A Cognitive View of Relevant Implication

Abstract

Relevant logics provide an alternative to classical implication that is capable of accounting for the relationship between the antecedent and the consequence of a valid implication. Relevant implication is usually explained in terms of information required to assess a proposition. By doing so, relevant implication introduces a number of cognitively relevant aspects in the de nition of logical operators. In this paper, we aim to take a closer look at the cognitive feature of relevant implication. For this purpose, we develop a cognitively-oriented interpretation of the semantics of relevant logics. In particular, we provide an interpretation of Routley-Meyer semantics in terms of conceptual spaces and we show that it meets the constraints of the algebraic semantics of relevant logic.

Introduction

Paradoxes of classical material implication often show a mismatch between our intuitions concerning valid patterns of reasoning and the formalization of implication provided by classical logic. Debates on the nature of implication can be traced back to the very origin of modern logic, involving for instance Brentano, Husserl, and Frege. Turning to contemporary developments of mathematical logic, the problem of the logical properties of implication has been approached by providing systems that aims to mend classical logic from inference patterns that are not motivated on the basis of a speci c view of reasoning.

Since in any logical system, the implication has the important role of encoding the properties of logical inference, by rejecting the properties of classical implication, one is often lead to rejecting classical logic. For instance, intuitionistic logic criticizes the non-constructive nature of classical implication. For that reason, intuitionists designed an alternative logic that rejects inference by contradiction and the law of the excluded middle. Moreover, relevant logic criticizes the lack of connection between the premises and the conclusion of a logical inference made explicit by some valid formula of classical logic, e.g., A ! (B ! A)|once A holds, one can infer that any B entails A|or (A ! B) _ (B ! A)|every pair of propositions can be connected by means of an implication. By keeping track of the antecedent-consequent connection, relevant logic prevents these paradoxes.

Furthermore, classical implication does not model any sort of relationship between the knowing subject and the matter of the proposition. The truthconditional de nition of the classical implication A ! B is given in terms of those states of a airs such that either the state of a airs corresponding to A does not hold or the state of a airs corresponding to B holds. Prosaically, A ! B is true whenever A is false or B is true. The relationship between the antecedent and the consequent of a classical implication can be understood only in terms of mere co-occurrence between the states of a airs of the corresponding propositions. The knowing subject is construed as a spectator of an independent reality that displays itself. A number of approaches to non-classical logics can be categorized as proposals to make logical implication sensitive to cognitively relevant aspects. For instance, intuitionistic logic models the abstract concept of a knowing subject and intuitionistic semantics is better understood in terms of proof-conditions instead of truth-conditions, where a proof is intended to model the activity of a knowing subject [ 20 ] with respect to propositions. A signi cant number of nonclassical logics are motivated by the idea of taking into account the activity of the knowing subject, e.g., just to mention a few, justi cation logics [ 8 ], prooftheoretical semantics [ 21 ], and number of relevant and substructural logics [ 13, 5 ]. Each of this approaches stresses that the information required to asses the status of a proposition is an essential part of the meaning of the proposition.

We place our analysis within the tradition of relevant logics [ 13, 2 ], a family of logics that have been traditionally interpreted as logics of information [ 12, 1, 13 ]. In particular, the analysis of relevant implication aims to investigate the connection between the information contained in the antecedent and the information contained in the consequence. Although relevant logics are e ective in preventing paradoxes of material implication, a drawback is that their algebraic semantics has been criticized on the ground that it lacks any strong intuitive motivation [ 5 ]. To cope with that, a number of approaches to relevant logics provided an intuitive reading of the semantics. From the point of view of cognition, the most interesting approach is due to Mares [ 13 ] who interprets deduction in relevant logics in terms of situated inference. Intuitively, a situation contains information that is relevant to make a proposition hold, thus situations are truth-makers of propositions. In this paper, we provide a version of the semantics of relevant logic based on a notion of situation de ned in terms of the theory of conceptual spaces [ 9 ], a theory on how we conceptualize the reality and how we reason on this conceptualization. Our aim is to motivate the idea of situated inference provided by Mares by means of the rich theory of cognition formalized by means of conceptual spaces. The exhibition of a concrete instance of the semantics of relevant logics based on a well developed model of cognition has a double impact: (i) it provides a clean cognitive interpretation of relevant logics; and (ii) it shows that relevant logics capture cognitively important aspects of inferences.

The paper is organized as follows. Sections 2 and 3 introduce the background on relevant logic and conceptual spaces. Section 4 informally describes the interpretation of the semantics of relevant logic in terms of conceptual spaces, while Section 5 provides the formal construction. Section 6 concludes the paper. 2

Relevant logic

We introduce a minimal background on the relevant logic R [ 2, 13, 7 ]. We con ne ourself to the implicative fragment of R that, by slightly abusing the notation, 1. A ! A 2. (A ! B) ! ((B ! C) ! (A ! C)) 3. A ! ((A ! B) ! B) 4. (A ! (A ! B)) ! (A ! B) we still label by R. Let Atom be a set of propositional atoms and p 2 Atom, the language of R is inductively de ned by:

LR := p j A ! A The axioms for R are presented in Table 1 while its Hilbert system is introduced as usual through the notion of derivation `R. The base case states that `R , where is an axiom in Table 1. The rule of modus ponens is then added: if `R A, `R A ! B, then `R B. By reasoning in R, a number of paradoxes of classical implication are blocked. For instance, the monotonicity of the entailment A ! (B ! A), which is an axiom in classical logic. Its meaning is: if A holds, then every B entails A, regardless the relevance of B for assessing A. Accordingly, in relevant logics that axiom is not valid. Moreover, in case we also assume a disjunction in our language, (A ! B) _ (B ! A) is not a theorem of R. 2.1

Routley-Meyer Semantics

We present the model of substructural logic in terms of ternary relations, that is due to Routley and Meyer [ 15, 18 ]. Ternary relations can be viewed as a generalization of (relational) Kripke semantics for intuitionistic and modal logics. Let S be a set of points and R S3. Moreover, let 1 2 S be a designated element. We de ne the following notations: { R2(xy)zw i there is an u 2 S such that Rxyu and Ruzw; { x y i R1xy.

De nition 1 (Substructural frame)). A substructural frame S = (S; 1; R) is a set S, with 1 2 S, equipped with a ternary relation R such that: A1. x x (R1xx) A2. Rxxx A3. if R2(xy)zw, then R2(xz)yw

(if there is u s.t. Rxyu and Ruzw, then there is v s.t. Rxzv and Rvzw)

A4. if Rxyz, then Ryxz

A5. if Rxyz and x w, then Rwyz

A valuation in a substructural frame is de ned by v : Atom ! P(S). The valuation is required to satisfy the following heredity condition: for every p 2 Atom, if x 2 v(p) and x y, then y 2 v(p). The valuation extends to any formula of R, by the semantics of implication: { s j= A ! B i for all r,t such that Rsrt, if r j= A, then t j= B.

Heredity has to extend to complex formulas, and it is easy to check that it is the case. The concept of truth in a model is de ned by evaluating propositions at the particular designed state 1.

De nition 2 (Substructural model). A substructural model (S; v) is a substructural frame S equipped with a valuation v that satis es heredity on atoms. A formula A is true in a substructural model (S; v) i 1 j= A. Moreover, A is valid i it is true in every substructural model (S; v).

This semantics is su cient to show that the logic R is sound and complete with respect to substructural models. The motivation for introducing a ternary relation R is that it is needed for the semantics of implication: R relates the states that are making A ! B, A, and B hold. Although the semantics based on ternary relations has been criticized for its abstract nature, there is a number of possible intuitive reading of R, cf. [ 5 ]. One of the reading of R groups the rst two components of the relation, R[xy]z, and can be read as \the combination of information in x and y is in z". This interpretation has been analyzed in more details by Mares [ 13 ] in terms of situated inference. In very abstract terms, the valuation associates situations to formulas and s j= A holds whenever the information contained in situation s is relevant for A. The clause for implication states that A ! B holds at s if the information contained in s combined with the information contained in r produces information t that is relevant for B. We shall focus on this reading in order to provide a concrete cognitively-oriented interpretation of ternary relations semantics. 3

Conceptual spaces

Gardenfors [ 9 ] proposes a cognitive model of representations based on the notion of conceptual space. The theory of conceptual spaces is grounded on the notion of similarity: \[j]udgments of similarity (...) are central for a large number of cognitive processes (...) such judgments reveal the dimensions of our perceptions and their structures" ([ 9 ], p.5). Quality dimensions |e.g., temperature, weight, pitch, brightness|correspond to \the di erent ways stimuli are judged to be similar or di erent" ([ 9 ], p.6). They are modeled as (possibly discrete) sets of points that represent exact similarities between individuals. Those points represent the qualities of individuals: two individuals are located in the same point when they are (cognitively or empirically) indistinguishable with respect to the considered dimension, e.g., they have the same temperature, the same quality. Furthermore, dimensions have a geometrical structure that organizes their points according to the level of similarity between stimuli.

A set S of dimensions is integral if an individual located in one dimension is necessarily located also in all the other dimensions in S. For example, fhue, brightnessg is integral because if an individual has a hue it necessarily has a brightness (and viceversa). A set of dimensions is separable if it is not integral, e.g., fhue, pitchg. In Gardenfors's terminology, domains are maximal sets of integral dimensions. For example, the hue, chromaticness, and brightness dimensions that form the color domain fhue, chromaticness, brightnessg are integral and separable from any other dimension. Domains are central in the work of Gardenfors because, by means of the separability condition, they can be used to assign properties to individuals independently of other properties. For instance, in empirical terms, the weight and the color of an individual can be measured independently. The classi catory nature of the sensory systems is defended also by Matthen [ 14 ]. In these views, properties do not have a strong ontological connotation, they do not capture how the world is but how it appears to us through our sensory systems (or arti cial sensors).1 The properties and the conceptual spaces are understood relativistically: their structure depends on the underlying culture, on measurement methods and sensors (in science), or on interpretation of the behavior of subjects (in the case of phenomenology ). However the determinate-determinable relation, see [ 19 ], makes sense also in this case. Fully determinate properties, i.e., maximally resolving properties according to the sensors one dispose of, are represented by points in the domain. Vice versa, determinable properties, properties that abstract from the resolution of the sensor, are represented by regions, i.e., sets of points in the domain. For instance, `being scarlet' and `being crimson' can be seen as points, while `being red' as a region containing the previous two points. Natural properties are convex regions.

Conceptual spaces are de ned as collections of one or more domains and concepts are represented as regions in conceptual spaces. They are static theoretical entities \in the sense that they only describe the structure of representations" ([ 9 ], p.31). Natural concepts are sets of regions in di erent domains \together with an assignment of salience weights to the domains and information about how the regions in di erent domains are correlated" ([ 9 ], p.105).

Finally, an individual is represented as a point in a conceptual space, a vector of coordinates in the dimensions of the space. The points of the space can then be seen as the representations of possibilia, the set of all the possible individuals. 4

From conceptual spaces to substructural models

Our goal is to provide a cognitive interpretation of the relevant logic R. More speci cally, following the idea of Mares, we provide an interpretation of the substructural models of R (cf. De nition 2) in terms of the theory of conceptual spaces properly modi ed and simpli ed for our goal. In this section we informally present our idea while Section 5 contains the technical details.

We assume a nite and xed number N of (disjoint) domains. The ith domain is noted Di. The dimensions of the domains are not relevant for our task, then, to simplify our framework, we do not consider them. Consequently, we lose the original distinction between qualities and properties and all our domains are assumed as separable from the others. In addition, (fully) determinates, originally represented by points of a domain, are here singletons. In this way, both 1 Causation links between how the world is and how it appears to us can be considered. the determinates and the determinables (the regions) are elements of a domain Di. This move simpli es the formalization and is consistent with a mereological view of domains where determinates correspond to atomic regions (see [ 6 ]). Furthermore, is the only relation between regions we consider, no topological or geometrical relations are introduced.2 Finally, we represent the classi cation of objects3 under the properties in the domains but not their categorization under the concepts. Actually, concepts are not needed for our goal. This may appear as an oversimpli cation of the original theory of conceptual spaces. However, note that (i) our notion of domain is perfectly aligned with the original that can be seen as a limit case of the one of concept, i.e., regions in the domains are simple concepts; (ii) links between domains useful to de ne natural concepts are modeled via correlations (see below); (iii) the basic framework introduced here can be easily modi ed to take into account dimensions while categorization is an extension that could underline a new kind of implication (in addition to the ones we discuss in Section 6) to be addressed in future work.

The original idea of representing individuals as vectors of points (singletons in our case) each one belonging to a di erent domain is too strong for our aims. This view assumes a complete knowledge about the individuals, while we are interested in the acquisition of knowledge, information, or data, about individuals. We then weaken the original theory by allowing two kinds of partial knowledge about individuals: (i) the exact location into a domain is not known, i.e., one can only assign a determinable property to the individual, e.g., one knows it is red, but not the exact shade of red; (ii) one does not have any information about a given property, one does not even know if an individual is located in a given domain, e.g., if it has a color or not. Firstly, note that in (i) one may consider the maximal region of a domain. That means, for instance, that one only knows that the individual is colored. Secondly, (ii) contemplates the case of individuals that lack some properties, i.e., individuals are not necessarily located in all the domains. For instance, abstract individuals are not in space, while holes do not weight. However, we do not represent the impossibility to be located in a domain4 but only the lack of information (see below).

The assumption that the conditions of individuation of objects are purely conceptual has been criticized by Pylyshyn. In [ 16 ] he explores the idea that \[p]art of what it means to individuate something is to be able to keep track of its identity despite changes in its properties and location" ([ 16 ], p.33). The initial individuation and tracking of objects is not conceptual, i.e., it is not based on the classi cation under concepts, it is based on a lower level mechanism built into the visual system called FINST. We cannot enter here into the details of the approach. What is interesting for us is the link, provided by Pylyshyn, with the theory of object les [ 11 ]. One \can think of an object le as a way for informa2 Consequently, the structural relations of spaces, e.g., distances or orders, are here only used to build the taxonomy of properties. As discussed in Section 6, this structural information could be also used to represent relations among objects. 3 From here we use `object' and `individual' as synonymous. 4 That could be useful for approaching the semantics of negation. tion to be associated with objects that are selected and indexed by the FINST mechanism. When an object rst appears in view (...) a le is established for that object. Each object le has a FINST reference to the particular individual to which the information refers." ([ 16 ], p.38) The le allows us to group and maintain all the informations associated to the same individual (maybe acquired or updated through time), in particular \the one-place predicates that pertain to that object" ([ 16 ], p.39). An object le may be seen as an updatable frame-based description of an individual.5

Following this idea, we assume a xed set OB of objects that are described by objects les de ned as tuples ha; R1; : : : ; Rni where a 2 OB and Ri Di is a set of regions of Di. Firstly, object les are contextual, they depend on the chosen sets of domains and objects. Secondly, they collect all the known properties of a given object, i.e., all their known locations inside the domains. Intuitively, an object le represents the whole information about an object one has at a given stage, i.e., in an ontological perspective, the collection of states of a airs [ 3 ] relative to the same object. Thirdly, the Ri are sets of regions rather than simply regions. This extension of the original notion of location into domains is required to represent the process of making the acquired knowledge about an object explicit. As an illustrative example, assume that the color domain contains three subregions such that: scarlet red colored. In f = ha; fscarletgi the only explicit knowledge is the scarletness of a, whereas f 0 = ha; fscarlet; redgi adds the redness of a. By looking at the structure of the color domain, the knowledge in f 0 was already present in f , but only in an implicit form, i.e., f 0 is the result of an inference process, a cognitive abstraction activity. In mathematics, one can see this situation as the introduction of a new theorem. The theorem was implicit in the theory but, by making it explicit, we add, in some sense, information.6 Fouthly, we need to guarantee that object les contain consistent information, e.g., it is possible to have ha; fscarlet; redgi but not ha; fred; bluegi (if `being red' and `being blue' are disjoint). Finally, Ri = ; represents the total lack of information, discussed above, about the ith domain. In particular, f = ha; ;; : : : ; ;i represents just the existence of a 2 OB .

A situation can be seen as a set of object les for the objects OB with respect to the domains D1; : : : ; DN, i.e., as a the collection of states of a airs relative to the objects OB expressible with the same set of properties. Because the Ri in the object les may be the empty set or may represent determinable properties, in general the situations capture partial information about the objects. In particular, the situation 1 is the situation where all the object les have the form ha; ;; : : : ; ;i, i.e., the situation 1 represents only the terminological knowledge. 5 Note that we do not consider time, updating must be intended in terms of knowledge or information acquisition steps. 6 In an empirical scenario where one disposes of instruments with di erent resolutions, the previous situation could be seen as the acquisition of a new measure with a coarser resolution. We do not consider this interesting observational perspective where one could also acquire new measures with identical resolution, e.g., one would be able to distinguish ha; fscarlet; scarletgi from ha; fscarletgi.

Then, we model the reachability relation between situations in terms of updates of the information contained in a situation. Intuitively, given the situation s, t, and u, Rstu holds when the object les in u can be obtained by means of the ones in s and t through two possible types of updating: abstraction and correlation. Abstraction generalizes conceptualization within the same domain (e.g. from scarlet to red), i.e., it relies on the -structure of domains. Vice versa correlation individuates dependencies between distinct domains, for instance, it may relate colors and shapes. Induction, as understood by Gardenfors, is an example of correlation: \[t]he essential role of induction is to establish connections among concepts or properties from di erent domains " ([ 9 ], p.211). More specifically, the \inductive process corresponds to determining mappings between the di erent domains of a space. Using such a mapping, one can then determine correlations between the regions of di erent domains. The correlation between two properties F and G, expressed on the symbolic level by a universal statement of the form \all F s are Gs," would then just be a special case" ([ 9 ], p.228). We represent only the simple correlation between two properties by a pair of regions, the regions that represent these properties.

Finally, following the Routley-Meyer Semantics, the function of valuation v assigns to any atomic proposition a set of situations. 5

Conceptual spaces and relevant logic

We formally de ne the notions introduced in the previous section. A domain D is given by the set of all regions over a set of values D = fp1; : : : ; plg: D = P?(D) = P(D) n ;, where we exclude ; to avoid counterintuitive \null properties". In what follows, we x a set of N domains D1; : : : ; DN, denoted by D. We denote by ri1, ..., rin the elements of a domain Di. Elements rij are called regions of the domain. We sometimes use names for labeling regions. For instance, let D = fp1; p2; p3g, then D has as elements regions such as fp1g, fp2g, and fp1; p2g. We may then label scarlet = fp1g, crimson = fp2g and red = fp1; p2g.

Given a domain Di, we denote by Ri Di a set of regions in Di. (Tr2Ri r) 6= ;.

De nition 3 (Consistency). We say that Ri is consistent i if Ri 6= ;, then

Intuitively, as we will see, consistent sets of regions can be intended as nonexclusive properties that can in principle be ascribed to an object. In case the set of regions is empty, it represents the absence of information of type Di concerning that object. For instance, Ri = fscarlet = fp1g; red = fp1; p2gg is consistent, since the intersection of the regions in Ri is not empty, whereas Ri0 = fscarlet = fp1g; crimson = fp2gg is not. That is, we can say that an object is both scarlet and red, as for instance scarlet red, but we cannot say that it is both scarlet and crimson.

Moreover, we x a set OB = fa1; : : : ; alg of objects.

De nition 4 (Object les). An object le fa is a vector ha; R1; : : : ; Rni, where a 2 OB , Ri Di, such that each Ri is consistent.

The set of all object les depends on the choice of D and OB , so we denoted by OBF ODB .We can now introduce the de nition of situation.

De nition 5 (Situation). A situation s is a set of object les s

such that, for every object a 2 OB , there exist a unique object le fa 2 s. OBF DOB Then, we assume a number of correlations relating regions in di erent domains.

De nition 6 (Correlations). A set of correlations COR is a set of pairs of

regions (ril; rjm), where ri 2 Di and rm

l j 2 Dj , i; j 2 f1; : : : ; Ng and i 6= j. Moreover correlations satisfy the following conditions: Restricted transitivity if (ril; rjm) 2 COR, (rjm; rhn) 2 COR, and h 6= j, then (ril; rhn) 2 COR.

Correlation composition if (ril; rjm) 2 COR and rih ril, then (rih; rjm) 2 COR; if (ril; rjm) 2 COR and rjm rjh, then (ril; rjh) 2 COR.

Restricted transitivity states that if we can connect two regions in a number of steps, we can also connect them by composing the correlations in one single step. The condition h 6= j in the restricted transitivity excludes that we end up relating regions of the same domain. For instance, it prevents passing from (scarlet; round) and (round; crimson) to (scarlet; crimson). The rules for correlation composition state that if we correlate a concept with another, the correlation applies also to the subconcept of the rst one and to super-concept of the second one. For instance, if we say that red things are round, we also say that scarlet things are round. We do not put any further consistency constraint on correlations. The reason is that correlations are intended to represent factual, but not necessarily correct, mappings between concepts. For instance, we do not exclude from COR correlations that can end up in inconsistent outcomes, e.g. (round; scarlet) and (round; crimson). The point is that correlations express matters of fact, thus they are falsi able and in principle revisable. By contrast, conceptual information is xed and non-revisable.

We turn now to the interpretation of the ternary relation R in our setting. Intuitively, situations are related if they are reachable by means of an abstraction move or by means of a correlation link. Denote by fas the (unique) object le fa in situation s. Moreover, denote by Ras;i the set of regions of Di that in situation s are associated to object a. We are ready now to present our interpretation of the ternary relation in terms of reachability of situations.

De nition 7 (Reachability of situations). Let u, t and s situations in OBF DOB .

The situation u is reachable from t given s, i.e., Rstu, i :

R1. for all a 2 OB , for all Rau;i then Rau;i (Ras;i [ Rat;i);

i.e., all the data in s and t are imported in u; R2. for all a 2 OB , for all r 2 Rau;in(Ras;i [ Rat;i), r is obtained in one of the two following ways: Abstraction there exists r0 2 Ras;i [ Rat;i such that r0 r;

Correlation there exists r0 2 Ras;j [ Rat;h such that (r0; r) 2 COR.

Rstu imposes that the whole information in u is derived (by using conceptual knowledge or correlations) from the one in s and the one in t. R1 entails that the regions in s and t are preserved in u. R2 shows that all the new regions in u are derived from the ones in s and t by abstraction or by correlation. Note that, in principle, a situation could be updated through abstraction and correlation into something that is not a situation, i.e., into a set of inconsistent object les. For instance, suppose that (round; crimson) 2 COR and that scarlet and crimson are disjoint. Suppose D contains just two domains, e.g. colors and shapes. Thus, a situation s that contains ha; fscarlet; redg; froundgi can be updated, by means of the correlation (round; crimson), to a set of object les that contains ha; fscarlet; red; crimsong; froundgi, which violates consistency of the sets of regions that is required for object les. Since we are assuming that R is de ned on situations, i.e. sets of object les with consistent Ri-sets, the case above is excluded. This point shows a signi cant di erence between abstraction and correlation: abstraction guarantees consistency of the update, whereas correlation does not. This re ects the distinction between conceptual and factual knowledge. Once the conceptual relations are set and we have assumed that they are consistent, by abstraction we can only generalize on given data. By contrast, correlations introduce new data that may be inconsistent with previous ones.

We de ne the following relation of consistency between situations (Cst) De nition 8 (Consistent situations Cst). The two situations s and t are consistent, noted by Cst, i : C1. for i 2 f1; : : : ; Ng, Ras;i [ Rat;i is consistent (cf. De nition 3)

By means of De nition 7, we can infer that, if a situation u is reachable from t given s, then u is consistent both with s and with t and s is consistent with t.

Proposition 1. If Rstu, then Csu, Ctu, and Cst.

Proof. Assume Rstu, that entails by R1 that for every i and every object a, Ras;i [ Rat;i Rau;i. Thus, since Rau;i is consistent by de nition, then Ras;i [ Rat;i is consistent, so Cst. The other cases follows by noticing that Ras;i Rau;i and Rat;i Rau;i.

We conclude this paragraph by providing an interpretation of the element 1 of the substructural model. We de ne 1 as the situation in which we have no information about any object, i.e., 1 := fha; ;; : : : ; ;i j a 2 OBg. Every ha; ;; : : : ; ;i is an object le, that is, it satis es consistency. Moreover 1 is a situation, since for every object, there exist a unique object le in 1. 5.1

Conceptual spaces as models of R

We can now show that our view of situations provides a model of relevant logic. De nition 9 (Conceptual substructural model). A conceptual substructural model is given by (hS; COR; R; 1i; v), where hS; COR; R; 1i is a conceptual substructural frame: S is a set of situations de ned wrt. a domain D and a set of objects OB , COR is a set of correlation between regions of D, R S3 is a reachability relation, and 1 := fha; ;; : : : ; ;i j a 2 OBg. Moreover, v is a valuation that associates to atoms sets of situations, i.e., v : Atom ! P(S) such that heredity holds.

We only need to show that R and 1 satisfy the axioms of De nition 1.

Proposition 2. The reachability of situations R satis es axioms A1, A2, A3,

A4, and A5 of De nition 1.

Proof. We only show the details of the representative cases.

A1: R1ss. R1 trivially holds. R2 holds because Ras;i n(Ra1;i [ Ras;i) = ;. A2: If Rstu, then Rtsu. It is su cient to notice that the de nition of R is symmetric wrt. Ras;i and Rat;i.

A3: If R2(st)uw, then R2(su)tw. We need to show that if there exists an x such that Rstx and Rxuw, then there exists a y such that Rsuy and Rytw. Assume that there exists an x such that Rstx and Rxuw.

We show that there is a y such that Rsuy and Rytw. We set for every a and i, y Ra;i = Ras;i [ Rau;i.

Firstly, we show that Rsuy. We have that Ras;i [ Rau;i Ray;i = Ras;i [ Rau;i, thus R1 is ne. Since there is no other regions in Ray;i, we can conclude that R2 is also satis ed. Hence, Rsuy.

Then, we have to show Rytw. By assumption, Rax;i [ Rau;i Raw;i , thus we can deduce Ray;i [ Rat;i Raw;i. So R1 is satis ed.

y Suppose now that there is an r 2 Raw;inRa;i [Rat;i, that is r 2 Raw;inRas;i [Rau:i [Rat;i. Since by assumption Rxuw, every region r in w is obtained by abstraction or correlation from regions in x or u. If r is obtained by abstraction or correlation from a region in Rau;i, then we are done, since Rau;i Ray;i. If r is obtained from regions in x, then, by assumption Rstx, so r is obtained from regions that are either in s or t. We approach the following cases: (i) r is obtained by correlation from an r0 2 Rax;i and r0 is obtained from correlation from an r00 2 Ras;i. This means that (r0; r); (r00; r0) 2 COR, thus, by restricted transitivity, (r00; r) 2 COR. Therefore, if r 2 Raw;i n Ras;i [ Rau:i [ Rat;i, then there is an r00 2 Ras;i such that (r00; r) 2 COR, thus we conclude; (ii) r is obtained by correlation from an r0 2 Rax;i and r0 is obtained by abstraction from an r00 2 Ras;i. This means that r00 r0 and (r0; r) 2 COR, thus, by the rst rule of correlation compoistion, (r00; r) 2 COR. Therefore, for r 2 Raw;i n Ras;i [ Rau:i [ Rat;i, there is an r00 2 Ras;i such that (r00; r) 2 COR and we conclude again; (iii) r is obtained by abstraction from r0 in x and r0 is obtained by abstraction from r00 in s. Then r can be obtained by abstraction from r00 and we are done; (iv) r is obtained by abstraction from r0 in x and r0 is obtained by correlation (r00; r0) 2 COR from r00 in s. In this case, r0 r and (r00; r0) 2 COR, thus by the second rule of correlation composition we infer (r00; r) 2 COR and we conclude. Therefore, also R2 is satis ed, therefore Rytw.

A4: Rsss holds, since R1 trivially holds and Ras;i n(Ras;i [ Ras;i) = ;. A5: If Rstu and w s, then Rwtu. Recall that w s is de ned by R1ws. We show only the following case. Suppose r 2 Rau;i n Raw;i [ Rat;i and that r is obtained by means of correlation from r0. In case r0 2 Raw;i [ Rat;i, we are done. Otherwise, by assumption r0 2 Ras;i and (r; r0) 2 COR. Since R1ws, there are two cases. Firstly, there exists r00 such that (r00; r0) 2 COR. By restricted transitivity, we conclude that for r 2 Rau;i n Raw;i [ Rat;i, there exists an r00 in Raw;i such that (r00; r) 2 COR. Secondly, r0 is obtained by abstraction from r00 in Raw;i, in this case by correlation composition, we conclude.

It is important to notice that the provided interpretation in terms of situations does not trivialize the substructural model, namely R does not provide a model of intuitionistic or classical implication. To see that, we show that monotonicity does not hold in conceptual substructural models. In axiomatic terms, monotonicity corresponds to the validity of A ! (B ! A). In semantic terms, it corresponds to the following constraint on the ternary relation [ 7 ]: f1

Rstu ) R1su Consider a simple example where s = fha; ;i; hb; fscarletgig, t = fha; fscarletgi; hb; ;ig, and u = fha; fscarlet; redgi, hb; fscarletgig. In this case, although Rstu, neither R1su nor R1tu hold, i.e., both the information in s and t is needed for u. Therefore, (f1) does not hold in every conceptual substrctural model, thus A ! (B ! A) is not valid. 6

Conclusions and future work

We presented a concrete instantiation of the ternary relation model of the relevant logic R that is grounded on the framework of conceptual spaces.Our instantiation of the Routley-Meyer semantics provides a number of reasons to interpret relevant implication in terms of cognitively aware updates of knowledge. Besides the logical contribution, we believe that both the notion of situation and the one of reachability between situations provide a useful framework for separating conceptual and factual knowledge and for modeling knowledge acquisition. However, the cognitive plausibility of the interpretation of the inferential mechanism we proposed still lacks an empirical assessment.

Future work concerns two directions. Firstly, notice that the proposed framework provides an interpretation only to atomic propositions that reduce to the assignment of a (unary) property to an object, it does not consider relations among objects. The extension to relations de nable in terms of relations among intrinsic properties of the relata is quite trivial.7 Using conceptual spaces, this kind of relations can be represented by means of higher level properties (see [9, 7 Even though one has to decide whether relational information is encoded in the objects les|e.g., if REL(a; b) holds then one needs to add this information in both the object- le relative to a and b|or outside them. sect.3.10.1]). For instance, suppose to have the dimension length structured by the order relation . The relation shorter than can be represented by a region in the space of the pairs of length-values, i.e. the region of all pairs (l1; l2) such that l1 l2. Thus, an object x is shorter than an object y if the pair (length of x, length of y) belongs to this region. Gardenfors seems to suggest that this approach is general enough to represent all the (binary) relations: \[a] relation between two objects can be seen as a simple case of a pattern of the location of the objects along a particular quality dimension" [9, p.93]. However, some structural relations, e.g., part-whole relations, seem to require really complex spaces founded on several quality dimensions (see [ 17 ]). More importantly, it is not clear to us how some relations like eat or married to can be reduced to intrinsic properties of relata. Similarly for relational categories [ 10 ], i.e., properties that are de ned in relational terms, e.g., a carnivore is an animal that eats meat.

Secondly, in De nition 7, we have distinguished two possible ways of updating the information contained in a situation: abstraction and correlation. We have suggested that, intuitively, they correspond to two distinct types of processes: the rst abstracts from already given data, the second allows to indirectly discover new data, a sort of indirect measurements. Contrast the following sentences: i: \If a is scarlet, then a is red" and ii: \If a is scarlet, then a is round." In our model, (i) updates a situation s that contains, let say, ha; fscarletg; ;i into a situation t that contains ha; fscarlet; redg; ;i whereas (ii) is an update from s to a situation t0 that contains ha; fscarletg; froundgi. Both these updates add information that was implicit, but they qualitatively di er because the update in (i) impacts the same domain while the one in (ii) impacts a different domain. Furthermore, our intuition is that (i) holds just in virtue of `the scarletness of a', while (ii) holds in virtue of both `the scarletness of a' and `the roundness of a' (assuming that `being scarlet' and `being round' are both fully determinate properties). In terms of truth-makers (see [ 4 ]) this means that the two propositions `a is scarlet' and `a is red' share the same truth-maker (`the scarletness of a'). By contrast, the two propositions `a is scarlet' and `a is round' need two di erent truth-makers, i.e., only the second inference reveals the existence of an implicit truth-maker. This would suggest that the rst kind of reasoning, the abstraction, is a purely mental process that does not need veri cation. By contrast, the second kind of reasoning, the correlation, needs additional validation in terms of truth-makers. Actually this provides a partial justi cation of the asymmetry between the required consistency of the conceptual knowledge vs. the possibility to have inconsistent correlations. An interesting question is whether it is possible to distinguish the two process in terms of inferential patterns, that is, we ask whether it is meaningful to de ne two kinds of implications, one corresponding to the sole updating by abstraction, and one corresponding to updating by correlation. We leave for future work the axiomatization of these two types of implications that, in our framework, can be characterized by two distinct reachability relations: one that only permits updates by abstraction, the other that only permits updates by correlations.

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