Abstract
According to Vázquez and Liz (Found Sci 16(4): 383–391, 2011), Points of View (PoV) can be considered in two different ways. On the one hand, they can be explained following the model of propositional attitudes. This model assumes that the internal structure of a PoV is constituted by a subject, a set of contents, and a set of relations between the subject and those contents. On the other hand, we can analyze points of view taking as a model the notions of location and access. If we choose to follow the second approach, instead of the first one, the internal structure of a PoV is not directly addressed, and the emphasized features of PoV are related to the function that PoV are intended to have. That is, PoV are directly identified by their role and they can solely be understood as ways of accessing the world that bring some kind of perspective about it. Having this in mind, we would like to propose a notation that explains how to understand such access as a sort of models (that can allow the creation of concepts), independently of whether the precise PoV under consideration is impersonal or non-impersonal, its kind of content, and its subjective or objective character. First, we will present an account of some previous approaches to the study of points of view. Then, we will analyze what kind of structure the world is assumed to posses and how the access to it is possible. Third, we will develop a notation that explains PoV as qualitative dimensions by means of which it is possible to valuate objects and states of the world.
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Notes
Like them, we will use PoV to denote singular as well as plural.
Cf. Vázquez and Liz (2011:385) for a more detailed development of these important distinctions that specify what a PoV is and some possible relations between different PoV.
In different personal communications, they talked about the possibility of developing a metaphysical approach to hybrid points of view, for instance. To appeal to the notion of PoV would also allow, according to their view, different developments in temporal or epistemic logics.
This is not the unique approach to model that supposes that, first, models have some kind of reality and, second, they have some causal powers. See Magnani (2012), for example.
This is an important distinction. As we will see below, when we analyze the account defended by Moline against Baier (Sect. 2.1), it is the only way through which we can distinguish between a Kantian approach on ‘to follow the rule’ (in which the notion of principle or maxim determine every possible action) and a non-deterministic approach on it (in which the important thing is not to adopt a particular rule or PoV but to act according to a particular rule or PoV).
We believe that these three kinds of claims, stated by using the notion of PoV, are directly related to the possibility of understanding other PoV from a given PoV. We will come to this topic in Sect. 4 below.
In a similar way, Brandom (1982) develops a brief account of PoV referred to moral practical reasoning. In his view, he confronts the formalist approach to moral judgments that look at all considerations as directly relevant with his pragmatic account about considering only the relevant statements from the PoV under consideration. According to Brandom, a PoV consists of two different elements. On the one hand, the PoV have a specification of a set of sentences expressing directly relevant considerations. On the other hand, they have a maxim that determines a preferred order of directly relevant circumstances. From there, those who adopt a particular PoV:
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(a)
Develop a particular practical deliberation through which to determine what sets of sentences are directly relevant (or assign probabilities to directly relevant considerations),
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(b)
Assign truth-values to conditionals whose antecedents describe the various contemplated actions that are being compared and whose consequences express directly relevant considerations,
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(c)
Apply the maxim to compare the directly relevant consequences and, then, yield a recommended course of action.
It is obvious that a set of inferential practices that could not be expressed entirely in terms of a particular PoV might be captured by two (or more) PoV according to a hierarchical decision-maker for adjudicating disputes between them (331). The notion of PoV, then, provides a tool to pass from reasons-on-balance to presumptive reasons without need to appeal to ceteris paribus clauses (324). Even though Brandom’s position analyzes PoV according to the model of propositional attitudes (that is, according to their inner structure and content), we think that it is extremely important because the results of adopting a particular PoV can be presented as a reason-on-balance from a restricted perspective without denying the feasibility of the action appraisal provided from other perspectives. We will come back to this below (see Footnote 10 and Sect. 3).
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(a)
We would like to acknowledge the clarification of this difficulty in Moline’s position to M. Liz in a personal communication.
As one of the referees pointed out, there is an apparent inconsistency in this point of Hautamäki’s system. On the one hand, it seems that in Hautamäki’s system PoV are primitive and therefore cannot be defined as a set of worlds. On the other hand, as Hautamäki points out, “a PoV selects a set of possible worlds, namely, those worlds, which have a structure or features presupposed by the PoV. In these worlds, the PoV is satisfied” (Hautamäki 1983b:226). In this sense, PoV are primitive with respect to the set of worlds that satisfy the PoV and the apparent inconsistency is solved.
This logic is based in the seminal work of Chellas (1980) on propositional modal logic. In some sense, this work is also related with the context of conditionalization in Deontic logic, also proposed by Chellas (1974). As Brandom (1982:324) says, a judgment tells us something about what range of possibly countervailing considerations has been taken into account under certain circumstances, but it is a question of particular valuation to determine the particular appropriate action. In a similar way, a PoV selects a set of possible worlds that have the structure or features assumed by the PoV. That is, a set of possible worlds that is consistent with a particular PoV. Then, a PoV is identified with a non-empty set of possible worlds that satisfy the conditions indicated by the particular PoV (Hautamäki 1983b:224–226). See also Appendix D in Prior (1957:140–145).
A determination basis \(\text{ D}=(\text{ D}(\text{ i}))_{\mathrm{I}}\) partitions a set of entities \(E\) so that the state function of \(E\) is an injection from \(E\) to the logical space XD (Hautamäki 1986:30). Informally, it generates a Cartesian product of determinables, in base to which the characteristics of an object can be expressed as a vector of determinates in a particular framework. For more precise definitions of these mathematical notions, we refer the interested reader to Hautamäki (1986:38) and the references therein (see in particular Note 6 in page 60).
It is important to notice that under Hautamäki’s approach, PoV are not limited to define features of one, and only one, object (entity of the world) at a time. We emphasize this characteristic of Hautamäki’s account precisely because we think that it is an important characteristic of his logical system that can be clarified with the notation we will develop in Sect. 3 below. Using this notation, we will establish the basis according to which define a total space of objects of the world to which a PoV would have access.
Hautamäki makes an interesting point in connection to this. He writes: “Since the lattice of partial functions is continuous, the finite nature of a point of view will not prove to be a fatal restriction: every total function is approximated by its finite parts” (Hautamäki 1986:9). This can be understood in the following way, since the number of determinables that a PoV can perceive is finite, in principle it is never possible to achieve the total perception of reality. However, the larger the number of determinables that are perceived, the better approximation to reality the PoV has. In Hautamäki’s words “it is the supremum of its approximations” and “it is the union of the points of view that it contains” (Hautamäki 1986:79, 80).
In fact, Hautamäki (1983a) defines a Kripke-style model for this language as an structure \(<\)W,I,R,S,V\(>\) where W is a non-empty set of possible worlds, I is a non-empty set of PoV, R and S are relations in WxI or subsets of (WxI)x(WxI) and V is a valuation function from FxWxI to {0,1} such that
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(i)
V(\(\lnot \) p,w,i) = 1 iff V(p,w,i) = 0
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(ii)
V(p&q,w,i) = 1 iff V(p,w,i) = V(q,w,i) = 1
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(iii)
V(Lp,w,i) = 1 iff V(p,w’,i) = 1 for all w’ such that \(<\)w,i’\(>\)R\(<\)w’,i\(>\)
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(iv)
V(Ap,w,i) = 1 iff V(p,w,i’) = 1 for all i’ such that \(<\)w,i\(>\)S\(<\)w,i’\(>\) Then, Hautamäki can also define Rp as \(\lnot A \lnot p\).
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(i)
Someone can insist in the fact that if we consider that truth-values can change, we will face the problem of semantic contradiction. But, according to Hautamäki (1983a:195), this is not the case. We need to consider this kind of contradiction as dialectical more than semantic, like a kind of epistemological antinomy. Unlike semantic contradiction in which two conflicting propositions are mutually exclusive, dialectical contradiction offers two different interpretations of the same object where one view complete each other. Therefore, this kind of contrapositions is understood as complementary oppositions. See also Hautamäki (1983b).
The discussion about the determinate/determinable distinction is not new. For instance, as Johnson (1921:173) reminds us, it is the traditional account of the principles of logical division, where a class is represented as consisting of sub-classes that are mutually exclusive, collectively exhaustive, and based upon a ‘fundamentum divisionis’. However, we will disagree with this classical account on divisions and defend a direct definition of determinables as objective aspects of the objects of the world, in contraposition to the definition of determinables solely based on the notion of membership of determinates to the determinable in an extensional way. For a similar account see, for instance, Prior (1949).
In contrast to Fine (2011:184), we do not suppose that we have access to all possible scenarios. Though he claims the resemblance relation between determinates, his approach seems to lose the specificity required to access to the determinables. That is, given that we only have access to determinables that appear reflected in our image of the world, we only have access to the determinables after these are determined. Then, the only permitted answer to what this image is will be the set of the total information provided by determinates as physical aspects of objects to which we have access and, by definition, this information is restricted.
This point is also highly controversial. For instance, Gillet and Rives (2005) advocate to the contrary for the non-existence of determinables: We only have a world of absolute determinates. Also Johansson (2000) argues that there are absolutely determinate mind-independent properties. Nevertheless, they disagree about the causal status of determinables.
Notice the relation with Brandom’s and Hautamäki’s approaches, where the formalism considers all the possibilities but the pragmatic vision, the filter, only takes into account the ones that are relevant to the PoV, see Footnotes 7 and 10.
This is a very controversial topic. For instance, as defined by Johnson (1921:174), determinables are abstract names that stand for adjectives that conform sets of determinates. In a different way, Searle (1959:154) considers determinables as abstract nouns that cover a whole range of characterizing terms (determinates, adjectives) that are all “in the same line of business”. Prior (1949) also analyzes determinables as an extensional kind of qualities. We instead agree with Fine (2011), who affirms this distinction cannot be solely based on the notion of extensionality or co-determinability (which holds among attributes that belong in the same determinable) because this avoids the question of what exactly a determinable is. We will try, then, to reformulate the nature of determinables through the analysis of its relation with determinates.
We use this example because it is the example used by Johnson (1921:176) to analyze the traditional account of divisions.
We would like to emphasize that ‘man’ might not be perceived as multidimensional from every PoV, and this depends strongly on each PoV’s filtered world. Notice that ’man’ can be perceived as multidimensional in an appropriate filtered world that is not necessarily specified; however, if the latter is specified, we can say that ’man’ is multidimensional in that filtered world.
Unlike the classical distinction between determinable and determinate advocated by Johnson.
The notion of state space goes back from the classical debate on scientific methodology. For instance, according to van Fraassen a physical theory uses a mathematical model in order to represent the behavior of a certain class of physical systems as defined through the specification of the set of states of which it is capable. These states are represented through elements of a mathematical space, or state space (1970:328). See also van Fraassen (1972). Carnap (1950:76) originally used this notion in a similar way applied to families of related properties. In a posterior work, he classifies primitive attributes also into families such that the attributes of a family are related to each other by belonging to the same general kind (or modalities) according to this state space (Carnap 1971:43).
In this respect, one could measure the difference between objects of the world using the notion of metric, as a distance function (with a numeric value that subsumes the attributes-pair) introduced by Carnap (1971:50–51 y 78–79) in connection to families of related properties.
The finiteness of O (by assumption) implies that this writing consists of a finite list of elements. Notice that then, describing an element by the complementary is always possible even if it might be unreasonably lengthy.
Notice that the definitions of ultraviolet and infrared follow from the standard human PoV, as they are the wavelengths outside the visible spectrum.
When all a, b and c have zero intensity we obtain the darkest color (black, no light), and full intensity of each of them, that is \(\text{ a}=\text{ b}=\text{ c}=1\) gives white color. Whenever \(\text{ a}=\text{ b}=\text{ c}\), the result is a shade of gray, darker or lighter depending on the actual value of the intensity. If any two of the coefficients a, b, and c are equal and the third is 0 (meaning that the corresponding color is absent in the mix) we obtain secondary colors: cyan is green and blue, magenta is red and blue, and yellow is red and green. The intensity of the color will depend on the value of the non-zero coefficients. Finally, when one of the coefficients is bigger than the others, the color is a hue near this primary color (reddish, greenish, or bluish).
Since the actual values will be irrelevant for our purposes, we have arbitrarily chosen to average the extreme values of the ranges previously mentioned.
Notice that the coefficients a, b, and c can be thought of as determinates of the determinables R, G, and B. The determinable will be determined once the intensity (the value of the coefficient) is fixed. However, it is important to stress that even if R, G, and B are chosen in the example (and hence one might think they can be considered determinables with the particular election of a wavelength being a determinate) they are fixed parameters that define the PoV of the normalized RGB model. Consequently, they are not determinables from the PoV of the normalized RGB model.
Newton, in his book Opticks, divided the spectrum into seven named colors: red, orange, yellow, green, blue, indigo, and violet. He chose seven colors out of a belief, derived from the ancient Greek sophists, that there was a connection between the colors, the musical notes, the known objects in the solar system, and the days of the week. Actually, the human eye is relatively insensitive to indigo’s frequencies and some otherwise well-sighted people cannot distinguish indigo from blue and violet.
For instance, if the filtered world can be endowed with the structure of a vector space, it can be proved (as can be found in any manual of Linear Algebra) that there is a minimum number of elements in a system of generators and that the elements of such a system are independent among them (and it is called a basis).
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Acknowledgments
We would like to thank Manuel Liz, Margarita Vázquez, Steven Hales, David Sosa, and two anonymous referees for useful comments and suggestions to previous drafts of this paper. It was developed under the framework provided by the Research Project “Points of View and Temporal Structures” (FII2011-24549), funded by the Spanish Government.
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Charro, F., Colomina, J.J. Points of View Beyond Models: Towards a Formal Approach to Points of View as Access to the World. Found Sci 19, 137–151 (2014). https://doi.org/10.1007/s10699-013-9325-z
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DOI: https://doi.org/10.1007/s10699-013-9325-z