Title: Objective Logic of Consciousness 1 Authors: Venkata Rayudu Posina and Sisir Roy 2 ORCID IDs: 0000-0002-3040-9224 and 0000-0002-5328-2713 3 Article Category: Original Research Article 4 Keywords: Category theory; Contradiction; Memory; Model; Negation; Perception; Sensation; 5 Theory; Truth value object. 6 Word Count: 8777 7 8 Affiliations: SR is Senior Homi Bhabha Visiting Professor, National Institute of Advanced 9 Studies, Bengaluru. VRP is an Independent scientist. 10 Emails: posinavrayudu@gmail.com; sisir.sisirroy@gmail.com 11 2 Abstract 12 We define consciousness as the category of all conscious experiences. This immediately raises 13 the question: What is the essence in which every conscious experience in the category of 14 conscious experiences partakes? We consider various abstract essences of conscious experiences 15 as theories of consciousness. They are: (i) conscious experience is an action of memory on 16 sensation, (ii) conscious experience is experiencing a particular as an exemplar of a general, (iii) 17 conscious experience is an interpretation of sensation, (iv) conscious experience is referring 18 sensation to an object as its cause, and (v) conscious experience is a model of stimulus. 19 Corresponding to each one of these theories we obtain a category of models of conscious 20 experiences: (i) category of actions, (ii) category of idempotents, (iii) category of two sequential 21 maps, (iv) category of brain-generalized figures, and (v) functor categories with intuition as base 22 and conceptual repertoire as exponent, respectively. For each theory of consciousness we also 23 calculate its truth value object and characterize the objective logic intrinsic to the corresponding 24 category of models of consciousness experiences. 25 3 Introduction 26 What is consciousness? Consciousness, according to Koch, ―is everything you experience. It is 27 the tune stuck in your head, the sweetness of chocolate mousse, the throbbing pain of a 28 toothache, the fierce love for your child and the bitter knowledge that eventually all feelings will 29 end‖ (Koch, 2018, p. S9). This raises two foundational questions: 30 1. What is the nature of conscious experiences? 31 2. What is the nature of consciousness? 32 How are we to think of the totality of conscious experiences i.e., consciousness? How are we to 33 think of the constituents of consciousness i.e., conscious experiences? One obvious answer: 34 Conscious experiences are objects of the category of all conscious experiences and 35 consciousness is the category of conscious experiences. In other words, every conscious 36 experience has the essence of the category of conscious experiences, whatever the essence(s) 37 maybe. This characterization is in the spirit of asserting that a chair is an object of the category 38 of chairs. 39 Let us consider a visual experience: a face. A first-order approximation would represent the 40 experience as a point in a feature-space or as a set of features i.e., Face = {eyes, nose, mouth} 41 (Fodor, 1998). Sensory features are obviously structured, unlike the structureless elements of sets 42 (Lawvere and Rosebrugh, 2003, p. 1). Equally importantly, sensory features of a visual object 43 are related to one another in specific ways resulting in a cohesive object that is conscious 44 experience, which cannot be modelled as a set with its zero internal cohesion (Lawvere and 45 4 Schanuel, 2009, p. 146). Elementism, notwithstanding the Gestalt demonstrations (Albright et 46 al., 2000, p. S34), continues to be the default terminology as in analysing ―perceptual experience 47 into a collection of simple sensory elements‖ (Albright, 2013a, p. 19). Along similar lines, mind 48 is defined as a set of brain functions (Bunge, 1981, p. 68; Kandel, 2013, p. 546). The claim that 49  mind is a set' is repeatedly asserted in the textbook Principles of Neural Science (Kandel et al., 50 2013, p. 5, 334, 384), which takes on added significance in light of its pedagogical value in 51 training neuroscientists. Of course, this terminology does not reflect any failure to recognize that, 52 in terms of the above example of face perception, the constituent eyes, nose, and mouth, unlike 53 the structureless elements of a set, are figures of various shapes; these figures constituting a face 54 are related to one another in specific ways (cf. Croner and Albright, 1999). Nevertheless, it does 55 highlight the absence and the significance of having a conceptual repertoire that fits the reality of 56 conscious experiences. Here we put forward mathematical category as a construct suited for the 57 study of consciousness (Lawvere, 1994; Lawvere and Schanuel, 2009, p. 21, 135-148). In line 58 with the commonplace understanding of the notion of category, a mathematical category consists 59 of objects all of which partake in the essence that is characteristic of the category; since every 60 object of the category partakes in the essence, the transformations of objects preserve the essence 61 (e.g. in the category of dogs, a transformation of an young dog into an old dog preserves the 62 ―dogness‖). We find that defining conscious experience as an object of the category of conscious 63 experiences, instead of as cohesion-less set of structure-less elements, provides the conceptual 64 repertoire-basic shapes, figures, and incidence relations-needed to reason about the essence of 65 conscious experiences and the essence-preserving transformations of conscious experiences. 66 67 5 Theory of Conscious Experiences 68 What is the essence of conscious experiences? Continuing with our example of face perception, 69 an experience of a face can be said to consist of figures of various shapes: two eye-shaped 70 figures, one nosed-shaped figure, and one mouth-shaped figure. Of these shapes, we can say that 71 eye, nose, and mouth are the basic shapes, and their incidence relations determine the mutual 72 relations between various basic-shaped figures constituting the face (Lawvere and Schanuel, 73 2009, pp. 82-83, 250-253, 369-371). When considering conscious experience in general, we may 74 treat sensory features (e.g. colour, shape), modalities (visual, tactile, etc.), and emotion, among 75 others, as basic shapes. For illustration, anger (in conscious experience) can be considered as an 76 emotion-shaped figure (in the experience) just as redness can be thought of as a colour-shaped 77 figure. The mutual relations between basic shapes, say, emotion and colour, determine the 78 mutual relations between figures of the corresponding shapes (anger and redness). 79 Basic shapes along with their incidence relations constitute the abstract essence or theory of the 80 category of conscious experiences (Lawvere, 2003, p. 215, 217; Lawvere, 2004a, pp. 10-12; 81 Lawvere and Rosebrugh, 2003, pp. 154-155, 235-236; Lawvere and Schanuel, 2009, pp. 149-82 151, 369-371). First, every experience has the essence [of the category of conscious experiences] 83 given by the basic shapes and their incidence relations. Next, every experience can be 84 represented as a structure formed of basic-shaped figures and their mutual relations induced by 85 the incidences of basic shapes (see Fig. 4 in Posina, Ghista, and Roy, 2017). Since every 86 experience has the essence of experiences, transformations of experiences are required to 87 preserve the essence of experiences, and as such are natural transformations (Lawvere and 88 Schanuel, 2009, p. 378). Geometrically speaking, natural transformations  do not tear' the 89 6 structure transformed (ibid, p. 210). Philosophically, a natural transformation is: Becoming 90 consistent with Being (e.g. biological growth; Posina, 2016). 91 What are we to make of the totality of all conscious experiences along with their essence-92 preserving transformations? Objects along with essence-preserving morphisms of objects form a 93 category. With experiences as objects [with a given structural essence] and essence-preserving 94 transformations of experiences as structure-preserving morphisms of objects, consciousness-the 95 totality of conscious experiences-can be construed as a category of conscious experiences. Note 96 that any experience can remain the same (identity transformation). If I went from sad to happy 97 and from happy to detached, then I went from sad to detached (composition of transformations of 98 experiences). Along these lines, other axioms and laws, which are required to be satisfied in 99 order for us to talk about a category of experiences, can be verified (Lawvere and Schanuel, 100 2009, p. 21, 149-160). Within this categorical framework, the structure of consciousness is an 101 external reflection of the structural essence of conscious experiences (Lawvere, 1972, p. 10). 102 More immediately, a category embodies a mode of cohesion (Lawvere and Schanuel, 2009, p. 103 146), which is the most basic attribute of conscious experience. For example, parts of a body 104 (hands, legs, etc.) have a mode of cohesion, which is different from the mode of cohesion of 105 parts of a perceptual object (colour, shape). Note that  part' is both itself and its relationship to 106 the whole (Lawvere, 1994, p. 53). Formally, a part of an object is not merely a subobject, but a 107 monomorphism specifying the inclusion of the subobject into the object (Lawvere and Schanuel, 108 2009, p. 335). 109 As illustrations of theory of a category and its basic shapes, we present simple theories 110 (abstract essences) of conscious experiences (in the spirit of Lawvere, 1999). More explicitly, the 111 7 mathematical method, according to F. William Lawvere, ―consists of taking the main structure 112 [of an object] by itself as a first approximation to a theory of the object, i.e. mentally operating as 113 though all further structure of the object simply did not exist‖ (Lawvere, 1972, pp. 9-10). An 114 example of an abstract theory of conscious experiences is  particular as an exemplar of a general' 115 (cf. categorical perception; Albright, 2013b, pp. 628-630; Grossberg, 1976), whose models form 116 a category of idempotents (Lawvere and Schanuel, 2009, pp. 99-106), with exemplar as its basic 117 shape. The truth value object of the category of idempotents has three global truth values. With 118  interpretation of sensation' (Albright and Stoner, 1995; Croner and Albright, 1999; Schlack and 119 Albright, 2007) as a theory of conscious experiences, we obtain a category of two sequential 120 processes as the category of conscious experiences. Here, the basic shapes are physical stimuli, 121 neural sensation of stimuli, and conscious interpretation of sensation. With conscious experience 122 as an object of the category of two sequential functions, we find that the objective logic intrinsic 123 to consciousness has four truth values (Posina, Ghista, and Roy, 2017, pp. 172-174). We also 124 consider  action of memory on sensation' (Albright, 2012, Fig. 5, 8; Hopfield, 1982; Lawvere 125 and Schanuel, 2009, p. 218),  referring sensation to an object as its cause' (Albright, 2015, p. 22; 126 Lawvere and Rosebrugh, 2003, pp. 125-126, 148-152), and  model of stimulus' (Chalmers, 127 2006; Posina, Ghista, and Roy, 2017) as theories of conscious experiences. 128 Given a category of experiences, how do we abstract the theory (essence) of experiences? 129 Theorization begins with measurements of properties of the objects of the given category. 130 Oftentimes, we find that there is small subcategory of properties (and their determinations) 131 within the category of all properties that constitutes the abstract essence shared by all objects of 132 the given category. This abstract essence in which every object of a given category partakes is 133 8 the theory of the given category (Lawvere, 1994, pp. 44-47; Lawvere and Rosebrugh, 2003, pp. 134 154-155; Lawvere and Schanuel, 2009, pp. 149-150; see also Fig. 5 in Posina, Ghista, and Roy, 135 2017). In geometric terminology, we consider a subcategory of basic shapes and their incidence 136 relations, and examine if figures with objects in the subcategory as shapes are adequate to 137 completely characterize every object of the category and tell apart transformations between 138 objects (Lawvere, 1994, p. 49; Lawvere and Schanuel, 2009, pp. 369-371). In the following we 139 focus on the calculation of truth value objects corresponding to various theories of conscious 140 experiences and the subsequent characterization of the objective logic intrinsic to various 141 categories of models of conscious experiences (ibid, pp. 335-357). 142 143 Action of Memory on Sensation 144 Conscious experience of a given physical stimulus can be thought of as an action of memory on 145 the sensation elicited by the stimulus (for a vivid illustration of the action of memory on 146 sensation, see Fig. 5 and 8 in Albright, 2012). A formalization of conscious experience as an 147 action of memory on sensation is provided by Hopfield (1982). Here sensation S is a 1 × n 148 feature vector, with each one of the elements of the vector S representing the activity of each one 149 of the n feature-selective neurons. Memory M, or the n × n synaptic weight matrix, is a result of 150 associative learning, and can be expressed as a product of the sensation S with its transpose S T , 151 i.e. 152 M = S T × S 153 9 Conscious experience C(S) corresponding to sensation S is: 154 C(S) = S × M. 155 For a given memory, conscious experiences corresponding to various sensations have the 156 structure of idempotents (as discussed in detail in appendix A1). Categorical perception, wherein 157 particulars (stimuli) are perceived as exemplars of a general (category; Albright, 2013b, pp. 628-158 630), also has the structure of idempotents. The abstract essence or theory of the category of 159 idempotents consists of one basic shape: exemplar, along with an idempotent endomap as the 160 structural map. Unlike the classical Boolean logic of sets, we find that the truth value object of 161 the category of idempotents consists of three truth values. Also, two dual forms of negation-162 not, non-can be defined (Lawvere, 1986, 1991). We find that double negation can be greater or 163 less than identity depending on the exact nature of negation. Furthermore, the category of 164 idempotents admits logical contradiction (Lawvere, 2003, p. 214-215; or boundary operation 165 defined as the intersection of a part with its negation; Lawvere, 1994, p. 48; Lawvere and 166 Rosebrugh, 2003, p. 201). 167 168 Interpretation of Sensation 169 Conscious experience involves two sequential processes of sensation (of stimulus) followed by 170 interpretation of the sensation. A classic illustration of the two sequential processes involved in 171 conscious experience is R. C. James's image (Miller, 1999). When looking at the image one 172 initially sees black and white blobs of various sizes and shapes, which subsequently, in light of 173 10 the concept DALMATIAN, is perceptually interpreted as a dog. That conscious experience is 174 mediated by the two processes of sensation followed by interpretation is well-established in 175 various perceptual modalities (e.g. Albright and Stoner, 1995; Croner and Albright, 1999). Thus 176 the abstract theory of consciousness consists of three basic shapes i.e. objects (Physical Stimuli, 177 Neural Codes, and Conscious Experiences) and two incidence relations i.e. maps (sensation and 178 interpretation) organized as shown below: 179 Physical Stimuli –sensation→ Neural Codes –interpretation→ Conscious Experiences 180 The truth value object of the category of models of conscious experiences of the above theory of 181 consciousness consists of four global truth values (Fig. 6c in Posina, Ghista, and Roy, 2017 182 depicts the internal diagram of the truth value object; see Appendix A2 in Posina and Roy, 2018 183 for the calculation of the truth value object; see also Linton, 2005). 184 185 Brain-generalized Figures 186 Everyday experience of effectively interacting with objects of conscious experience indicates 187 that conscious experience of objects is recovery of the objects based on the sensation elicited by 188 the objects (i.e. constructing an object isomorphic to the cause of sensation; Albright, 2015, p. 189 22; Lawvere and Rosebrugh, 2003, pp. 125-126, 148-152). In reconstructing objects in conscious 190 experience, we encounter the possibility of not only the commonplace isomorphism between 191 objects and conscious experience of objects, but also illusory conscious experiences with no 192 correspondent underlying objects. We now present a mathematical framework rich enough to 193 11 capture both veridical perception and illusions (Lawvere, 2004b). Given a stimulus A, we define 194 sensation p as a brain V-valued property of the stimulus A, i.e. 195 p: A → V 196 Next, we consider brain V-valued properties of sensations, i.e. 197 q: V A → V 198 which may be considered as a special case of interpretation of sensation, i.e. a neural measure of 199 sensation. Perceived element is defined as a brain V-generalized point of A satisfying naturality 200 conditions (discussed in detail in appendix A2). Within this mathematical framework, we find 201 that defining brain V as neuronal states of  firing' and  not firing', i.e. as a two-element property 202 type, can give rise to illusions. However, defining brain V in terms of all possible changes in 203 neuronal firing rate, i.e. as a three-element set V = {decreased firing rate, constant firing rate, 204 increased firing rate}, ensures illusion-free perception, or isomorphism between elements of a 205 stimulus set and perceived elements. As we have been emphasizing, physical stimuli, neural 206 sensations, and conscious experiences are much more structured than structureless sets; as such 207 we are working on refining the mathematical framework to accommodate conscious experience 208 defined as brain-generalized figure of stimuli (for basic shapes and their incidences of the 209 categories corresponding to physical stimuli and neural sensation). 210 211 212 12 Model of Physical Stimulus 213 Of the various abstract essences (theories) of conscious experiences, the most basic 214 characterization of conscious experience is: conscious experience of a stimulus is a model of the 215 stimulus (Chalmers, 2006). This immediately suggests functorial semantics, which provides a 216 mathematical account of constructing models of particulars, as an abstract theory of conscious 217 experience (Lawvere, 1994, 2004a). Given a category of physical stimulus, a model or conscious 218 experience of the stimulus is calculated by abstracting the essence (mental concepts) of its brain-219 valued properties i.e. sensation. Thus abstracted mental concepts are then interpreted into a 220 background category of intuition to obtain models of the physical stimuli or conscious 221 experiences. The objective logic of conscious experiences construed as functor categories, with 222 intuition as base and mental concepts as exponent, is not classical (Posina, Ghista, and Roy, 223 2017). Furthermore, subjectivity (understood as viewpoint; cf. Sen, 1993) is captured by the 224 framework of functorial semantics; more specifically, the mathematical construct of monad 225 determines how a category of particulars is [subjectively] generalized into the adjoint pair of 226 functors: mental concepts (theories) and conscious percepts (defined as functorial interpretation 227 of concepts into a background of intuition or models; Eilenberg and Moore, 1965; Lawvere, 228 1994, 2004a). As such, functorial semantics is the objective logic (as defined in Lawvere, 1994, 229 p. 43) of consciousness. 230 231 232 233 13 Conclusions 234 We defined conscious experience as an object of the category of conscious experiences, which 235 aligns with the intuitions engendered by our everyday experiences with objects (cf. a table is an 236 object of a category of tables). It is fascinating to note that the most advanced scientific 237 understanding of object (as an object of a category of objects; Lawvere, 2015) is in accord with 238 our ordinary experience. The category of conscious experiences provides the conceptual 239 repertoire-basic shapes, figures, and incidences-needed to develop an adequately explicit 240 theory of conscious experience. Given that the objective logic intrinsic to conscious experiences 241 is not classical for a variety of abstract essences of consciousness that we considered, it would be 242 interesting to compare the objective logic of consciousness with quantum logic that was found to 243 better account for cognition compared to the classical logic (Roy, 2016). 244 245 Acknowledgments 246 We dedicate our paper to the memory of Professor B. V. Sreekantan. VRP is truly grateful to 247 Professors: Andrée C. Ehresmann and F. William Lawvere for invaluable help in learning 248 category theory. Thanks also to Professor Narasimhan Marehalli and Dr. Ruadhan O'Flanagan 249 for helpful discussions. One of the authors (Sisir Roy) greatly acknowledges Homi Bhabha Trust 250 for financial assistance for this work. 251 14 Appendices 252 A1. Category of Idempotents 253 Let's consider, within the framework of Hopfield model (1982), a neural network consisting of 254 two neurons coding for two features. For a given stimulus, a neuron can respond with a decrease, 255 constant, or increase in its firing rate. So, we have nine 1 × 2 vectors as possible sensations S: 256 S1 = [-1 -1] 257 S2 = [-1 0] 258 S3 = [-1 +1] 259 S4 = [0 -1] 260 S5 = [0 0] 261 S6 = [0 +1] 262 S7 = [+1 -1] 263 S8 = [+1 0] 264 S9 = [+1 +1] 265 We define memory as a synaptic weight matrix M with entries given by associative learning: 266 mij = si × sj 267 15 where (subscripts) i, j index the two neurons. For example, memory M1 of sensation S1 is: 268 M1 = S1 T × S1 269 (T denotes transpose), which is a 2 × 2 weight matrix. Substituting the values of sensation S1, we 270 find: 271 272 Next, we define perception P as an action of memory M on sensation S: 273 P = S × M 274 With memory M = M1, we find the percepts resulting from the action of memory M1 on the nine 275 sensations: 276 P1 = S1 × M1 = 2[-1 -1] 277 P2 = S2 × M1 = [-1 -1] 278 P3 = S3 × M1 = [0 0] 279 P4 = S4 × M1 = [-1 -1] 280 P5 = S5 × M1 = [0 0] 281 P6 = S6 × M1 = [+1 +1] 282 P7 = S7 × M1 = [0 0] 283 1 1 1 1 M1 = 16 P8 = S8 × M1 = [+1 +1] 284 P9 = S9 × M1 = 2[+1 +1] 285 Note that P1, P5, and P9 are fixed-points (with, say, S1 as initial state and P1 (= S1) as final state in 286 the language of dynamical systems), while sensations S1, S2, and S4 perceived as P1; sensations 287 S3, S5, and S7 perceived as P5; sensations S6, S8, and S9 perceived as P9. The dynamics of action 288 of memory on sensation has the structure of idempotents as displayed below: 289 290 291 292 293 The above neural network can be formalized with action P of memory M on sensation S defined 294 as a map: 295 P: S × M → S 296 For a neural network of N neurons, memories M are N × N matrices, which when thought of 297 endomaps N → N, and with matrix multiplication as composition (of endomaps; MacLane, 1998, 298 p. 11), form a monoid (Lawvere and Rosebrugh, 2003, p. 167). The N × N matrix with all 299 diagonal elements as 1 serves as monoid identity (ibid, p. 77), while the N × N matrix with all 300 S4 S2 S1 S8 S6 S9 S7 S5 S3 17 entries as 0 is a constant C, since C × M = C for all memories M. Since this is the only constant 301 of the monoid of memories, the category of actions (Lawvere and Schanuel, 2009, p. 360): 302 P: S × M → S 303 forms a topos of idempotents (ibid, p. 367), i.e. a category with truth value object. The truth 304 value object of the category of idempotents has three truth values: 305 V = {false, u, true} 306 equipped with an idempotent endomap: 307 v: V → V 308 defined as 309 v(false) = false, v(u) = true, and v(true) = true. 310 Categorical perception, wherein a particular (rose) is perceived as an exemplar of a general 311 (flower), also has the structure of idempotents (see also Lawvere and Schanuel, 2009, p. 106): 312 313 314 315 Given the splitting of idempotent endomaps into section-retract pairs, the category of 316 idempotents can also be characterized in terms of the opposite pair of section-retract maps, 317 lily rose flower 18 wherein the basic shapes are particulars and generals, while sorting and exemplifying are the 318 incidence relations between the two basic shapes (Kathpalia, Posina, and Nagaraj, 2017). 319 We now calculate the truth value object of the category of idempotents. Truth value object of 320 the category of idempotents is an object of the category of idempotents (just as in the case of 321 sets, where a two-element set 2 = {false, true} is the truth value object of the category of sets). 322 An object of the category of idempotents (modelled in the category of sets) is a set A equipped 323 with an idempotent endomap 324 a: A → A 325 satisfying 326 a ◦ a = a. 327 where  ◦' denotes composition of maps. A map f from an idempotent a to an idempotent b 328 (where b: B → B satisfying b ◦ b = b) is a function 329 f: A → B 330 satisfying 331 f ◦ a = b ◦ f. 332 The truth value object of a category is defined as an object representing every part 333 (monomorphism) of any object of the category. The truth value object can be calculated in terms 334 of the inverse images of parts of basic shapes along incidence relations (structural maps). Basic 335 19 shapes along with incidence relations constitute the abstract essence or the theory of a given 336 category. What are the basic shapes and their incidences constituting the theory of category of 337 idempotents? The theory of idempotents consists of a generic idempotent E, along with an [non-338 identity] idempotent endomap e on E (as shown below): 339 340 341 342 343 The basic shape E along with the incidence relation e together constitute the theory subcategory 344 E of the category of idempotents. The basic shape E has three parts false, u, and true (shown 345 below), which, by the definition of truth value object, correspond to three E-shaped figures (truth 346 values) in the truth value object of the category of idempotents. 347 348 349 350 The idempotent endomap v: V → V, where V = {false, u, true}, is given by the inverse images of 351 the three parts along the incidence relation e: E → E. The inverse image of the part false along e 352 is false; the inverse image of the part u along e is true; and the inverse image of the part true 353 E E e E E E true G E u 0 E false 20 along e is also true. Putting it all together we obtain the truth value object v (depicted below) of 354 the category of idempotents. 355 356 357 Based on the truth value object v, logical operations negation, AND, and OR can be defined. 358 Given a part P of an object A (of the category of idempotents), the familiar negation not(P) is 359 defined as the largest part of A whose intersection with P is empty. Dually, another negation 360 non(P) is defined as the smallest part of A whose union with P is the whole object A. In the 361 category of sets, both negation operations-not, non-are identical. However, in the category of 362 idempotents, these two negation operations can give different results (as shown below). The 363 negation operation non allows logical contradiction: P AND non(P), which is boundary in 364 geometric terminology (Lawvere, 1986, 1991; Lawvere and Rosebrugh, 2003, p. 201). 365 366 367 368 369 Furthermore, double negation can be bigger or smaller than identity depending on the nature of 370 negation as shown below: 371 true v: V → V u false P non(P) not(P) non(P ) P non(P) AND P 21 372 373 374 375 376 A2. Illusion-free Perception 377 Consider a set A and a set V of values of properties of A. A function 378 p: A → V 379 is a V-valued property of the set A. The set of all V-valued properties of A (or more broadly, the 380 set of all functions from the domain set A to the codomain set V) is the map set V A . The set V A 381 of properties, in turn, has properties, which are functionals 382 q: V A → V 383 where the functional q is a V-valued property of the set V A of all V-valued properties of A. Our 384 objective is to reconstruct the set A from the set V (VA) of properties of its V-valued properties 385 (Lawvere and Rosebrugh, 2003, pp. 125-126, 148-152). Towards this end, we define generalized 386 point. First recollect that a point of a set A is a function 387 a: 1 → A 388 P not(not(P)) not(P) P non(P) non(non((P)) 22 where 1 = {*} is the terminal set of the category of sets, i.e. a single-element set; hence points of 389 a set correspond to its elements. Since elements of set completely characterize a set, we define 390 generalized point so as to establish a 1-1 correspondence between points and generalized points 391 (perceived elements). 392 A V-generalized point of the set A is a functional 393 q: V A → V 394 such that for each V-valued property of A 395 p: A → V 396 and for every endomap of the property type V 397 e: V → V 398 the following equation is satisfied: 399 q (e ◦ p) = e (q (p)) 400 where  ◦' denotes composition of functions and parentheses denote evaluation of functions. 401 Let's first consider the left-hand side 402 q (e ◦ p) 403 The V-valued property of A 404 p: A → V 405 23 is an element of the set V A of all V-valued properties of A. Furthermore, since the codomain V 406 of p is same as the domain V of 407 e: V → V 408 we can compose them to obtain a composite 409 e ◦ p = A → V → V = A → V 410 which is also an element of the set V A of all V-valued properties of A; and hence when the 411 functional 412 q: V A → V 413 is evaluated at  e ◦ p' gives an element  v' of the set V of values 414 q (e ◦ p) = v 415 Let us now consider the right-hand side 416 e (q (p)) 417 Once again 418 p: A → V 419 is a V-valued property of A, i.e. an element of the set V A of all V-valued properties of A. Hence 420 evaluating the functional 421 q: V A → V 422 at p gives a value 423 24 q (p) 424 which is an element of the set V of values. Hence the endomap 425 e: V → V 426 when evaluated at the element  q (p)' of domain set V gives an element of V, i.e. 427 e (q (p)) = v' 428 Summing up, the functional 429 q: V A → V 430 is a V-generalized point of A (perceived element of A) if for every 431 p: A → V 432 and for every 433 e: V → V 434 we find that 435 q (e ◦ p) = e (q (p)) 436 or in terms of our above example 437 v = v'. 438 25 Returning to our main objective, i.e., establishing an isomorphism between points and 439 perceived elements (generalized points) involves finding a property type V such that for every 440 set A, the V-generalized points of A, i.e. the functionals 441 q: V A → V 442 satisfying 443 q (e ◦ p) = e (q (p)) 444 for every 445 p: A → V 446 and for every 447 e: V → V 448 are in 1-1 correspondence with the points 449 a: 1 → A 450 of the set A. 451 We give an example of perceived elements (generalized points) corresponding to points of a set. 452 Let A = {a1, a2} be the object of our investigation and V = {0, 1} be the property type. There are 453 two points in A 454 a1: 1 → A 455 and 456 26 a2: 1 → A 457 Now let's calculate the number of generalized points. First, there are four functions from A to V, 458 i.e., four V-valued properties of A, p: A → V (V A = 2 2 = 4) 459 p1: A → V; p1(a1) = 0, p1(a2) = 0 460 p2: A → V; p2(a1) = 1, p2(a2) = 0 461 p3: A → V; p3(a1) = 0, p3(a2) = 1 462 p4: A → V; p4(a1) = 1, p4(a2) = 1 463 Thus, 464 V A = {p1, p2, p3, p4} 465 Next, there are 16 functionals, q: V A → V (V (VA) = 2 (22) = 16) 466 q1: V A → V; q1(p1) = 0, q1(p2) = 0, q1(p3) = 0, q1(p4) = 0 467 q2: V A → V; q2(p1) = 1, q2(p2) = 0, q2(p3) = 0, q2(p4) = 0 468 q3: V A → V; q3(p1) = 0, q3(p2) = 1, q3(p3) = 0, q3(p4) = 0 469 q4: V A → V; q4(p1) = 1, q4(p2) = 1, q4(p3) = 0, q4(p4) = 0 470 q5: V A → V; q5(p1) = 0, q5(p2) = 0, q5(p3) = 1, q5(p4) = 0 471 q6: V A → V; q6(p1) = 1, q6(p2) = 0, q6(p3) = 1, q6(p4) = 0 472 q7: V A → V; q7(p1) = 0, q7(p2) = 1, q7(p3) = 1, q7(p4) = 0 473 27 q8: V A → V; q8(p1) = 1, q8(p2) = 1, q8(p3) = 1, q8(p4) = 0 474 q9: V A → V; q9(p1) = 0, q9(p2) = 0, q9(p3) = 0, q9(p4) = 1 475 q10: V A → V; q10(p1) = 1, q10(p2) = 0, q10(p3) = 0, q10(p4) = 1 476 q11: V A → V; q11(p1) = 0, q11(p2) = 1, q11(p3) = 0, q11(p4) = 1 477 q12: V A → V; q12(p1) = 1, q12(p2) = 1, q12(p3) = 0, q12(p4) = 1 478 q13: V A → V; q13(p1) = 0, q13(p2) = 0, q13(p3) = 1, q13(p4) = 1 479 q14: V A → V; q14(p1) = 1, q14(p2) = 0, q14(p3) = 1, q14(p4) = 1 480 q15: V A → V; q15(p1) = 0, q15(p2) = 1, q15(p3) = 1, q15(p4) = 1 481 q16: V A → V; q16(p1) = 1, q16(p2) = 1, q16(p3) = 1, q16(p4) = 1 482 Of these 16 functionals, there are only two V-generalized points of A corresponding to the two 483 points of A = {a1, a2}. The two V-generalized points of A are 484 q11: V A → V; q11(p1) = 0, q11(p2) = 1, q11(p3) = 0, q11(p4) = 1 485 and 486 q13: V A → V; q13(p1) = 0, q13(p2) = 0, q13(p3) = 1, q13(p4) = 1 487 i.e. they both satisfy 488 q (e ◦ p) = e (q (p)) 489 Let's consider the functional 490 28 q11: V A → V; q11(p1) = 0, q11(p2) = 1, q11(p3) = 0, q11(p4) = 1 491 In order for the functional q11 to be a V-generalized point of A, it has to satisfy 492 q11 (e ◦ p) = e (q11 (p)) 493 for all four p: A → V and all four endomaps e: V → V. They are 494 e1: V → V; e1(0) = 0, e1(1) = 0 495 e2: V → V; e2(0) = 1, e2(1) = 0 496 e3: V → V; e3(0) = 0, e3(1) = 1 497 e4: V → V; e4(0) = 1, e4(1) = 1 498 So, the set of all endomaps of property type V is V V = {e1, e2, e3, e4}. 499 Thus, there are 16 cases we have to evaluate to show that the functional 500 q11: V A → V; q11(p1) = 0, q11(p2) = 1, q11(p3) = 0, q11(p4) = 1 501 is a V-generalized point of A. 502 Case 1: (p1, e1) 503 q11 (e1 ◦ p1) = e1 (q11 (p1)) 504 LHS: q11 (e1 ◦ p1) = q11 (p1) = 0 505 RHS: e1 (q11 (p1)) = e1 (0) = 0 506 Case 2: (p2, e1) 507 29 q11 (e1 ◦ p2) = e1 (q11 (p2)) 508 LHS: q11 (e1 ◦ p2) = q11 (p1) = 0 509 RHS: e1 (q11 (p2)) = e1 (1) = 0 510 Case 3: (p3, e1) 511 q11 (e1 ◦ p3) = e1 (q11 (p3)) 512 LHS: q11 (e1 ◦ p3) = q11 (p1) = 0 513 RHS: e1 (q11 (p3)) = e1 (0) = 0 514 Case 4: (p4, e1) 515 q11 (e1 ◦ p4) = e1 (q11 (p4)) 516 LHS: q11 (e1 ◦ p4) = q11 (p1) = 0 517 RHS: e1 (q11 (p4)) = e1 (1) = 0 518 q11: V A → V; q11(p1) = 0, q11(p2) = 1, q11(p3) = 0, q11(p4) = 1 519 Case 5: (p1, e2) 520 q11 (e2 ◦ p1) = e2 (q11 (p1)) 521 LHS: q11 (e2 ◦ p1) = q11 (p4) = 1 522 RHS: e2 (q11 (p1)) = e2 (0) = 1 523 Case 6: (p2, e2) 524 30 q11 (e2 ◦ p2) = e2 (q11 (p2)) 525 LHS: q11 (e2 ◦ p2) = q11 (p3) = 0 526 RHS: e2 (q11 (p2)) = e2 (1) = 0 527 Case 7: (p3, e2) 528 q11 (e2 ◦ p3) = e2 (q11 (p3)) 529 LHS: q11 (e2 ◦ p3) = q11 (p2) = 1 530 RHS: e2 (q11 (p3)) = e2 (0) = 1 531 Case 8: (p4, e2) 532 q11 (e2 ◦ p4) = e2 (q11 (p4)) 533 LHS: q11 (e2 ◦ p4) = q11 (p1) = 0 534 RHS: e2 (q11 (p4)) = e2 (1) = 0 535 q11: V A → V; q11(p1) = 0, q11(p2) = 1, q11(p3) = 0, q11(p4) = 1 536 Case 9: (p1, e3) 537 q11 (e3 ◦ p1) = e3 (q11 (p1)) 538 LHS: q11 (e3 ◦ p1) = q11 (p1) = 0 539 RHS: e3 (q11 (p1)) = e3 (0) = 0 540 Case 10: (p2, e3) 541 31 q11 (e3 ◦ p2) = e3 (q11 (p2)) 542 LHS: q11 (e3 ◦ p2) = q11 (p2) = 1 543 RHS: e3 (q11 (p2)) = e3 (1) = 1 544 Case 11: (p3, e3) 545 q11 (e3 ◦ p3) = e3 (q11 (p3)) 546 LHS: q11 (e3 ◦ p3) = q11 (p3) = 0 547 RHS: e3 (q11 (p3)) = e3 (0) = 0 548 Case 12: (p4, e3) 549 q11 (e3 ◦ p4) = e3 (q11 (p4)) 550 LHS: q11 (e3 ◦ p4) = q11 (p4) = 1 551 RHS: e3 (q11 (p4)) = e3 (1) = 1 552 q11: V A → V; q11(p1) = 0, q11(p2) = 1, q11(p3) = 0, q11(p4) = 1 553 Case 13: (p1, e4) 554 q11 (e4 ◦ p1) = e4 (q11 (p1)) 555 LHS: q11 (e4 ◦ p1) = q11 (p4) = 1 556 RHS: e4 (q11 (p1)) = e4 (0) = 1 557 Case 14: (p2, e4) 558 32 q11 (e4 ◦ p2) = e4 (q11 (p2)) 559 LHS: q11 (e4 ◦ p2) = q11 (p4) = 1 560 RHS: e4 (q11 (p2)) = e4 (1) = 1 561 Case 15: (p3, e4) 562 q11 (e4 ◦ p3) = e4 (q11 (p3)) 563 LHS: q11 (e4 ◦ p3) = q11 (p4) = 1 564 RHS: e4 (q11 (p3)) = e4 (0) = 1 565 Case 16: (p4, e4) 566 q11 (e4 ◦ p4) = e4 (q11 (p4)) 567 LHS: q11 (e4 ◦ p4) = q11 (p4) = 1 568 RHS: e4 (q11 (p4)) = e4 (1) = 1 569 Thus, the functional 570 q11: V A → V; q11(p1) = 0, q11(p2) = 1, q11(p3) = 0, q11(p4) = 1 571 is a V-generalized point of A. Along similar lines, we can show that the functional 572 q13: V A → V; q13(p1) = 0, q13(p2) = 0, q13(p3) = 1, q13(p4) = 1 573 is another V-generalized point of A. These two perceived elements (generalized points) 574 correspond to the two points 575 33 a1: 1 → A 576 a2: 1 → A 577 of the set A = {a1, a2}. 578 Next, we give an example of a functional (one of the remaining 14 functionals of the total 16 579 functionals) which is not a generalized point. Consider the functional 580 q12: V A → V; q12(p1) = 1, q12(p2) = 1, q12(p3) = 0, q12(p4) = 1 581 In order to be a V-generalized point of A, the functional 582 q12: V A → V 583 must satisfy 584 q12 (e ◦ p) = e (q12 (p)) 585 for all p in V A = {p1, p2, p3, p4} and for all e in V V = {e1, e2, e3, e4}, i.e., for all 16 cases we 586 evaluated earlier. Let's consider the case of p = p1 and e = e1. We have to check for the equality 587 q12 (e1 ◦ p1) = e1 (q12 (p1)) 588 LHS: q12 (e1 ◦ p1) = q12 (p1) = 1 589 RHS: e1 (q12 (p1)) = e1 (1) = 0 590 Since LHS is not equal to RHS, the functional 591 q12: V A → V; q12(p1) = 1, q12(p2) = 1, q12(p3) = 0, q12(p4) = 1 592 34 is not a V-generalized point of A. 593 Now we spell out how each point of a set gives rise to a generalized point (perceived element). 594 Consider a set A and a type V. Since generalized point is a functional 595 q: V A → V 596 we first construct a functional called evaluation functional for each element  a' of the set A. 597 Recollect that the elements of domain set V A of the functional q are V-valued properties of A, 598 i.e. 599 p: A → V 600 and the functional q assigns to each p in V A an element  v' of the codomain set V of the 601 functional q. An evaluation functional corresponding to an element  a' of the set A is defined as 602 qa: V A → V 603 with 604 qa(p) = p(a) 605 Now we show that this evaluation functional is a generalized point, i.e. satisfies 606 qa(e ◦ p) = e (qa(p)) 607 where 608 e: V → V 609 is an endomap on the type V of property. 610 35 LHS: qa(e ◦ p) = (e ◦ p) (a) = e(p(a)) 611 RHS: e (qa(p)) = e(p(a)) 612 Thus, the evaluation functional corresponding to each point of a set is a generalized point 613 (Lawvere and Rosebrugh, 2003, p. 150). We now give an example of this general result. 614 Consider a set A = {a1, a2, a3} and a property type V = {0, 1}. The set V A of all V-valued 615 properties of A consists of 8 functions, i.e. 616 V A = {p1, p2, p3, p4, p5, p6, p7, p8} 617 where 618 p1: A → V; p1(a1) = 0, p1(a2) = 0, p1(a3) = 0 619 p2: A → V; p2(a1) = 1, p2(a2) = 0, p2(a3) = 0 620 p3: A → V; p3(a1) = 0, p3(a2) = 1, p3(a3) = 0 621 p4: A → V; p4(a1) = 1, p4(a2) = 1, p4(a3) = 0 622 p5: A → V; p5(a1) = 0, p5(a2) = 0, p5(a3) = 1 623 p6: A → V; p6(a1) = 1, p6(a2) = 0, p6(a3) = 1 624 p7: A → V; p7(a1) = 0, p7(a2) = 1, p7(a3) = 1 625 p8: A → V; p8(a1) = 1, p8(a2) = 1, p8(a3) = 1 626 Let us now consider a point 627 36 a1: 1 → A 628 and the corresponding evaluation functional 629 qa1: V A → V 630 defined as 631 qa1(p) = p(a1) 632 for all p in V A , i.e. 633 qa1(p1) = p1(a1) = 0 634 qa1(p2) = p2(a1) = 1 635 qa1(p3) = p3(a1) = 0 636 qa1(p4) = p4(a1) = 1 637 qa1(p5) = p5(a1) = 0 638 qa1(p6) = p6(a1) = 1 639 qa1(p7) = p7(a1) = 0 640 qa1(p8) = p8(a1) = 1 641 Now we have to show that this evaluation functional 642 qa1: V A → V 643 37 satisfies 644 qa1 (e ◦ p) = e (qa1 (p)) 645 for all elements (V-valued properties of A) p in V A = {p1, p2, p3, p4, p5, p6, p7, p8}, and for all 646 elements (endomaps of property type V) e in V V = {e1, e2, e3, e4} defined as 647 e1: V → V; e1(0) = 0, e1(1) = 0 648 e2: V → V; e2(0) = 1, e2(1) = 0 649 e3: V → V; e3(0) = 0, e3(1) = 1 650 e4: V → V; e4(0) = 1, e4(1) = 1 651 Thus we have to test for the equality 652 qa1 (e ◦ p) = e (qa1 (p)) 653 in 32 cases. They are: 654 Case 1: (p1, e1) 655 qa1 (e1 ◦ p1) = e1 (qa1 (p1)) 656 LHS: qa1 (e1 ◦ p1) = qa1 (p1) = 0 657 RHS: e1 (qa1 (p1)) = e1 (0) = 0 658 Case 2: (p2, e1) 659 qa1 (e1 ◦ p2) = e1 (qa1 (p2)) 660 38 LHS: qa1 (e1 ◦ p2) = qa1 (p1) = 0 661 RHS: e1 (qa1 (p2)) = e1 (1) = 0 662 Case 3: (p3, e1) 663 qa1 (e1 ◦ p3) = e1 (qa1 (p3)) 664 LHS: qa1 (e1 ◦ p3) = qa1 (p1) = 0 665 RHS: e1 (qa1 (p3)) = e1 (0) = 0 666 Case 4: (p4, e1) 667 qa1 (e1 ◦ p4) = e1 (qa1 (p4)) 668 LHS: qa1 (e1 ◦ p4) = qa1 (p1) = 0 669 RHS: e1 (qa1 (p4)) = e1 (1) = 0 670 Case 5: (p5, e1) 671 qa1 (e1 ◦ p5) = e1 (qa1 (p5)) 672 LHS: qa1 (e1 ◦ p5) = qa1 (p1) = 0 673 RHS: e1 (qa1 (p5)) = e1 (0) = 0 674 Case 6: (p6, e1) 675 qa1 (e1 ◦ p6) = e1 (qa1 (p6)) 676 LHS: qa1 (e1 ◦ p6) = qa1 (p1) = 0 677 39 RHS: e1 (qa1 (p6)) = e1 (1) = 0 678 Case 7: (p7, e1) 679 qa1 (e1 ◦ p7) = e1 (qa1 (p7)) 680 LHS: qa1 (e1 ◦ p7) = qa1 (p1) = 0 681 RHS: e1 (qa1 (p7)) = e1 (0) = 0 682 Case 8: (p8, e1) 683 qa1 (e1 ◦ p8) = e1 (qa1 (p8)) 684 LHS: qa1 (e1 ◦ p8) = qa1 (p1) = 0 685 RHS: e1 (qa1 (p8)) = e1 (1) = 0 686 Case 9: (p1, e2) 687 qa1 (e2 ◦ p1) = e2 (qa1 (p1)) 688 LHS: qa1 (e2 ◦ p1) = qa1 (p8) = 1 689 RHS: e2 (qa1 (p1)) = e2 (0) = 1 690 Case 10: (p2, e2) 691 qa1 (e2 ◦ p2) = e2 (qa1 (p2)) 692 LHS: qa1 (e2 ◦ p2) = qa1 (p7) = 0 693 RHS: e2 (qa1 (p2)) = e2 (1) = 0 694 40 Case 11: (p3, e2) 695 qa1 (e2 ◦ p3) = e2 (qa1 (p3)) 696 LHS: qa1 (e2 ◦ p3) = qa1 (p6) = 1 697 RHS: e2 (qa1 (p3)) = e2 (0) = 1 698 Case 12: (p4, e2) 699 qa1 (e2 ◦ p4) = e2 (qa1 (p4)) 700 LHS: qa1 (e2 ◦ p4) = qa1 (p5) = 0 701 RHS: e2 (qa1 (p4)) = e2 (1) = 0 702 Case 13: (p5, e2) 703 qa1 (e2 ◦ p5) = e2 (qa1 (p5)) 704 LHS: qa1 (e2 ◦ p5) = qa1 (p4) = 1 705 RHS: e2 (qa1 (p5)) = e2 (0) = 1 706 Case 14: (p6, e2) 707 qa1 (e2 ◦ p6) = e2 (qa1 (p6)) 708 LHS: qa1 (e2 ◦ p6) = qa1 (p3) = 0 709 RHS: e2 (qa1 (p6)) = e2 (1) = 0 710 Case 15: (p7, e2) 711 41 qa1 (e2 ◦ p7) = e2 (qa1 (p7)) 712 LHS: qa1 (e2 ◦ p7) = qa1 (p2) = 1 713 RHS: e2 (qa1 (p7)) = e2 (0) = 1 714 Case 16: (p8, e2) 715 qa1 (e2 ◦ p8) = e2 (qa1 (p8)) 716 LHS: qa1 (e2 ◦ p8) = qa1 (p1) = 0 717 RHS: e2 (qa1 (p8)) = e2 (1) = 0 718 Case 17: (p1, e3) 719 qa1 (e3 ◦ p1) = e3 (qa1 (p1)) 720 LHS: qa1 (e3 ◦ p1) = qa1 (p1) = 0 721 RHS: e3 (qa1 (p1)) = e3 (0) = 0 722 Case 18: (p2, e3) 723 qa1 (e3 ◦ p2) = e3 (qa1 (p2)) 724 LHS: qa1 (e3 ◦ p2) = qa1 (p2) = 1 725 RHS: e3 (qa1 (p2)) = e3 (1) = 1 726 Case 19: (p3, e3) 727 qa1 (e3 ◦ p3) = e3 (qa1 (p3)) 728 42 LHS: qa1 (e3 ◦ p3) = qa1 (p3) = 0 729 RHS: e3 (qa1 (p3)) = e3 (0) = 0 730 Case 20: (p4, e3) 731 qa1 (e3 ◦ p4) = e3 (qa1 (p4)) 732 LHS: qa1 (e3 ◦ p4) = qa1 (p4) = 1 733 RHS: e3 (qa1 (p4)) = e3 (1) = 1 734 Case 21: (p5, e3) 735 qa1 (e3 ◦ p5) = e3 (qa1 (p5)) 736 LHS: qa1 (e3 ◦ p5) = qa1 (p5) = 0 737 RHS: e3 (qa1 (p5)) = e3 (0) = 0 738 Case 22: (p6, e3) 739 qa1 (e3 ◦ p6) = e3 (qa1 (p6)) 740 LHS: qa1 (e3 ◦ p6) = qa1 (p6) = 1 741 RHS: e3 (qa1 (p6)) = e3 (1) = 1 742 Case 23: (p7, e3) 743 qa1 (e3 ◦ p7) = e3 (qa1 (p7)) 744 LHS: qa1 (e3 ◦ p7) = qa1 (p7) = 0 745 43 RHS: e3 (qa1 (p7)) = e3 (0) = 0 746 Case 24: (p8, e3) 747 qa1 (e3 ◦ p8) = e3 (qa1 (p8)) 748 LHS: qa1 (e3 ◦ p8) = qa1 (p8) = 1 749 RHS: e3 (qa1 (p8)) = e3 (1) = 1 750 Case 25: (p1, e4) 751 qa1 (e4 ◦ p1) = e4 (qa1 (p1)) 752 LHS: qa1 (e4 ◦ p1) = qa1 (p8) = 1 753 RHS: e4 (qa1 (p1)) = e4 (0) = 1 754 Case 26: (p2, e4) 755 qa1 (e4 ◦ p2) = e4 (qa1 (p2)) 756 LHS: qa1 (e4 ◦ p2) = qa1 (p8) = 1 757 RHS: e4 (qa1 (p2)) = e4 (1) = 1 758 Case 27: (p3, e4) 759 qa1 (e4 ◦ p3) = e4 (qa1 (p3)) 760 LHS: qa1 (e4 ◦ p3) = qa1 (p8) = 1 761 RHS: e4 (qa1 (p3)) = e4 (0) = 1 762 44 Case 28: (p4, e4) 763 qa1 (e4 ◦ p4) = e4 (qa1 (p4)) 764 LHS: qa1 (e4 ◦ p4) = qa1 (p8) = 1 765 RHS: e4 (qa1 (p4)) = e4 (1) = 1 766 Case 29: (p5, e4) 767 qa1 (e4 ◦ p5) = e4 (qa1 (p5)) 768 LHS: qa1 (e4 ◦ p5) = qa1 (p8) = 1 769 RHS: e4 (qa1 (p5)) = e4 (0) = 1 770 Case 30: (p6, e4) 771 qa1 (e4 ◦ p6) = e4 (qa1 (p6)) 772 LHS: qa1 (e4 ◦ p6) = qa1 (p8) = 1 773 RHS: e4 (qa1 (p6)) = e4 (1) = 1 774 Case 31: (p7, e4) 775 qa1 (e4 ◦ p7) = e4 (qa1 (p7)) 776 LHS: qa1 (e4 ◦ p7) = qa1 (p8) = 1 777 RHS: e4 (qa1 (p7)) = e4 (0) = 1 778 Case 32: (p8, e4) 779 45 qa1 (e4 ◦ p8) = e4 (qa1 (p8)) 780 LHS: qa1 (e4 ◦ p8) = qa1 (p8) = 1 781 RHS: e4 (qa1 (p8)) = e4 (1) = 1 782 Thus the evaluation functional 783 qa1: V A → V; qa1(p1)=0, qa1(p2)=1, qa1(p3)=0, qa1(p4)=1, qa1(p5)=0, qa1(p6)=1, qa1(p7)=0, qa1(p8)=1 784 satisfying 785 qa1 (e ◦ p) = e (qa1 (p)) 786 (for all p: A → V and for all e: V → V) is a V-generalized point of A corresponding to the point 787 a1: 1 → A 788 Along the same lines, we can show that each of remaining two points of the set A = {a1, a2, a3} 789 give rise to corresponding V-generalized points of A. 790 With property type V = 2 (two-element set), although there is a V-generalized point of A 791 corresponding to each point of A, there can also be generalized points that do not correspond to 792 any points of A, which we may call illusions (or ghost points). In order to obtain a 1-1 793 correspondence between points of any set A and V-generalized points of A, i.e. isomorphism 794 between objects and perceived objects, we need a 3-element set as the property type V. One of 795 our objectives is to calculate objects analogous to the 3-element set (in the category of sets) in 796 categories that are reflective of reality such as the category of categories. We plan to approach 797 this goal by calculating the basic types of knowing (objects analogous to 3-element set in the 798 46 category of sets) in more structured categories such as dynamical systems, functions, graphs, and 799 actions. 800 47 References 801 Albright, T. D. (2012) On the perception of probable things: Neural substrates of associative 802 memory, imagery and perception. Neuron 74, 227-245. 803 Albright, T. D. (2013a) The Veiled Christ of Cappella Sansevero: On art, vision and reality. 804 Leonardo 46, 19-23. 805 Albright, T. D. 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