aa r X i v : . [ c s . L O ] F e b Order effects in dynamic semantics
Peter beim Graben ∗ Department of German Language and LinguisticsHumboldt-Universit¨at zu Berlin, Germany
Abstract
In their target article, Wang and Busemeyer (2013) [A quantum question or-der model supported by empirical tests of an a priori and precise prediction.
Topics in Cognitive Science ] discuss question order effects in terms of in-compatible projectors on a Hilbert space. In a similar vein, Blutner recentlypresented an orthoalgebraic query language essentially relying on dynamicupdate semantics. Here, I shall comment on some interesting analogies be-tween the different variants of dynamic semantics and generalized quantumtheory to illustrate other kinds of order effects in human cognition, such asbelief revision, the resolution of anaphors, and default reasoning that resultfrom the crucial non-commutativity of mental operations upon the beliefstate of a cognitive agent.
Keywords:
Question order effects, belief revisions, anaphor resolution,default reasoning, generalized quantum theory, dynamic semantics ∗ Department of German Language and LinguisticsHumboldt-Universit¨at zu BerlinUnter den Linden 6D – 10099 BerlinPhone: +49-30-2093-9632Fax: +49-30-2093-9729
Email address: [email protected] (Peter beim Graben)
URL: (Peter beim Graben)
Preprint submitted to Topics in Cognitive Science August 2, 2018 . Introduction
In their target article, Wang and Busemeyer (2013) discuss question ordereffects in terms of incompatible, i.e. non-commuting, projectors on a Hilbertspace. In their model, a person’s belief state is expressed by a vector ina linear space that is equipped with a scalar product while the answers toyes/no-questions correspond to orthogonal subspaces in Hilbert space. Aquestion is answered by projecting the current belief state vector either ontothe question’s yes or no subspace. When answer subspaces to different ques-tions do not coincide, the sequence of projections matters and questions areincompatible to each other.In a similar vein, Blutner (2012) presented an (ortho-) algebraic approachfor a query language that is not only able to explain question order effectsbut also allows the analysis of conditional questions of the form “If Maryreads this book, will she recommend it to Peter?” (Blutner, 2012). AlsoBlutner’s approach essentially relies upon a Hilbert space representation ofbelief states where questions induce a decorated partition into orthogonal an-swer subspaces. Yet, Blutner (2012) explicitly constructs his query languageas a “version of update semantics” (Blutner, 1996; Veltman, 1996) where the“meaning of a sentence is not its truth condition but rather its impact onthe hearer” (Kracht, 2002).In my commentary on the target article of Wang and Busemeyer, Ishall further elaborate the interesting analogies between the different vari-ants of update semantics (Blutner, 1996; Veltman, 1996), dynamic semantics (G¨ardenfors, 1988; beim Graben, 2006; Kracht, 2002), and dynamic logics (Groenendijk and Stokhof, 1991; Staudacher, 1987) on the one hand and generalized quantum theory (Atmanspacher et al., 2002), respective quantumdynamic logic (Baltag and Smets, 2011) on the other hand in order to il-lustrate some other kinds of order effects in human cognition, such as beliefrevision, the resolution of anaphors, and default reasoning that essentially re-sult from the non-commutativity of mental operations upon a person’s beliefstates. 2 . Generalized Quantum Theory In generalized (or “weak”) quantum theory, Atmanspacher et al. (2002) con-sider a set X as a general state space and functions ( morphisms ) Mor( X ) = { A | A : X → X } , transforming a state x ∈ X into another state y ∈ X through y = A ( x ) . (1)Particular functions from a subset A ⊆
Mor( X ) are called observables . Ob-servables can be concatenated, i.e. iteratively invoked, such that ( B ◦ A )( x ) = B ( A ( x )) = B ( y ), for all x ∈ X . This observable product AB = A ◦ B is as-sociative: A ( BC ) = ( AB ) C , but in general not commutative: AB = BA .Only when AB = BA , observables are called compatible , otherwise they arecalled incompatible .Atmanspacher et al. (2002) supply a number of further axioms describingthe properties of such observables and their impact upon the state space X .One of these axioms introduces a neutral element , such that ◦ A = A ◦ = A (2)for all A ∈ A . Another axiom additionally introduces a zero observable ∈ A and a zero state o ∈ X , such that ( x ) = o (3) A ( o ) = o (4) A = A = (5)for all x ∈ X and A ∈ A .An important class of observables P ⊂ A are projectors which are idem-potent A = AA = A . (6)Applying a projector A to a state x ∈ X yields another state y = A ( x ) = A ( x ) = A ( A ( x )) = A ( y ) that does not change under subsequent applicationsof A anymore. The projected state y = A ( y ) is hence an eigenstate of A .3 . Classical Dynamic Semantics Regarding the state space X of generalized quantum theory as the set of epis-temic states of a cognitive agent, yields an instantiation of dynamic updatesemantics (Blutner, 1996; G¨ardenfors, 1988; Veltman, 1996) in the followingway: Elements x, y, z ∈ X are called epistemic states , or belief states whileobservables A, B ∈ A become interpreted as epistemic operators . By restrict-ing observables only to commutative and idempotent operators, one obtains propositions . Their (commutative) composition can then be identified withlogical conjunction A ∧ B = AB = BA = B ∧ A . (7)An important notion in dynamic semantics is that of acceptance . A propo-sition A ∈ P is said to be accepted in state x ∈ X (or x is accepting A ),if A ( x ) = x . (8)That means, the state x is an eigenstate of A . Because propositions areidempotent, the state y = A ( x ) always accepts A . Thus, Eq. (1) receives astraightforward interpretation as information update.Furthermore, logical consequence (or stability in Blutner (1996)) is definedas follows: A proposition B is called a logical consequence of a proposition A , if B ∧ A = A ∧ B = A . (9)In this case, y = A ( x ) entails B ( y ) = B ( A ( x )) = A ( x ) = y , such that B isaccepted whenever A is accepted in an epistemic state (but not vice versa).The given system can be equipped with other logical connectives such asnegation ( ¬ A ) or disjunction ( A ∨ B ). G¨ardenfors (1988) has proven thatthe resulting calculus is equivalent to intuitionist logics which can be furtherextended to classical propositional logics. Another important extension isBayesian update semantics where states are interpreted as probability dis-tributions ρ over propositions. Then the impact of a proposition A upon abelief state ρ is expressed by Bayesian conditionalization ρ A ( B ) = ρ ( B ∧ A ) ρ ( A ) =: ρ ( B | A ) (10)4f the distribution with respect to A (van Benthem et al., 2009; G¨ardenfors,1988; beim Graben, 2006).
4. Non-classical Dynamic Semantics
Classical dynamic semantics comprises epistemic operators that are commu-tative and idempotent propositions. Moreover, such systems are monotonic,as propositions which already have been accepted remain accepted duringthe updating of epistemic states. This follows from commutativity: Let A beaccepted in state x (i.e. A ( x ) = x ) and let B ( x ) = y , such that B is learnedduring the updating from x to y . Then A ( y ) = A ( B ( x )) = ( A ∧ B )( x ) =( B ∧ A )( x ) = B ( A ( x )) = B ( x ) = y , saying that A and B are both acceptedin the updated state y . However, this account is not appropriate when belief states have to be revisedby new evidence. Belief revision processes are in general not commutative andhence non-monotonic such that order effects become ubiquitous. G¨ardenfors(1988) introduces a belief-revision operator as a mapping ∗ : P → A \ P assigning a revision A ∗ ∈ A \ P to a proposition A ∈ P . This revisiondynamics has to obey several minimality axioms.In order to illustrate this process, consider an agent in a belief state x thataccepts the proposition A =“the moon consists of blue cheese” (beim Graben,2006). Its revision is hence A ∗ = “the moon does not consist of blue cheese”.Another proposition might be B = “the moon consists of stone”. Since x accepts A , the application of B , B ( x ), leads to the zero state o ∈ X ofgeneralized quantum theory, that becomes now interpreted as the absurd state accepting every proposition (G¨ardenfors, 1988). Therefore also BA = “themoon consists of blue cheese and of stone” is accepted in o . This state doesnot change under the revision A ∗ , hence ( A ∗ B )( x ) = o . On the other hand theproduct BA ∗ applied to x yields B ( y ) where y = A ∗ ( x ) accepts the revisionof A . Therefore, BA ∗ ( x ) = o because BA ∗ = “the moon does not consist ofblue cheese, it rather consists of stone” can be consistently accepted. Thus5 ∗ B = BA ∗ , i.e. belief revisions and propositions do generally not commuteand are hence incompatible to each other.Belief revision in a probabilistic, Bayesian framework requires condi-tionalization with respect to a minimally altered probability distribution ρ ∗ which involves several technical peculiarities such as epistemic entrench-ment (Baltag and Smets, 2008; G¨ardenfors, 1988; beim Graben, 2006). Inthe framework of quantum cognition, however, Bayesian conditionalizationis replaced by the L¨uders-Niestegge rule (L¨uders, 1950; Niestegge, 2008) ρ A ( B ) = ρ ( ABA ) ρ ( A ) (11)(see also Blutner (2009); Blutner et al. (2013)) resulting from the non-commutativityof Hilbert space projections (Atmanspacher et al., 2002), where ρ ( A ) = |h ψ | A | ψ i| gives the quantum probability in state vector | ψ i . Therefore, belief revisionseems to be a good candidate for quantum probability models in dynamicsemantics (Engesser and Gabbay, 2002). Another important example for order effects in dynamic semantics is the reso-lution of anaphors. For this aim, Staudacher (1987) and Groenendijk and Stokhof(1991) have independently developed models of dynamic predicate logics,where quantifiers, such as “there exists an x ” or “for all x ”, and anaphors,e.g. pronouns, are described by epistemic operators acting upon model the-oretic valuations (see also Kracht (2002)).As an instructive example we consider three propositions A =“John satat the table”, B =“George came in”, and C =“he was wearing a hat” Equation (11) holds for self-adjoint projectors on Hilbert space. For general operatorsthat are not necessarily self-adjoint, one had (Atmanspacher et al., 2002) ρ A ( B ) = ρ ( A ∗ BA ) ρ ( A ∗ A ) . It might be tempting to speculate about the possible relationship between the belief revi-sion operator “ ∗ ” in dynamic semantics and the algebra involution “ ∗ ” in quantum theory. CBA = “John sat at the table; George came in;he was wearing a hat” and
CAB = “George came in; John sat at the table; hewas wearing a hat”. In the first case, the pronoun “he” refers to “George”,while it refers to “John” in the second case. These anaphors therefore haveto be described as non-commutative operators as well.
Finally, Blutner (1996) and Veltman (1996) have observed that by relaxingthe stability condition of logical consequence in (9), dynamic logics becomesnon-monotonic. This allows the treatment of default operations, such as“may” or “normally”. Veltman (1996) presented a nice example for such anordering effect in default reasoning: Let A =“Somebody is knocking at thedoor”, B =“Maybe it’s John”, and C =“It’s Mary”. Then the composition CBA =“Somebody is knocking at the door. Maybe it’s John. It’s Mary.”makes perfect sense, while
BCBA =“Somebody is knocking at the door.Maybe it’s John. It’s Mary. Maybe it’s John.” does not.
5. Conclusion
Classical dynamic semantics formalizes propositional logics in terms of com-mutative and idempotent epistemic operators that constitute a monotonicsystem of belief updating dynamics. By contrast, belief revision, the res-olution of anaphors and non-monotonic reasoning in default logics requirenon-commutative operations.In probabilistic dynamic semantics, updating is expressed by means ofBayesian conditionalization, whereas the description of belief revision pro-cesses requires rather peculiar mechanisms that could probably be more nat-urally expressed by means of quantum probability theory.Other types of cognitive order effects such as anaphor resolution or de-fault reasoning have been successfully described by extensions of dynamicsemantics including predicate calculus or non-monotonicity. To my presentknowledge, probabilistic generalizations of these models utilizing quantum7robability theory have not yet been devised. This might be a promisingdirection for future research.
Acknowledgements
I gratefully acknowledge support from the German Research Foundation(DFG) through Heisenberg fellowship GR 3711/1-1 and by the Franklin Fet-zer Trust.
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