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Combinatorial Bitstring Semantics for Arbitrary Logical Fragments

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Abstract

Logical geometry systematically studies Aristotelian diagrams, such as the classical square of oppositions and its extensions. These investigations rely heavily on the use of bitstrings, which are compact combinatorial representations of formulas that allow us to quickly determine their Aristotelian relations. However, because of their general nature, bitstrings can be applied to a wide variety of topics in philosophical logic beyond those of logical geometry. Hence, the main aim of this paper is to present a systematic technique for assigning bitstrings to arbitrary finite fragments of formulas in arbitrary logical systems, and to study the logical and combinatorial properties of this technique. It is based on the partition of logical space that is induced by a given fragment, and sheds new light on a number of interesting issues, such as the logic-dependence of the Aristotelian relations and the subtle interplay between the Aristotelian and Boolean structure of logical fragments. Finally, the bitstring technique also allows us to systematically analyze fragments from contemporary logical systems, such as public announcement logic, which could not be done before.

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Notes

  1. See http://www.logicalgeometry.org.

  2. A similar technique, that puts less emphasis on the semantics of the formulas, is Pellissier’s [45] setting approach. Seuren’s [53–55] valuation spaces and Schang’s [50] question-answer semantics are also related, but mathematically less developed than the bitstring approach proposed in the present paper.

  3. Given Boolean algebras \(\mathbb {A} = \langle A,\wedge _{\mathbb {A}},\vee _{\mathbb {A}},\neg _{\mathbb {A}},\bot _{\mathbb {A}},\top _{\mathbb {A}}\rangle \) and \(\mathbb {B} =\langle B,\wedge _{\mathbb {B}},\vee _{\mathbb {B}},\neg _{\mathbb {B}},\bot _{\mathbb {B}},\top _{\mathbb {B}}\rangle \), a Boolean algebra isomorphism \(f\colon \mathbb {A}\to \mathbb {B}\) is a bijection f:A → B that preserves Boolean structure, i.e. such that \(f(x \wedge _{\mathbb {A}} y) = f(x) \wedge _{\mathbb {B}} f(y)\) and \(f(\neg _{\mathbb {A}}x) = \neg _{\mathbb {B}}f(x)\). Using these properties, one can also show that \(f(\bot _{\mathbb {A}}) = \bot _{\mathbb {B}}\), \(f(\top _{\mathbb {A}}) = \top _{\mathbb {B}}\), \(f(x \vee _{\mathbb {A}} y) = f(x) \vee _{\mathbb {B}} f(y)\), etc. Usually, we will omit the subscripts, and simply write f(x ∧ y) = f(x)∧f(y), etc.

  4. When the system S is clear from the context, we will often leave it implicit, and simply talk about ‘contrary’ instead of ‘ S-contrary’, and write C instead of C S , etc.

  5. The ¬(φ ∧ ψ) part in Definition 1 specifies whether the formulas can be true together; similarly, given the equivalence of φ ∨ ψ and ¬( ¬ φ ∧ ¬ ψ), the φ ∨ ψ part in Definition 1 specifies whether the formulas can be false together.

  6. It should be emphasized that unlike Moretti [41] and Schang [51], who use bitstrings to encode the Aristotelian relations holding between formulas, the current paper uses bitstrings to encode the formulas themselves.

  7. Again, when the bitstring length n is clear from the context, we will often leave it implicit, and simply talk about ‘contrary’ instead of ‘n-contrary’, and write C instead of C n , etc.

  8. To avoid terminological confusion, note that we use the term Boolean isomorphism for a mapping between two fragments, regardless of their size and/or Boolean structure (see Definition 4). By contrast, the term Boolean algebra isomorphism is used for a mapping between two Boolean algebras (this is the usual notion of a bijective homomorphism between Boolean algebras; see, for example, Givant and Halmos [24, Chapter 12].

  9. For example, if R x is x-contrariety, then R y is the corresponding relation of y-contrariety.

  10. Depending on the values of x and y, there are thus four ‘types’ of Aristotelian and Boolean isomorphisms: (i) from formulas of S 1 to formulas of S 2, (ii) from formulas of S 1 to bitstrings of length n 2, (iii) from bitstrings of length n 1 to formulas of S 2, and (iv) from bitstrings of length n 1 to bitstrings of length n 2. Finally, note that it immediately follows from this definition that if two fragments are Boolean isomorphic, they have the same Boolean closure (up to Boolean algebra isomorphism).

  11. Many authors refer to this same notion as logical independence, e.g. see Hughes [28], Béziau [2], Seuren [53] and Read [48]. Furthermore, Smessaert and Demey [60] provide an alternative, positive characterization of unconnectedness as the combination of two other relations, viz. non-contradiction and non-implication.

  12. See Smessaert and Demey [60, 64] for the original formulation and proof of this theorem.

  13. The JSB hexagon in Fig. 2a is named after Jacoby [29], Sesmat [52] and Blanché [4], and the SC hexagon in (b) after William of Sherwood [33, 34] and Czeżowski [14]. The Boolean differences between these two types of diagrams are studied in Smessaert [58]. The distinction between strong and weak JSB hexagons was introduced by Pellissier [45], and will be studied in more detail in Section 5.1. Finally, the diagram in Fig. 2c is called an ‘unconnected-4’ hexagon because it contains exactly 4 pairs of unconnected formulas; it has recently been studied by Seuren [54] and Smessaert and Demey [59].

  14. Since Aristotelian diagrams typically contain only contingent formulas, the definition of being Boolean closed is restricted to contingent Boolean combinations. For example, even if a diagram contains formulas φ and ¬ φ, the condition of being Boolean closed does not require that it also contain their tautological disjunction φ ∨ ¬ φ and contradictory conjunction φ ∧ ¬ φ.

  15. A Buridan octagon is a type of Aristotelian diagram that was first studied by the medieval philosopher John Buridan [28, 48], and can be shown to contain two SC and two U4 hexagons [59, 63]. We will return to this type of Aristotelian diagram in Section 5.2.

  16. The 6 bitstrings with identical values are 1001, 1000, 0001, 1110, 0111 and 0110; the 8 bitstrings with different values are 1101, 1100, 0101, 0100, 1011, 1010, 0011 and 0010. Of course, the top- and bottom elements 1111 and 0000 also have identical values in their second and third bit positions, but as usual, these are ignored in Aristotelian diagrams (recall Footnote 14), which explains the numerical discrepancy between the two groups.

  17. For example, by collapsing the second and third bit positions, the bitstrings 1000 and 0110 for □p and ♢p ∧ ♢ ¬ p in RDH are compressed into the bitstrings 100 and 010 in Fig. 2a, respectively.

  18. We will write [b] i = [b] j to express the condition that a bitstring b has the same value in bit positions i and j. The complementary condition [b] i ≠ [b] j is satisfied by bitstrings with different values in positions i and j. Using this notation, the 6 strong JSB hexagons inside RDH correspond to the conditions [b]1 = [b]2, [b]1 = [b]3, [b]1 = [b]4, [b]2 = [b]3, [b]2 = [b]4 and [b]3 = [b]4, and the 6 complementary Buridan octagons correspond to the complementary conditions [b]1 ≠ [b]2, [b]1 ≠ [b]3, [b]1 ≠ [b]4, [b]2 ≠ [b]3, [b]2 ≠ [b]4 and [b]3 ≠ [b]4. Other subdiagrams of RDH turn out to correspond to other, more complex conditions on bitstrings.

  19. In general, it holds for all bitstrings b 1, b 2 ∈ {0,1}n, Aristotelian relations R and permutations π:{0,1}n→{0,1}n that R(b 1, b 2) iff R(π(b 1),π(b 2)). Hence, if a logical fragment \(\mathcal {F}\) can be represented by means of a bitstring isomorphism \(\beta \colon \mathcal {F}\subseteq \mathbb {B}_{n}\to \{0,1\}^{n}\), it can equally well be represented by means of \(\pi \circ \beta \colon \mathcal {F}\subseteq \mathbb {B}_{n}\to \{0,1\}^{n}\).

  20. For example, the conjunction (p ∧ q) ∧ ( ¬ p ∨ ¬ q) ∧ (p ∨ q) ∧ ( ¬ p ∧ ¬ q) is C P L-inconsistent, while the conjunction (p ∧ q) ∧ ¬( ¬ p ∨ ¬ q) ∧ (p ∨ q) ∧ ¬( ¬ p ∧ ¬ q) can be simplified to p ∧ q.

  21. We thus require that \(\mathcal {F}_{1} \cup \mathcal {F}_{2} = \mathcal {F}\). In concrete cases it will also typically hold that \(\mathcal {F}_{1}\cap \mathcal {F}_{2}=\emptyset \), but this is not strictly necessary.

  22. From a mathematical perspective, the meet of two partitions is well-defined because partitions form a lattice structure [5, 25]. From a more cognitive perspective, meets of partitions also play an important role in concept formation; for example, Seuren and Jaspers [56, p. 627] describe how the partition {male, female} “crosscuts” the partition {minor, adult}, thereby producing the new, more fine-grained partition {boy, man, girl, woman}. Finally, see [7] and [42] for a more visual-diagrammatic perspective on crosscutting partitions.

  23. In ongoing work, Demey [19] provides a systematic analysis of the effects of \(\beta _{\mathsf {S}}^{\mathcal {F}}\) outside the realm of \(\mathbb {B}(\mathcal {F})\).

  24. This is also manifest in the induction on formula complexity in the proof of Lemma 6: the base case is not about propositional atoms, as usual, but rather about formulas from \(\mathcal {F}\).

  25. For example, for b=1010∈{0,1}4 we define \(\varphi _{1010} := \bigvee \{\alpha _{i} \in \Pi _{\mathsf {S}}(\mathcal {F}) \mid [1010]_{i} = 1\} = \alpha _{1}\vee \alpha _{3}\), and thus \(\beta _{\mathsf {S}}^{\mathcal {F}}(\varphi _{1010}) = \beta _{\mathsf {S}}^{\mathcal {F}}(\alpha _{1}\vee \alpha _{3})=\beta _{\mathsf {S}}^{\mathcal {F}}(\alpha _{1})\vee \beta _{\mathsf {S}}^{\mathcal {F}}(\alpha _{3}) = 1000 \vee 0010 = 1010 = b\).

  26. We say that β is a permutation variant of \(\beta _{\mathsf {S}}^{\mathcal {F}}\) iff there is a permutation π:{0,1} n →{0,1} n such that \(\beta = \pi \circ \beta _{\mathsf {S}}^{\mathcal {F}}\); recall Footnote 19.

  27. For example, note that \(Pa \wedge \exists x\neg Px \in \mathbb {B}(\mathcal {F}^{\dag })-\mathbb {B}(\mathcal {F}^{\dag }_{**})\).

  28. Comparing the new partition \(\Pi _{\mathsf {FOL}}(\mathcal {F}^{\dag }_{**})\) with the original partition \(\Pi _{\mathsf {FOL}}(\mathcal {F}^{\dag })\), we see that the two original anchor formulas \(\alpha _{2},\alpha _{3} \in \Pi _{\mathsf {FOL}}(\mathcal {F}^{\dag })\), i.e. P a ∧ ¬∀x P x and ¬ P a ∧ ∃x P x, are collapsed into a single new anchor formula, viz. \((Pa \wedge \neg \forall x Px) \vee (\neg Pa \wedge \exists x Px) \equiv _{\mathsf {FOL}} \exists x Px \wedge \exists x \neg Px \in \Pi _{\mathsf {FOL}}(\mathcal {F}^{\dag }_{**})\).

  29. Comparing the new partition \(\Pi _{\mathsf {CPL}}(\mathcal {F}^{\ddag }_{*})\) with the original partition \(\Pi _{\mathsf {CPL}}(\mathcal {F}^{\ddag })\), we see that the original anchor formula \(\alpha _{2} \in \Pi _{\mathsf {CPL}}(\mathcal {F}^{\ddag })\), i.e. (p ∨ q)∧( ¬ p ∨ ¬ q), has been split into two new anchor formulas in \(\Pi _{\mathsf {CPL}}(\mathcal {F}^{\ddag }_{*})\), viz. p ∧ ¬ q and ¬ p ∧ q, in the sense that (p ∨ q)∧( ¬ p ∨ ¬ q)≡ C P L (p ∧ ¬ q)∨( ¬ p ∧ q).

  30. Note that we are only interested in the minimal number of bit positions (i.e. the minimal bitstring length) that is required to represent a fragment of a given size. After all, if a fragment can be represented by bitstrings of length n, then it can trivially also be represented by bitstrings of length k, for any k ≥ n.

  31. Theorem 4 assumes that \(|\Pi _{\mathsf {S}}(\mathcal {F})|\geq 2\). For the sake of completeness, note that if \(|\Pi _{\mathsf {S}}(\mathcal {F})|= 1\), then \(1\leq |\mathcal {F}| \leq 2\). To see this, note that \(|\Pi _{\mathsf {S}}(\mathcal {F})| = 1\) means that \(\Pi _{\mathsf {S}}(\mathcal {F}) = \{\top \}\), and thus \(\mathcal {F}=\{\top \}\) or \(\mathcal {F} = \{\bot \}\) or \(\mathcal {F}=\{\top ,\bot \}\).

  32. Theorem 5 assumes that \(|\mathcal {F}|\geq 2\). For the sake of completeness, note that if \(|\mathcal {F}|= 1\), then \(1\leq |\Pi _{\mathsf {S}}(\mathcal {F})| \leq 2\). To see this, suppose that \(\mathcal {F}=\{\varphi \}\); if φ is S-contingent, then \(\Pi _{\mathsf {S}}(\mathcal {F}) = \{\varphi ,\neg \varphi \}\), otherwise \(\Pi _{\mathsf {S}}(\mathcal {F}) = \{\top \}\).

  33. Demey and Smessaert [21, Subsection 5.2] and Demey [17] systematically study the logic-sensitivity of some other well-known Aristotelian diagrams, and its effects on the bitstrings that are used to represent those diagrams.

  34. The existential import of syllogistics is here formalized by including ∃x S x as an axiom (so we have S Y L=F O L∪{∃x S x}). Another formalization involves adding ∃x S x as a conjunct to the categorical statements: for φ∈{A,I,E,O}, put φ imp!: = φ ∧ ∃x S x [10, Definition 4], but then more needs to be said about the contradictions—for example, although A and O are contradictory to each other, A imp! = ∃x S x ∧ ∀x(S x → P x) is not contradictory to O imp! = ∃x S x ∧ ∃x(S x ∧ ¬ P x), but rather to ¬∃x S x ∨ ∃x(S x ∧ ¬ P x), which is equivalent to Chatti and Schang’s [10, Definition 5] O imp?.

  35. Observations such as these have led Seuren [55, p. 505–506] to call F O L an “impoverished system”, since some “logical (meta)relations are lost”. However, even though the relations may be lost in the concrete Aristotelian square shown in Fig. 4b, they are not lost ‘in general’, in the sense that F O L still has other formulas standing in those relations. For example, the formulas ∀x S x and ∀x ¬ S x are F O L-contrary, but happen to be absent from the square in Fig. 4b.

  36. The same situation also arises in modal logic, where systems containing the D-axiom ♢⊤ (which is the modal counterpart of the existential import axiom ∃x S x) yield a classical square, but the minimal normal system K merely yields an ‘X of opposition’ [11].

  37. Note that although the I- and O-statement are not in \(\mathcal {F}\), they are the Boolean negations of the E- and A-statements, respectively, and thus they do belong to the Boolean closure \(\mathbb {B}(\mathcal {F})\). The bitstring approach developed in Section 3 will thus allow us to assign bitstrings not only to the A- and E-statements, but also to the I- and O-statements (cf. Theorem 1).

  38. That bitstrings of length 3 do not suffice in the case of F O L should not come as a big surprise, since the F O L-‘cross’ in Fig. 4b contains unconnected formulas, and it is well-known that unconnectedness can only be represented by bitstrings of length at least 4 (recall Footnote 12).

  39. In Peirce’s logical writings [26], we already find what essentially amounts to a F O L-based bitstring semantics for the categorical statements in terms of bitstrings of length 4 (CP 2.456), immediately followed by the observation that this does not yield a classical square of opposition, but rather a ‘cross’ (CP 2.460).

  40. The difference between the two pairs of bitstring mappings is also manifest in their typographic rendering: in Example 7 we have different values for the fragment parameter in superscript (\(\beta _{\mathsf {FOL}}^{\mathcal {F}^{\dag }}\) vs. \(\beta _{\mathsf {FOL}}^{\mathcal {F}^{\dag }_{**}}\)), whereas in the present section we have different values for the logic parameter in subscript (\(\beta _{\mathsf {FOL}}^{\mathcal {F}}\) vs. \(\beta _{\mathsf {SYL}}^{\mathcal {F}}\)).

  41. Equivalently, one can also define a JSB hexagon to be strong iff the conjunction of the formulas on its subcontrariety triangle is an S-contradiction, i.e. ⊧ S ¬( ¬ α ∧ ¬ β ∧ ¬ γ).

  42. Since \(p\wedge \Diamond \neg p \in \mathcal {F}_{2}\) but \(p\wedge \Diamond \neg p \notin \mathbb {B}(\mathcal {F}_{1})\), it holds that \(\mathcal {F}_{2} \not \subseteq \mathbb {B}(\mathcal {F}_{1})\), and we thus cannot define bitstrings \(\beta _{\mathsf {S5}}^{\mathcal {F}_{1}}(\varphi )\) for \(\varphi \in \mathcal {F}_{2}\). Finally, note the analogy between \(\mathcal {F}_{1}\) and \(\mathcal {F}_{2}\) here and \(\mathcal {F}^{\dag }_{**}\) and \(\mathcal {F}^{\dag }\) in Example 7.

  43. Two diagrams are said to belong to the same Aristotelian family iff they are Aristotelian isomorphic to each other.

  44. In this section, we will write Q 1 Q 2 for the formula Q 1 x Q 2 y R(x,y) and Q 1 Q 2¬ for the formula Q 1 x Q 2 y ¬ R(x,y), with Q 1, Q 2 ∈ {∀, ∃}. For example, ∀x∃y R(x,y) and ∃x∀y ¬ R(x,y) will be abbreviated as ∀∃ and ∃∀¬, respectively. Finally, note that John Buridan himself already made use of formulas involving multiple quantifiers to construct a Buridan octagon. For example, he considered sentences of the form “of every human, every donkey runs” [48, p. 107], which can be formalized as ∀x(human(x)→∀y((donkey(y)∧own(x,y))→run(y))).

  45. Note, in particular, that the conjunctions on rows 5 and 8 are F O L-consistent.

  46. Note that \(\Pi _{\mathsf {S5}}(\mathcal {F}_{1}) = \Pi _{\mathsf {S5}}(\mathcal {F}_{1}^{b})\); the reason for this is that although \(\mathcal {F}_{1}^{a} \not \subseteq \mathcal {F}_{1}^{b}\), it does hold that \(\Pi _{\mathsf {S5}}(\mathcal {F}_{1}^{b})\) is a refinement of \(\Pi _{\mathsf {S5}}(\mathcal {F}_{1}^{a})\) (in the sense of Lemma 5), and hence \(\Pi _{\mathsf {S5}}(\mathcal {F}_{1}) = \Pi _{\mathsf {S5}}(\mathcal {F}_{1}^{a}) \wedge _{\mathsf {S5}} \Pi _{\mathsf {S5}}(\mathcal {F}_{1}^{b}) = \Pi _{\mathsf {S5}}(\mathcal {F}_{1}^{b})\) (also recall Footnote 22).

  47. Assume that the Buridan octagons in cases 2 and 3 have the formulas {α,β 1, β 2, γ} and \(\{\alpha ^{\prime },\beta _{1}^{\prime },\beta _{2}^{\prime },\gamma ^{\prime }\}\) as their respective left-hand-sides (see Fig. 8a). A concrete Boolean isomorphism ι between these Buridan octagons looks as follows: ι(α) = ¬ γ ′, \(\iota (\beta _{i}) = \neg \beta _{i}^{\prime }\) for i∈{1,2}, ι(γ) = ¬ α ′, and ι( ¬ φ) = ¬ ι(φ ′) for φ∈{α,β 1, β 2, γ}. Informally, ι thus maps the left-hand-side of the first Buridan octagon onto the right-hand-side of the second one, and vice versa.

  48. The first subtype, in which α ≡ β 1∧β 2 and γ ≡ β 1∨β 2, has also been called a rhombicube [59, 62, 63], because it can be represented by bitstrings of length 4, and can thus be embedded inside the Aristotelian rhombic dodecahedron (which visualizes \(\mathbb {B}_{4}\)).

  49. The Béziau octagon is an Aristotelian diagram named after Béziau [2], and can be shown to contain a JSB hexagon and an SC hexagon.

  50. To better appreciate the difference between these two calculations, note that it follows from Theorems 4 and 5 that \(|\Pi _{\mathsf {S}}(\mathcal {F})| \leq 2^{|\mathcal {F}|}\), whereas \(|\mathbb {B}(\mathcal {F})| = 2^{|\Pi _{\mathsf {S}}(\mathcal {F})|} \leq 2^{(2^{|\mathcal {F}|})}\) (for any logical system S and fragment \(\mathcal {F}\)).

  51. Note the notational similarity between the public announcement operators 〈!p〉/ [!p] and the ordinary modal operators ♢/ □, respectively. See [15, 16] for a more precise explanation as to why 〈!p〉 is existential in nature and [!p] universal.

  52. A similar situation arises in applications of Aristotelian diagrams in artificial intelligence. In a recent series of papers, Dubois, Prade and various co-authors have discovered that a single ‘cube of opposition’ (i.e. an Aristotelian octagon) can be used to describe various knowledge representation formalisms, such as formal concept analysis, modal logic, rough set theory, Sugeno integrals and several others. Consequently, they state that “This discovery leads to a new perspective on many knowledge representation formalisms, laying bare their underlying common features. The cube of opposition exhibits fruitful parallelisms between different formalisms, which leads to highlight some missing components present in one formalism and currently absent from another.” [22, p. 2933]. Yet another, more subtle example can be found in [62], which compares two competing accounts of the subjective quantifiers many and few. It is shown that on both accounts, these quantifiers induce a quadripartition (and thus require bitstrings of length 4) and yield a Buridan octagon, but one of them yields a better correlation between logical and lexical complexity, as reflected in the slightly different ways in which the subjective quantifier expressions are mapped onto the bitstrings.

  53. Recall the lingua franca quotation given in Section 1.

  54. It should be emphasized that despite its fully formal nature, the bitstring approach developed here is also perfectly applicable to natural language sentences and expressions. (To appreciate the importance of this observation, recall from Section 1 that linguistics is one of the primary fields of application for Aristotelian diagrams, and indirectly thus also for bitstrings.) After all, the crucial idea of the present approach is that of a fragment inducing a partition, which is independent of whether that fragment consists of formal or natural language sentences (also see Footnote 22). For example, [56] have shown that several elementary lexical fields (i.e. lexically coherent fragments of natural language expressions, such as {married, husband, wife, single}) induce tripartitions, and thus correspond to bitstrings of length 3. More recently, [49] has shown that other, more complex lexical fields (e.g. measure adjectives and gradable adjectives) induce quadripartitions, and thus correspond to bitstrings of length 4.

  55. If S is fully Boolean in nature, but also has additional language elements (e.g. modal operators, quantifiers, etc.), it might be interesting to develop a notion of bitstring semantics that also takes those additional elements into account (rather than just focusing on the Boolean structure of S). However, a potential disadvantage of such a fine-grained bitstring mapping might be that it remains too close to the system S itself, and will thus no longer allow us to draw interesting comparisons across different logical systems and applications (cf. Footnote 52). Thanks to an anonymous referee for an interesting discussion on this issue.

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Acknowledgments

Thanks to Jan Heylen, Alessio Moretti, Fabien Schang, Margaux Smets and an anonymous referee for their comments on earlier versions of this paper. The first author holds a Postdoctoral Fellowship of the Research Foundation – Flanders (FWO).

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  1. Center for Logic and Analytic Philosophy, KU Leuven, Leuven, Belgium

    Lorenz Demey

  2. Research Group on Formal and Computational Linguistics, KU Leuven, Leuven, Belgium

    Hans Smessaert

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Demey, L., Smessaert, H. Combinatorial Bitstring Semantics for Arbitrary Logical Fragments. J Philos Logic 47, 325–363 (2018). https://doi.org/10.1007/s10992-017-9430-5

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