Analogy and cognitive architecture: Two kinds of systematicity, one kind of (universal) construction

Analogy and cognitive architecture: Two kinds of systematicity, one kind of (universal) construction Steven Phillips (steve@ni.aist.go.jp) Mathematical Neuroinformatics Group, Human Technology Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8568 JAPAN Abstract Cognitive science recognizes two kinds of systematicity: (1) as the property where certain cognitive capacities imply certain other related cognitive capacities (Fodor & Pylyshyn, 1988); and (2) as the principle that analogical mappings based on col- lections of connected relations are preferred over relations in isolation (Gentner, 1983). These two kinds of systematicity were shown to derive from one type of (universal) construction (Phillips, 2014), using category theory (Mac Lane, 2000). Un- derlying both forms of systematicity is a kind of optimization. We provide an informal summary of this result, and suggest an extension to address other (semantic) aspects of analogy. Keywords: analogy; cognitive architecture; systematicity; category theory; universal construction Introduction Cognitive science recognizes two kinds of systematicity. One kind of systematicity is the property of cognition where the capacity for certain cognitive abilities implies the capacity for certain other cognitive abilities, i.e. capacity is distributed around equivalence classes of cognitive abilities (Fodor & Pylyshyn, 1988). Another kind of systematicity is the pref- erence for analogical mappings based on collections of con- nected relations over relations in isolation (Gentner, 1983). Whether these two kinds of systematicity are aspects of a deeper property of cognition is hitherto unknown. In previous work (Phillips & Wilson, 2010, 2011, 2012, 2014), we explained Fodor and Pylyshyn’s kind of system- aticity using the category theory notion of universal construc- tion (Mac Lane, 2000). We also showed that these two kinds of systematicity are derived from one type of universal con- struction as an optimization of cognitive resources (Phillips, 2014). We provide an informal summary of this result (after recalling the two notions of systematicity), and suggest ex- tensions to address some other (semantic) aspects of analogy. Systematicity (Fodor and Pylyshyn) A remarkable property of human cognition is the distribution of cognitive capacities, where the capacity for certain cogni- tive abilities implies the capacity for certain other cognitive abilities. For example, suppose one is shown pairs of geo- metric shapes such as a square to the left of a triangle. If one has the ability to infer square as the left shape in the pair (square, triangle), then one also has the ability to infer trian- gle as the left shape in the pair (triangle, square), assuming that squares and triangles are individually recognizable. This property is generally referred to as systematicity (Fodor & Pylyshyn, 1988), and is characterized more broadly as hav- ing capacity c 1 if and only if c 2 (McLaughlin, 2009), i.e. as equivalence classes of cognitive capacities. For cognitive science, a major challenge has been to pro- vide a theory of cognitive architecture that explains system- aticity: i.e. why certain cognitive capacities are distributed in a particular, non-arbitrary way (Fodor & Pylyshyn, 1988; Fodor & McLaughlin, 1990; Aizawa, 2003). Cognitive archi- tecture is the collection of basic processes and their modes of composition that together provide the basis for a slew of cognitive abilities, from recognizing concrete physical ob- jects to reasoning about abstract mathematical structures. The problem with the major theoretical frameworks, including classicism (symbols systems) and connectionism (neural net- works), is that systematicity does not necessarily follow from the core principles and assumptions of their proposed theo- ries. Although classical and connectionist theories are suf- ficiently general to derive models supporting the systematic- ity property, they are insufficiently specific to rule out mod- els (derived from the same core principles and assumptions) that do not support systematicity (in those same cognitive do- mains). Further, ad hoc (essentially, arbitrary) assumptions are needed that take up the explanatory slack to exclude those classical or connectionist models that do not support system- aticity, and so neither classical nor connectionist theory fully explains systematicity (Aizawa, 2003). Systematicity (Gentner) In the context of analogy, systematicity is the observation that when matching source and target domains of (relational) knowledge people match systems of (higher-order) relations in preference to isolated (lower-order) relations (Gentner, 1983). This observation is embodied as the systematicity principle, or assumption, in the structure mapping theory of analogy (Gentner, 1983). Structure mapping theory supposes that relational knowledge consists of a system of concepts ar- ranged in tree-like structures. Three kinds of concepts are distinguished: objects, attributes and predicates. Object con- cepts are things like John; they are constants, which are nodes in a concept tree from which there are no more branches. At- tributes and predicates are concepts that express some propo- sition about other concepts. An attribute expresses a propo- sition about a single concept, e.g., Male(John) expresses the proposition that John is a male, and its concept tree structure has Male (attribute) as the root concept node and John (ob- ject) at its only branch. Predicates are concepts that express propositions about two or more concepts, e.g., Loves(John, Mary) expresses the proposition that John loves Mary, and its concept tree structure has Loves (predicate) as the root node, John at its first branch and Mary at its second branch. Later, we consider objects, attributes and predicates as instances of