Logic, language and the brain Michiel van Lambalgen Cognitive Science Center Amsterdam http://staff.science.uva.nl/˜michiell
Aim and program • aim: explain the use of computational logic in cognitive science – the domain is language comprehension and production – show how logical modelling leads to testable predictions, both for behaviour and brain imaging – show how logical modelling connects to biological issues, e.g. neural substrate of linguistic processing, and evolutionary considerations • lecture 1: time, tense and biology • lecture 2: the event calculus • lecture 3: verb tense and closed world reasoning • lecture 4: predictions for EEG • lecture 5: executive function and behavioural predictions for autism and ADHD; neural network architecture
Warming-up: tense/aspect and goals • consider ‘Mary was writing a letter when her sister spilled coffee over the paper’ • the syntactic structure of ‘write a letter’ seems to suggest a transitive verb with a direct object • but ‘a letter’ is not a direct object in the sense of ‘a ball’ in ‘kick a ball’ – e.g. it need not exist, or only partially • whether it can be assumed to exist depends on tense/aspect • it will be fruitful to view ‘a letter’ as goal to be achieved • ‘The semantics of tense and aspect is profoundly shaped by concerns with goals, actions and consequences . . . temporality in the narrow sense of the term is merely one facet of this system among many.’ (Steedman, Temporality )
Introducing the event calculus • language comprehension was characterised as a mapping discourse �→ discourse model • the discourse model contains causal information imported from world knowledge • the mapping discourse �→ discourse model is non-monotonic • the discourse model will be viewed as the minimal model (w.r.t. well- founded semantics) of a (constraint) logic program which consists of – axioms for causality – clauses expressing the meaning of the lexical items in the discourse – ‘goals representing the sentences in the discourse’ • the backbone of this logic program is furnished by the event calculus , a theory of causation developed by Kowalski in a legal context and by Shanahan to apply to robotics
Event calculus: general logical characteristics • formulated in many-sorted predicate logic; primitive predicates for causal concepts, connected by axioms • how can such a formalism ever be computationally feasible? • the logical reflex: look at modal logics, considered as subsystems of predicate logic (modal formulas correspond to predicate logical formu- las involving a single binary R ) • which are expressively rich qua iterability of the modal operators, but the language itself is poor • another option: rich language, but restrictions on the recursive defini- tion of wffs • (representational versus procedural semantics)
Event calculus: ontology • obviously the event calculus is about events, but there is a distinction in the event calculus between different kinds of events (‘perfect’ and ‘imperfect’ nominals – PToE ch. 12) – action/event types: e , e ′ . . . (for example ‘break’, ‘ignite’) [perhaps a further distinction between actions and events is necessary – gov- erned by separate axioms?] – (there are good reasons for having both event types (‘lightning’) and tokens (‘lightning on August 7, 2008, 8.25am’); e.g. perfect nominalisation yields event types ) – implicitly time-varying properties or fluents : f , f ′ . . . (for example ‘being broken’, ‘walking’), possibly with parameters – one can obtain these from imperfect nominalisation • event types (or tokens?) cause changes in time-varying properties (instantaneous change (Hume)) • sometimes a fluent causes another fluent to change: pushing in ‘push a cart’ changes the position of the cart – continuous change (Kant)
Jean-Yves Girard on event ontology Il y a d’autres intuitions de base qui ont ´ et´ e ´ evacu´ ees par la logique, ainsi la distinction essentielle entre parfait et imparfait , distinction rendu en fran¸ cais par le choix des temps, en russe par le changement de verbe. Cette nuance n’existe pas dans le monde v´ eriste.’ (Girard, La logique etrie du cognitif ) comme g´ eom´ (There are other basic intuitions that have been kicked out by logic, for example the essential distinction between perfective and imperfective aspect, a distinction captured in French by verb tenses, and in Slavic languages by verb pairs. This subtle distinction does not exist in logics obsessed with truth.)
Event calculus: auxiliary ontology • individual objects (‘John’) – although many individuals will be mod- elled as fluents, not as objects • (objects can be viewed as temporally extended events) • instants of time, interpreted as ‘real numbers’ – technically variables for time take values in a ‘real-closed field’ • (a ’real-closed field’ (Tarski) is a model of the set of axioms for the real numbers in the language <, + , × (e.g. ‘a polynomial of odd degree has a root’) – these axioms are complete ) • this choice does not reflect an ontological commitment to a particular structure of time (e.g. a continuum of points ): there are also many countable structures satisfying the axioms for real-closed fields, in some of these all ‘reals’ are computable, and hence approximable • various other real quantities for e.g. position, velocity, degree of some quality (such as state of completion of a house in the process of being built) [with the same proviso as for time]
Event calculus: logical aspects • instants of time, interpreted as ‘real numbers’ – technically variables for time take values in a ‘real-closed field’ • a ’real-closed field’ (Tarski) is a model of the set of axioms for the real numbers in the language <, + , × (e.g. ‘a polynomial of odd degree has a root’) – these axioms are complete • completeness follows from quantifier elimination : every quantified for- mula in this language is equivalent to a Boolean combination of poly- nomial equalities and inequalities (‘constraints’) • (gives good decision procedure) • most importantly: definable sets have a very simple structure – e.g. all definable subsets of the real line are finite unions of intervals • (technically: definable sets are semi-algebraic)
Event calculus: primitive predicates for instantaneous change • relations and functions such as <, + , × over the reals • event calculus predicates for instantaneous (Humean) change 1. Initially ( f ) (‘fluent f holds at the beginning of the discourse’) 2. Happens ( e, t ) (‘event type e has a token at t ’) 3. Initiates ( e, f, t ) (‘the causal effect of event type e at time t is the fluent f ’) 4. Terminates ( e, f, t ) (‘the causal effect of event type e at time t is the negation of the fluent f ’) 5. Clipped ( s, f, t ) (roughly, ‘an event type terminating f has a token between times s and t ’) 6. the ‘ truth predicate ’ HoldsAt ( f, t ) (see below)
More on event types and fluents • in standard first order logic there is an absolute distinction between terms and formulas • terms are constructed from variables ( x, y, z, x 1 , . . . ), constants ( a, b, c, a 1 , . . . ) and function symbols ( f, g, . . . ) for each arity; e.g. f ( x 1 , a ) is a term • formulas are built up from atomic formulas (see below) using the logical operations ¬ , ∧ , ∨ , ∀ , ∃ • an atomic formula is constructed from predicates A ( x 1 , . . . , x n ) by substitution of terms t 1 , . . . , t n for the variables x 1 , . . . , x n • what is not allowed is a ‘formula’ of the form A ( B ( x, b ) , t ) , i.e. where a formula is substituted for a variable • event types and fluents are terms which can be seen as codes for formulas via reification (also called G¨ odelization) – what is this?
More on event types and fluents • events – e.g. shaking hands, the destruction of natural habitats – seem to act like terms somehow derived from natural language ex- pressions • verb tenses seem to need a transformation of V(erb)P(hrases) into various kinds of events • hence if one treats natural language formally, i.e. as a formal lan- guage with a formal semantics, one needs to have a transformation of formulas into terms • this transformation must be iterable: one can say (1) Halting the destruction of natural habitats will prove to be diffi- cult. • furthermore there must actually be two such transformations – from ‘ x destroys natural habitats’ to ‘the destruction of natural habitats’ [perfect nominal] – from ‘ x destroys natural habitats’ to ‘destroying natural habitats’ [imperfect nominal]
More on event types and fluents • there is a general procedure to transform formulas into terms: G¨ odel numbering – originally devised for treating self-reference • standard notation: if ϕ is a formula, then � ϕ � is its G¨ odel number • in AI this procedure is called reification • we still have to bring in the distinction between perfect (‘noun-like’) and imperfect (‘verb-like’), which has to do with time – in a ‘verb-like’ nominal time is an internal argument • assume all verbs come with a variable over time (not over events, as in Davidson): destroy ( x, y, t ) • the imperfect nominal corresponds to � destroy ( x,y,t) � • the perfect nominal corresponds to � ∃ t destroy ( x,y,t) � • in the event calculus, fluents are formed analogous to � destroy ( x,y,t) � , event types analogous to � ∃ t destroy ( x,y,t) �
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