Notions of predication based on extensional and intensional subsumption, as described by Woods, are related to Kahneman's systems of thinking fast and slow. Path-based reasoning with links is applied to predication over not only individuals but (following Carlson) kinds and stages/time. Predications, fast and slow, are formulated in monadic secondorder logic over strings, analyzed in Goguen and Burstall's institutions.
1. logical reasoning, which is rigorous and formal, but often counterintuitive, and which has algorithms that match expressions, substitute values for variables, and invoke rules, and 2. associative networks, which are structured and intuitive, but typically informal; however, they support efficient algorithms that follow paths through links to draw conclusions [Woods, 2010, page 625]. These traditions offer different perspectives on concepts and subsumption v between concepts. The custom in logic is to interpret a concept C as a set [[C]] of C-instances, called its extension (under [[ ]]), with C subsumed by a more general concept C0, C v C0, when every C-instance is a C0-instance
C v C0 under [[ ]] [[C0]] (1) [Baader et al., 2003]. Rejecting the reduction of a concept to its extension, Woods advocates a notion of intension that “is as much a psychological issue as a logical issue” [Woods, 2007, page 83]. Woods steps away from arbitrary instances given by some interpretation [[ ]] to a carefully crafted conceptual taxonomy, for an intensional subsumption quicker to compute than the extensional notion specified by (1).1 Woods 1Analyzing intension as a function from indices (or points of reference) to extensions [Carnap, 1947; Montague, 1974] arguably only compounds (1), multiplying notions of instance by indices. hypothesizes the bulk of reasoning is of the quick “recognize and react” variety, implicating the efficient algorithms of associative networks, as opposed to the ponderous mechanisms of logic. In a similar vein, the psychologist Daniel Kahneman argues that thinking operates largely under a fast system, breakdowns in which trigger a second slower system that would otherwise lie dormant [Kahneman, 2011]. Woods and Kahneman independently suggest commonsense reasoning is often easy but at times hard.
The fast/slow intensional/extensional contrasts are taken
up together below, starting in x2 with Formal Concept
Analysis [Ganter and Wille, 1999] for a tight conceptual pairing of
extension and intension (shortened there to extent and intent).
Well-known complications in predication point to a
loosening of that pairing, paving paths in x3 to logical systems in x4
that Goguen and Burstall [1992] call institutions. Institutions
based on monadic second-order logic over strings are defined,
providing a uniform path-based account of predication over
kinds and stages in the sense of Carlson [
Formal Concept Analysis and partiality To bring out a notion of intension buried in purely extensional accounts, FCA defines a context to be a triple hO; A; H i consisting of (i) a set O of objects d; d0; : : :, (ii) a set A of attributes a; a0; : : :, and (iii) a binary relation H O A specifying the attributes a in A that an object d in O has (so dH a can be read: object d has attribute a).
Now fix a context hO; A; H i. The extent of a set A attributes is the set A of AH := fd 2 O j (8a 2 A) dH ag of objects that have every attribute in A, while the intent of a set D O of objects is the set
DH := fa 2 A j (8d 2 D) dHag of attributes that every object in D has. The notions of extent and intent constitute an antitone Galois connection inasmuch as for every D O and A A,
D
AH () A
DH : (2) The inclusions in (2) are strengthened into equalities in defining a concept to be a pair (D; A) such that D = AH and A = DH : Equivalently, a concept is a subset A of A such that A = (AH )H (replacing D by AH , to focus on attributes). Reducing extension to extent, extensional subsumption (as described in x1) between concepts A and A0 is just inclusion of extents which, in view of (2), is the converse of inclusion of intents A vH A0 ()
AH
A0H A vH A0 ()
A0
A: We can turn an attribute or object x 2 A [ O unambiguously into a concept concept(x) := fxgH (fxgH )H if x 2 O if x 2 A assuming A \ O = ;, and then for x0 2 A [ O, define x is aH x0
concept(x) vH concept(x0) so that for d 2 O and a 2 A, and for d0 2 O, The =)-half of the last biconditional is the inference rule d is aH a ()
dHa d is aH d0 () fd0gH d H f g : d0Ha
d is aH d0
dHa
for property inheritance. Shortcomings of (3) are
longstanding concerns in Knowledge Representation, a common
example being that (a1) to (a3) leave out penguins.
(a1) bird H fly (i.e., birds fly)
(a2) tweety is aH bird (i.e., Tweety is a bird)
(a3) tweety H fly (i.e., Tweety flies)
To accommodate exceptions, let us replace is aH by a binary
relation IS (left unspecified for the moment) and, following
Reiter [
M(dHa) d IS d0
dHa with M pronounced “it is consistent to assume” (left implicit in the :-notation for justifications in Default Logic). Under (4), (a3) follows from (a1) and tweety IS bird only with M(a3), which fails when there is information to the contrary of (a3). Now, assuming a comes with a contrary attribute a 2 A, (3) (4) a plausible candidate for M(dHa) is the negation :(dHa) expressing the absence of contrary information. Were it the case that dHa () :(dHa) (5) (4) would be vacuous, as :(dHa) reduces to the conclusion dHa of (4). But (5) cannot hold in a context hO; A; Hi with a bird that flies and another that doesn’t,
neither bird H fly nor bird H fly exposing a sense in which H is partial, and pushing us beyond H if, as commonsense demands, we are to make anything of (4). 3
Varieties of predication and causal paths What could birds fly mean when plainly some birds don’t? The linguist Greg Carlson contrasts two views of generic sentences, an inductive approach based on observed instances, and a “rules and regulations” view emphasizing not so much their “episodic instances but rather the causal forces behind those instances” [Carlson, 1995, page 225]. The latter causal approach (which Carlson favors) is broadly in line with the proposal made in Steedman [2005] that causality and goaldirected action lie at the heart of temporal semantics. A simple example of goal-directed action is expressed by the predication die(tweety), which overturns the temporal proposition alive(tweety). The proposition alive(tweety) is inertial inasmuch as it persists in the absence of a force overturning it — or to bring out the similarity with (4), alive(tweety)@t t S t0
:opp(alive(tweety)@t) alive(tweety)@t0 (6) where (i) alive(tweety)@t says “alive(tweety) holds at time t” (ii) t S t0 says “t is succeeded (temporally) by t0” (iii) opp( ) says “some force opposes ” whence :opp( ) says “no force opposes .”
For rigour, let us associate with every attribute a 2 A, a distinct unary relation symbol Pa, allowing us to encode an FCA-relation H O A as a fPaga2A-model [[ ]] over the universe/domain O interpreting Pa as the subset
[[Pa]] = fd 2 O j dHag of O, so that d 2 [[Pa]]
() for all d 2 O. But the point of the relation symbols Pa is not to recreate a particular relation H O A but to move beyond such relations, as described by (4) and (6) above. And central to these moves are the relations IS in (4) and S in (6), for which we introduce a binary relation symbol S. We can then reformulate (4) as the sentence dHa saying a is inherited through S in the absence of information a to the contrary (assuming a 2 A). S inverts IS in ih[a] for uniformity with S in (6), which we generalize to a as the inertial requirement ir[a] := 8x8y((Pa(y) ^ ySx ^ :Po(a)(y)) assuming an attribute o(a) 2 A for an a-opposing force. An obvious choice for o(alive(tweety)) is die(tweety), which describes an event that terminates the state alive(tweety). The distinction between events and states is critical to temporal semantics [Kamp and Reyle, 1993; Allen and Ferguson, 1994], with ir[a] suited as a requirement for a equal to alive(tweety) but not for die(tweety). Further evidence for the significance of the event/state divide is the 9=8-contrast illustrated by (a4) and (a5). (a4) Tweety flew in his first year. (a5) Tweety was flightless his first five weeks.
While some flight by Tweety in his first year is enough to
make (a4) true, (a5) specifies flightlessness at every instant of
his first five weeks. That is, (a4) claims of an interval that a
certain event happens within it, while (a5) claims of an
interval that a certain state holds at each instant in it. Inasmuch as
states holds at instants, while events happen over intervals, an
analogy can be drawn with individuals/kinds
state
event
instant
interval
individual
kind
:
Carlson [
As presupposition failures, sortal errors are commonly held to apply equally to negations, expressed in (5) and ih[a] by a, and not to be confused with :. To block the leap from (a6) to (a7) on the basis of ih(widespread), we must refrain from requiring ih[a] of kind-level predicates, just as we refrain from requiring ir[a] of predicates describing events (for which o(a) may not even be defined). That said, the present paper focuses on attributes a suited to ih[a] or to ir[a]. Let us agree to call the former inheritable, and the latter inertial.
Ensuring an inheritable attribute a satisfies ih[a] or an inertial attribute satisfies ir[a] may call for some repairs on [[Pa]]. The repair for ih[a] can be described through a fresh attribute a+ with Pa+ given by two rules
Pa(x) Pa+ (x)
Pa+ (y)
ySx Pa+ (x) :Pa(x) (7) extending Pa so that Pa+ (x) can be read there is an S-path to x avoiding Pa from some y such that Pa(y).
9y(Pa(y) ^ ySax) where Sa expresses the reflexive transitive closure of the restriction Sa of S to pairs (x1; x2) such that not Pa(x2) Sa :=
x1 x2(x1Sx2 ^ :Pa(x2)): Now, whether or not ih[a] is true at a model [[ ]], the aforementioned assumptions about Pa+ make ih[a+] true at [[ ]] with a+ = a. Put another way, ih[a] is satisfied by a model [[ ]]+ identical to [[ ]] except possibly at Pa, where As for the inertial sentence ir[a], we introduce, in place of a+ for ih[a], an attribute ao with Pao given by
an Sao-path to x exists from some y such that Pa(y) where Sao is the counterpart in ir[a] of Sa
Sao :=
x1 x2(x1Sx2 ^ :Po(a)(x1)): Then [[ ]]o satisfies ir[a] where [[Pa]]o := [[Pao ]]. Sa-paths and Sao-paths alike are causal, implementing the inheritance and inertial laws ih[a] and ir[a], respectively.
Looking back at the previous section, it is doubtful FCAintents are what Woods has in mind by intension (as marvellous as Galois connections are). Accordingly, we trade subsumption vH between intents of FCA-concepts for a predicate symbol S that has many interpretations, in line with Woods contention that intension is as much psychological as logical. Insofar as no single interpretation of S on its own will do, it is not unreasonable to keep these interpretations as simple as possible. Suppose, for example, [[S]] were given by some finite string d1 dn of n distinct objects di [[S]] = f(di; di+1) j 1 i ng and A were finite. Then the construction of [[ ]]+ to ensure ih[a] for a in some set Ih A of inheritable attributes can be implemented by a finite-state transducer transforming a string A1 An with
Ai = fa 2 A j di 2 [[Pa]]g to A01
A0i = fa 2 A j di 2 [[Pa]]+g: The transducer has subsets of Ih as states q, the empty set as the initial state, all states final, and transitions
q A!:A0 A0 \ Ih where A0 := A [ fa 2 q j a 62 Ag so that the current state consists of the inheritable attributes in the last set A0 of attributes returned. Moreover, for a 2 Ih with a 2 Ih and a = a, we have
[[Pa]]+ \ [[Pa]]+ = ; () [[Pa]] \ [[Pa]] = ; allowing us to set a+ equal to a+. As for ir[a] where a belongs to some set Ir A of inertial attributes, we can build a finite-state transducer taking A1 An to Ao1 Aon where
Aio = fa 2 A j di 2 [[Pa]]og using subsets of Ir as states q, and transitions q A!:Ao q0 where Ao := A [ q and q0 := fa 2 Ao \ Ir j o(a) 62 Ag.
Predications in flux: an MSO institution We step in this section from an FCA context hO; A; Hi up to an institution hSign; sen; Mod; j=i in the sense of [Goguen and Burstall, 1992], where contexts can be varied systematically via signatures. The definition of an institution uses a modicum of category theory, and is, on first exposure, a mouthful: (i) Sign is a category, with objects called signatures (ii) sen is a functor from Sign to the category of sets and functions, with elements of the set sen( ) called sentences (iii) Mod is a contravariant functor from Sign to the category of (small) categories and functors, with Mod( )-objects called -models (iv) j= is an indexed family fj= g 2jSignj of binary relations
j= jMod( )j sen( ) between -models and -sentences, indexed by signatures such that for every Sign-morphism ! 0, 0-model M and -sentence ',
() A context hO; A; Hi can be packaged as an institution with exactly one signature , trivializing the category-theoretic requirements above so that the functor sen amounts to and the indexed family j= to
j= = H: A far more interesting institution associated with hO; A; Hi pieces together finite fragments of hO; A; Hi on which the S-paths behind a+ and ao in x3 tread. Some preliminary definitions help make this precise. Given a finite subset A of A, (i) the fragment MSOA of Monadic Second-Order logic over strings [e.g., Libkin, 2004] is given by a binary predicate symbol S (expressing succession) and a unary predicate symbol Pa for each a 2 A (ii) an MSOA-model h[n]; Sn; f[[Pa]]ga2Ai with domain [n] := f1; : : : ; ng; interpreting S as the successor (plus 1) relation on [n], and each Pa as a subset [[Pa]] of [n] can be identified with the string A1 An of subsets
Ai := fa 2 A j i 2 [[Pa]]g of A consisting of attributes a in A with i in the interpretation [[Pa]] of the unary predicate for a.2
An, the equations [[Pa]] = fi 2 [n] j a 2 Aig (a 2 A) take us back to h[n]; Sn; f[[Pa]]ga2Ai, justifying the identification of MSOA-models with strings of subsets of A.
MSOA-sentences and MSOA-satisfaction j=A are defined as usual in classical predicate logic so that, for example, A1
An j=A 9x(Pa(x) ^ 8y:ySx) () for all strings A1 An of subsets of A. A fundamental result is the Bu¨chi-Elgot-Trakhtenbrot theorem [Libkin, 2004, page 124]: the set
MSOA(') := fs 2 (2A) j s j=A 'g of strings satisfying an MSOA-sentence ' is regular, and conversely, every regular language over the alphabet 2A (of subsets of A) can be obtained this way from some MSOAsentence '. The use of the powerset 2A as the alphabet of strings constituting MSOA-models departs from the custom of A in statements of the Bu¨chi-Elgot-Trakhtenbrot theorem but is crucial when forming an institution where A varies [Fernando, 2016].3
Let IA be the institution where Sign is the set Fin(A) of finite subsets A of A partially ordered by (for a category), and for every A 2 Fin(A), (i) an A-sentence is an MSOA-sentence, an A-model is a string over the alphabet 2A and j=A is MSOAsatisfaction (ii) for A A0 2 Fin(A), sen(A; A0) is the inclusion of MSOA-sentences in MSOA0 -sentences, while Mod(A0; A) intersects a string A01 A0n 2 (2A0 ) componentwise with A
Mod(A0; A)(A01
A0n) := (A01 \ A) (A0n \ A): Next, we bring in an FCA-context hO; A; Hi to define for every d 2 O and A 2 Fin(A), the set
A[d] := fa 2 A j dHag of attributes in A that d has, which we then use to map a string d1 dn 2 O to the string
A[d1 dn] := A[d1]
A[dn] over the alphabet 2A. Let I(O; A; H) be the institution with the same signature category and sentence functor as IA, but with a string d1 dn 2 On as an A-model that j=Asatisfies an MSOA-sentence ' precisely if A[d1 dn] j=A ' (in MSOA). That is, I(O; A; H) picks out the A-models A1 An in IA given by strings d1 dn and H
Ai = fa 2 A j diHag: But H might be transformed to a different FCA-context, as we saw in the transformations in x3 of [[ ]] to [[ ]]+ and [[ ]]o to satisfy the MSO-sentences
3The reason is that the structure h[n]; Sn; [[Pa]]a2Ai encoded by a string A1 An over 2A has A-reduct h[n]; Sn; [[Pa]]a2Ai, encoded by the string (A1 \ A) (An \ A) over 2A.
A and for each a 2 Ih, a 2 Ih and a = a with
ySx saying: y subsumes x.
Inertia ir[a] is suited to attributes a belonging to a set Ir such that (Inr) Ir A and for each a 2 Ir, o(a) 2 A a = a with Ir; a 2 Ir and ySx saying: y is followed by x.
We can use a monadic second-order variable X to pick out a path, defining the MSOfa;ag-formula patha(X) patha(X) := 8x(X(x) (Pa(x)_ 9y(X(y) ^ ySx ^ :Pa(x)))) by inverting the rules (7) for Pa+ with X in place of Pa+ . Then ih[a+] follows from the MSOfa;ag-formula 8x(Pa+ (x) 9X(X(x) ^ patha(X))) (10) reducing Pa+ (x) to the possibility of putting x in a set X such that patha(X). Similarly, for inertia ir[a] and ao, we invert the rules (8) for Pao with X in place of Pao for pathoa(X) := 8x(X(x) (Pa(x)_ 9y(X(y) ^ ySx ^ :Po(a)(x)))) to derive ir[ao] from the reduction 8x(Pao (x) 9X(X(x) ^ pathoa(X))) (11) of Pao to pathoa. The finiteness of [[S]] (or more specifically [[S]]-chains) is crucial to push through the arguments for ih[a+] and ir[ao] above (extracting Sa- and Sao-paths that reach Pa from patha(X) and patha(X)).
For a in Ih or Ir, it is natural to assume a does not co-occur with a
nc[a] := 8x:(Pa(x) ^ Pa(x)) banning contradictions in [[Pa]] \ [[Pa]]. The reduction (10) above yields the equivalence of nc[a] and nc[a+] nc[a] nc[a+] provided Pa does not discriminate between S-predecessors of the same object 8x8y8y0(ySx ^ y0Sx ^ Pa(y) Pa(y0)) which follows from the uniqueness of S-predecessors 8x8y8y0(ySx ^ y0Sx y = y0): Otherwise, nc[a] may hold while nc[a+] fails because an object in [[Pa]] has the same S-successor as another in [[Pa]] d1[[S]]d and d2[[S]]d with d1 2 [[Pa]] and d2 2 [[Pa]]: The same applies to inertia ir[a] and a0; nc[a0] follows from nc[a] and the reduction (11), under the interpretation above of S as for some positive integer n. As observed at the end of x3, this interpretation of S allows us to build a finite-state transducer that computes [[Pa+ ]] for finitely many inheritable a in one pass from A1 down to An. Similarly for inertia and ao.
The institution IA accommodates different FCA-contexts with attribute set A, providing accounts at bounded granularities A 2 Fin(A) that can be refined or coarsened, as required. Assuming A is infinite, we can always extend A to a larger finite A-subset A0 A, leading, after repeated extensions, to infinite models at the limit (or, as detailed below, inverse limit). How smoothly can these extensions be carried out? As a partial answer, clause (iv) in the definition above of an institution provides the biconditional
()
known as the Satisfaction condition [Goguen and Burstall,
1992]. In the present institution IA, (9) unwinds to
whenever A A0 2 Fin(A), and for every MSOA-sentence
' and string A01 A0n of subsets of A0. In particular, for
a fixed sentence ', we can set A to the vocabulary of ',
voc('), defined to be the set of all attributes mentioned in
' (making voc(') the smallest subset A of A for which ' is
an MSOA-sentence). Satisfaction of ' by a string A01 A0n
of subsets of A0 2 Fin(A) with voc(') A0 then reduces to
satisfaction by the string
(A01 \ voc('))
(A0n \ voc('))
of subsets of voc('). In other words, the attributes we see in
' can — as far as the issue of satisfying ' (or not) is
concerned — be assumed to be all there is, in accordance with
the bias What you see is all there is (WYSIATI) from
Kahneman [2011]. Of course, “what you see” in ' depends on what
we put in '. And while, for example, ih[a] mentions only the
attributes a and a, many more attributes may appear once we
try to spell out what information to the contrary is buried in a
(not to mention a). In the simplest case, a might be false (i.e.,
8x:Pa(x)) turning ih[a] into
8x8y((Pa(y) ^ ySx)
but the point of ih[a] is to deal with more interesting cases. As
Reiter [
Pbird(x) (Pfly(x) (Ppenguin(x) _ Postrich(x) _ )) (12) The same open-endedness applies to the forces overturning an inertial attribute a, glossed over by the non-inertial attribute o(a) in ir[a]. In practice, simplifying assumptions are adopted (dropping, for example, in (12)) to facilitate reasoning. In situations where these assumptions fail, reasoning may break down. As Shanahan [2016] points out,
Because it sometimes jumps to premature conclusions, bounded rationality is logically flawed, but no more so than human thinking.
Elaborating on the attributes a; a; o(a) is an art in managing flaws, and recovering from missteps (slowing reasoning down).
Apart from the attributes in A, there is also the binary predicate symbol S, interpreted as the successor relation Sn on f1; 2; : : : ; ng for some integer n > 0 for either subsumption (Inh) or temporal precedence (Inr). Widespread views that time branches and is infinite, and similar claims about conceptual taxonomies raise the question: is it not problematic to assume that [[S]] is Sn for some integer n > 0? It is, if an A-model that interprets S as a finite successor relation is asked to carry on its own the burden of representing an infinite branching structure. But once we adjust our sights from a single context hO; A; H i for defining concepts to a multitude of satisfaction relations j=A capturing finite fragments of a multitude of contexts, the problem arguably dissolves. If time branches, it is because there is more than one A-model in Mod(A) describing a branch up to A. And if time is infinite, it is because no single signature A can represent the totality of signatures in Sign = Fin (A).
And exactly how might infinite branching structures emerge from these finite approximations? Very briefly, through an inverse limit construction linking vocabulary (A 2 Fin (A)) with ontology (Mod(A)). More precisely, we work with strings A1 An of subsets of A that are stutterless in that
Ai 6= Ai+1 for all 1 i < n (the intuition being that a stutter is some substring AiAi+1 such that Ai = Ai+1). Clearly, a string s of subsets of A is stutterless iff s = bc(s) where the block compression bc(s) of s is defined by bc(s) := ( s bc(As0) A bc(A0s0) if length(s) 1 if s = AAs0 if s = AA0s0 where A 6= A0 (implementing a form of the Aristotelian claim, no time without change). Next, for any A 2 Fin (A), let bcA : (2A) ! (2A) be the function intersecting a string A1 An 2 (2A) componentwise with A before destuttering bcA(A1
An) := bc((A1 \ A) (An \ A)) (recalling from the definition of IA that whenever A A0 2 Fin (A), Mod(A0; A) intersects a string in (2A0 ) componentwise with A). Now, the inverse limit
limfbcAg of the indexed family fbcAgA2F in(A) of functions is the set of functions f : Fin (A) ! (2A) such that f (A) = bcA(f (A0)) whenever A A0 2 Fin (A): This equality ensures that f provides a consistent system of A-approximations f (A) of a structure that is infinite insofar as for any positive integer n, there is an A 2 Fin (A) such that f (A) is a string of length > n. For branching in limfbcAg, we lift the prefix relation
on strings to a relation s s0
s0 = ss00 for some string s00 ()
A on limfbcAg by universal quantification f () (8A 2 Fin (A)) f (A)
For A equal to the set of rational numbers, we can express the Dedekind cut construction of a real number r 2 R as a function fr 2 limfbcAg to get a copy of the real line R from A restricted to the functions fr (for r 2 R).4
Compression bcA conditioned by A links the ontology [n] of an A-model h[n]; Sn; f[[Pa]]ga2Ai to the granularity A. Consider, for instance, the finite-state transducer above for inheritance, enforcing ih[a] for a 2 Ih (assumed finite, for convenience). Suppose on input A1 An, this transducer outputs the string T (A1 An). Applying bcIh to this output yields the block compression of T ’s output on input bcIh(A1 An) bcIh(T (A1
An)) = bc(T (bcIh(A1
An))): Similarly for inertia and its transducer To, bcA(To(A1
An)) = bc(To(bcA(A1
An))) where A = Ir [ fo(a) j a 2 Irg. Attention to granularity A pays off in bounding the search space of candidate A-models. But before requiring an A-model A1 An is equal to bcA(A1 An), we should be clearer about what the strings A1 An represent. Common to ih[a] and ir[a] is some form of Leibniz’ Principle of Sufficient Reason, no change (in Pa over S) without a reason,
PSR[a] := 8x8y((Pa(y) ^ ySx ^ :Pa(x)) xRay) where the reason Ra is reduced to a unary relation xRay :=
Pa(x) Po(a)(y) for inheritance (Inh) for inertia (Inr).
PSR[a] with xRay as Po(a)(y) is essentially an explanation closure axiom [Lifschitz, 2015]. The non-inertial attribute o(a) designates any force opposing a, while a serves in ih[a] as a differentia in a taxonomy, a path in which is picked out by an interpretation of S. By placing demands on the signature A of MSOA, these attributes provide a causal ontology for change. It is instructive to eliminate the negation : in PSR[a] and ih/ir[a] by moving Pa(x) to the right of the implication for a choice (Pa(y) ^ ySx) (Pa(x) _ xRay) between Pa(x) and xRay given Pa(y) and ySx. A bias towards Pa(x) leading to S-paths in x3 amounts to a minimisation assumption on reasons and on a domain [n] subject to bcA. Of course, we can eliminate a stutter in a string not just through bc but by adding an attribute n to A that names a unique position 8x8y(Pn(x) ^ Pn(y) x = y) (corresponding to a nominal in Description Logic). But this goes against the custom of using attributes for universals, 4Inverse limits aside, the focus of the present work is very much on the approximations from finite strings, which, under the prefix relation , do not form a complete partial order — a basic requirement on domains over which default reasoning is investigated in [Hitzler, 2004]. These finite strings enjoy a special status as compact elements in algebraic cpo’s. (I am indebted to a referee for bringing this paper to my attention.) rather than particulars, and the anti-nominalist thrust of an attribute-centered institution IA that constructs individuals and temporal instants through inverse limits over kinds and temporal intervals (conceived as basic, rather than as sets of instances and instants). 5
The account of predication above proceeds from
(a) an extensional conception of predication as instantiation,
analyzed in section 2 as extensional subsumption is aH ,
given a fixed FCA-context hO; A; Hi
to
(b) an institution IA in section 4 within which finite
fragments of various FCA-contexts are represented by
strings A1 An of finite subsets of A, the successor
relation in which, S, is an intensional alternative to (the
converse of) is aH , along which fast path-based
reasoning advocated by Woods [
Thanks to three Commonsense-2017 referees for comments. [Woods, 2010] W.A. Woods. The right tools: Reflections on computation and language. Computational Linguistics, 36(4):601–630, 2010.