Abstract
The aim of this paper is to discuss and extend some of Béziau’s (published and unpublished) results on the logical geometry of the modal logic S5 and the subjective quantifiers many and few. After reviewing some of the basic notions of logical geometry, we discuss Béziau’s work on visualising the Aristotelian relations in S5 by means of two- and three-dimensional diagrams, such as hexagons and a stellar rhombic dodecahedron. We then argue that Béziau’s analysis is incomplete, and show that it can be completed by considering another three-dimensional Aristotelian diagram, viz. a rhombic dodecahedron. Next, we discuss Béziau’s proposal to transpose his results on the logical geometry of the modal logic S5 to that of the subjective quantifiers many and few. Finally, we propose an alternative analysis of many and few, and compare it with that of Béziau’s. While the two analyses seem to fare equally well from a strictly logical perspective, we argue that the new analysis is more in line with certain linguistic desiderata.
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Notes
- 1.
It is well-known that ¬(¬φ∧¬ψ) is equivalent to φ∨ψ, but we choose to stick with the former notation, because it more clearly expresses the idea of φ and ψ being false together.
- 2.
The system S is assumed to have connectives expressing classical negation (¬), conjunction (∧) and implication (→), and a model-theoretic semantics (⊨).
- 3.
As limiting cases, the non-contingent bitstrings 0000 and 1111 can be called level 0 (L0) and level 4 (L4), respectively.
- 4.
Formally, a diagram or set of formulas is said to be Boolean closed iff whenever it contains formulas φ,ψ, it also contains their contingent Boolean combinations (¬φ, φ∧ψ, φ∨ψ). Note that this definition is restricted to contingent Boolean combinations, so if □p and ¬□p occur in a diagram, then it is not required for the diagram to be Boolean closed that it also contains the contradiction □p∧¬□p and the tautology □p∨¬□p.
- 5.
The process described above works not only for the S5 square (Fig. 2(b)), but also for the bitstring square (Fig. 2(c)). We thus obtain three bitstring JSB hexagons, the first one of which was already shown in Fig. 3(b). The two new ones are displayed simultaneously with their S5 counterparts in Fig. 4. For reasons of space, we will in the remainder of this paper no longer distinguish between S5 diagrams and bitstring diagrams, and decorate the Aristotelian diagrams sometimes with concrete formulas, sometimes with bitstrings, and sometimes with both simultaneously.
- 6.
These hexagons also show that, from the perspective of subalternation, classical negation occupies a position that is intermediate between paracomplete and paraconsistent negation. In Fig. 4(b), the classical negation (¬p) is entailed by the paracomplete negation (¬◊p), while in Fig. 4(c), it entails the paraconsistent negation (¬□p).
- 7.
Since there are 3 hexagons and each hexagon contains 6 formulas, one might expect the total number to be 3×6=18 formulas. However, this calculation ignores the fact that certain formulas occur in two distinct hexagons. In particular, the formulas ◊p and ¬◊p occur in hexagons (a) and (b), the formulas □p and ¬□p occur in hexagons (a) and (c), and the formulas p and ¬p occur in hexagons (b) and (c).
- 8.
Somewhat confusingly, Béziau [1] talks about the ‘stellar dodecahedron’ instead of the ‘stellar rhombic dodecahedron’, thereby suggesting that the solid he had in mind (but never actually drew) is the stellation of the ‘ordinary’ pentagonal (i.e. Platonic) dodecahedron, rather than that of a rhombic dodecahedron. This confusion resurfaces in Moretti’s remarks that Béziau’s solid is “obtained by constructing a pentagonal pyramid or spike over each of the 12 pentagonal faces of a dodecahedron” [19, p. 75, our emphases]. Accordingly, the figure given by Moretti [19, p. 76] shows the stellation of a pentagonal dodecahedron, rather than that of a rhombic dodecahedron (although he still calls it ‘Escher’s solid’ and attributes it to Béziau). In a more recent paper, Béziau does provide a figure of the stellar rhombic dodecahedron (without its S5-decoration) [5, p. 13], but still calls it the ‘stellar dodecahedron’.
- 9.
Also recall Footnote 7.
- 10.
Note, trivially perhaps, that these two formulas do not occur together in any of the four JSB hexagons considered above. After all, if a diagram contains these two formulas, then it cannot be Boolean closed.
- 11.
- 12.
The differences between these three visualisations are discussed in more detail in [27, Sect. 2].
- 13.
Going beyond the rhombic dodecahedron would thus require us to introduce bitstrings of length 5. This can certainly be done, but the Aristotelian diagrams will become exponentially larger. For example, as far as Boolean closed diagrams are concerned, we move from the rhombic dodecahedron (which has 24−2=14 vertices) to a diagram that has 25−2=30 vertices. (Recall Footnotes 3 and 4.)
- 14.
We will henceforth add a ‘1’ in subscript to the expressions many and few to refer to the semantic interpretation they receive in Béziau’s analysis. Similarly, in the next sections, we will add a ‘2’ in subscript to refer to our alternative analysis.
- 15.
In terms of bitstrings, the one-sided reading corresponds to a bitstring that has one transition in bit values, i.e. from 1 to 0 or vice versa (e.g. 1110), whereas the two-sided reading corresponds to a bitstring having two transitions in bit values (e.g. 0110).
- 16.
This distinction also applies to the modal operators, where one-sided possibility (◊p) is compatible with necessity, but two-sided possibility (◊p∧¬□p, usually called ‘contingency’) is not.
- 17.
- 18.
We only look at the minimal number of binary connectives, because every semantic value (bitstring) can be expressed in a number of syntactically different ways. For example, the bitstring 0010 can be expressed as ¬p∧◊p [0011 ∧ 1110] with one binary connective, but also as ¬p∧(□p∨(¬p∧◊p)) [0011 ∧ 1010] with three binary connectives.
- 19.
From a linguistic point of view, the semantics of many and few is notoriously complex. For example, Keenan writes: “we shall largely exclude many and few from the generalisations we propose since our judgements regarding their interpretations are variable and often unclear” [16, pp. 47–48].
- 20.
If we were to work with bitstrings of length 3, ignoring the subjective quantifiers and focussing on the six quantifier expressions in the top part of Table 2, some would be the L1 bitstring 010, whereas at least one would be the L2 bitstring 110.
- 21.
More specifically, three types of mismatches can be distinguished: (i) there is both a semantic and a lexical arrow but they point in opposite directions—e.g. between 1100 and 0100, (ii) there is a semantic arrow but no lexical arrow at all—e.g. between 0100 and 1101, and (iii) there is a lexical arrow but no semantic arrow at all—e.g. between 1100 and 0101.
- 22.
Notice, incidentally, that the difference in subscripts between Béziau’s many 1 and our many 2 nicely reflects this contrast between the one-sided and the two-sided readings.
- 23.
Recall Footnote 15.
- 24.
Using simple propositional reasoning, the expression many if not all can be shown to be equivalent to the expression many or all. Intuitions differ as to whether an expression of the form p if not q should be read as the conditional ¬p→q or rather as ¬q→p, but both readings are equivalent to the disjunction p∨q.
- 25.
Since any can be seen as the negation of no, the expression few if any is semantically equivalent to few if not no, and can thus also be shown to boil down to the disjunction few or no (recall Footnote 24). Additional linguistic evidence for this equivalence comes from translational equivalents such as Dutch weinig of geen and French peu ou pas. Furthermore, even in English the disjunctive semantics of few or no is lexicalised, albeit only for abstract and mass nouns, viz. as little or no.
- 26.
Note that there is no horizontal semantic arrow between the bitstrings 1100 and 0101, since neither of them entails the other one. There is no horizontal lexical arrow between them either, since their corresponding quantifier expressions have the same degree of lexical complexity (viz. a single Boolean operator). Because of this twofold absence, the correlation between semantic and lexical complexity is preserved in this case as well. Similar remarks apply to the case of 1010 and 0011.
- 27.
It should be emphasised that we take the term ‘Buridan octagon’ to refer to (the visualisation of) a certain constellation of Aristotelian relations, rather than to the particular form or content matter of the formulas involved. As a matter of fact, Buridan’s own use of his octagon was in describing the logical geometry of first-order modal logic, rather than that of the subjective quantifiers.
- 28.
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Acknowledgement
We thank Dany Jaspers, Alessio Moretti, Fabien Schang and Margaux Smets for their comments on earlier versions of this paper. The second author gratefully acknowledges financial support from the Research Foundation—Flanders (FWO).
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Smessaert, H., Demey, L. (2015). Béziau’s Contributions to the Logical Geometry of Modalities and Quantifiers. In: Koslow, A., Buchsbaum, A. (eds) The Road to Universal Logic. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-10193-4_23
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