible. For, on the one hand, there are the MATIVES ( RESULTATIVES ): John was crossing the street =⇒ 6 John crossed the street.
PERFOR -
(3)
On the other hand, there are verbs that denote ACTIVITIES (NON - RESULTATIVES): Mary was whistling the tune =⇒ Mary whistled the tune.
(4)
Accordingly, the progressive has been attempted to be captured in terms of INERTIA WORLDS [5]. They are those possible worlds in which some ‘natural courses of events’ will continue to hold. P2: PROG(p) is true at t iff in all INERTIA of the event e0 of p.
WORLDS
the event e of PROG(p) is part
1 [8] propose an interval semantics for computational tasks, not for natural-language applications. [17] suggests extending their proposal to aspectual constructions in natural language. See also ( [10, 14, 15, 16]). 2 There is no paradox arising here, so I will call what Dowty and Lascarides term the ‘imperfective paradox’ the imperfective puzzle.
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But now, suppose we have Scenario 3: John stepped onto the street inattentively while a bus was approaching him in high speed. Mere inertia worlds are thus not sufficient, since progressives are defeasible by whatever surprising facts may come to pass. Thus, one attempts the following amendment: P3: PROG(p) is true at t iff in all ‘normally-continuing worlds’ the event e of PROG(p) continues after the time t and is part of the event e0 of p. However, also this definition leads to well-known problems [11]. Problems with the semantics of the progressive It is clear that the strategy of refining the truth conditions for progressives, in terms of attempting to take all exceptions that may occur into account and otherwise clinging on to tense logical semantics, is bound to remain unsatisfactory.
3.
Interval semantics
What is it? Traditional modal temporal logic interprets formulas over TIME POINTS. However, for real life events, we need durations, intervals, continuants, differential equations, and other dynamic resources [9]. Furthermore, durations may be infinitesimally small. For example, in qualitative physics, the event Water level increased by 5 cm needs to be captured somehow. Computational tasks refer to events such as The robot carried out the manufacturing task or The unit is processing the information. In biology, evolution is, overall, a continuous event: Species are evolving. As a first approximation, one might resort to the distinction between OPEN vs. CLOSED INTERVALS to capture the essence of progressives [3]: P4: PROG(cross the street) is true at I iff there exists some larger interval I 0 such that I 0 is OPEN, I ⊆ I 0 , and John ∈ ||cross the street|| at I 0 .3 Now, it indeed appears that John is crossing the street =⇒ 6 John crossed / has crossed / had crossed the (5) street. On the other hand, however: John built a house in one month (I ∗ ) last year =⇒ John is building a house (6) at any I ⊆ I ∗ . 3 John
∈ ||cross the street|| means that the proper name John is in the extension of cross the street.
Ahti-Veikko Pietarinen / Sphere Semantics for Aspect
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Heterogeneous structures of intervals (i.e., those that have no subinterval property) S are ones in which we do not take ni=1 I over I 0 . Alternatively, intervals may be represented by a pair of moments ht1 ,t2 i in which a proposition is true [8]. Problems with Interval Semantics Interval semantics presents us with a gamut of problems: • There is no notion of time at the extremities of an open interval. • How to proceed from one open interval to another? How are they connected? • Open intervals fail to capture TELICITY, such as finish, complete, end, arrive. • There are no CULMINATIONS (John’s life be ending). We are thus unhappy with the simple dichotomy between open and closed intervals. But how to devise a semantics for processes and other expressions of continuants that would account for at least some sense of telicity, termination or culmination?
4.
Sphere semantics
Basic concepts My proposal is along the following lines. We replace the truthconditional notion of the interval semantics, ‘satisfaction by an interval I and interpretation M’, namely hM, Ii |= p, by ‘satisfaction by a SPHERE s’, hM, si |= p. Spheres, which are simple bounded regions of space, are the geometrical primitives of the system, and they give rise to extensional mereology just as Tarski’s geometry of solids do [19]. ‘Being a part of’ is the only relational primitive in sphere semantics. Unlike in interval logic, in spheres there are no unique, designated starting and ending points. Instead, there are the notions of INITIATION and EXIT via INTERNAL and EXTERNAL TANGENTIALITY. In sphere semantics, a well-defined notion of time exists at such tangents. The following are the basic concepts of sphere semantics: I NTERPRETATION M = hT,V i; T EMPORAL of’ relation;
STRUCTURE
T = hS , vi, where S is a set of spheres and v is a ‘part
VALUATION V : Φ → 2S , S ⊆ S ; P ROPER PART: s1 @ s2 , if s1 v s2 and s1 6= s2 , s1 , s2 ∈ S ;
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D ISJOINTNESS: s1 s2 if there is no s3 such that s3 v s1 and s3 v s2 . From these basic concepts we can define the notions of external and internal tangentiality, external and internal diametricity, and concentricity. For example, the external and internal tangentialities are as follows: E XTERNAL TANGENTIALITY: s0 is EXTERNALLY TANGENTIAL to s∗ if • s0 s∗ , and • for any s1 , s2 ∈ S such that s∗ v s1 , s∗ v s2 , s1 s0 and s2 s0 , s1 @ s2 or s2 @ s1 . I NTERNAL TANGENTIALITY: s0 is INTERNALLY TANGENTIAL to s∗ if • s∗ s0 , and • for any s1 , s2 ∈ S , such that s∗ v s1 , s∗ v s2 , s1 v s0 and s2 v s0 , s1 @ s2 or s2 @ s1 . For convenience, we use the ‘earlier-than’ ordering relation: O RDER RELATION: s1
