Abstract
In this study, we examine modern reading of the Square of Opposition by means of intensional logic. Explicit use of possible world semantics helps us to sharply discriminate between the standard and modal (‘alethic’) readings of categorical statements. We get thus two basic versions of the Square. The Modal Square has not been introduced in the contemporary debate yet and so it is in the centre of interest. Some properties ascribed by medieval logicians to the Square require a shift from its Standard to Modal version. Not inevitably, because for each of the two squares there exists its mate which can be easily confused with it. The discrimination between the initial and modified versions of the Standard and Modal Square enable us to separate findings about properties of the Square into four groups, which makes their proper comparison possible. The disambiguation so achieved leads to the solution of various puzzles often mentioned in recent literature.
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Notes
- 1.
Needles to remind the present reader of the recent wave of scholars, papers, books and events organized by J.-Y. Béziau which are focused on Squares, Hexagons and other figures displaying oppositions (cf. [3]).
- 2.
- 3.
One of several reasons for adoption of hyperintensions is this. According to possible world semantics (PWS), the meaning of all true mathematical sentences is one and the same, viz. the proposition true in all possible worlds. Consequently, the argument “Alice believes that 8 = 8. Therefore, Alice believes that 2 × 4 = √64.” is evaluated as valid, which is intuitively not, Alice cannot be omniscient. Obviously, there is a structuredness issue which is relevant to the invalidity of such arguments. Semantic theory of TIL solves the paradox of omniscience by discriminating between the proposition true in all possible worlds, which is not the meaning but denotation of the sentences, and its infinitely many constructions differing in their structure.
- 4.
Of course, it may happen that a construction is denoted by an expression which expresses a higher-order construction of the denoted construction. Moreover, expression can lack denotation or even meaning.
- 5.
On the right side of ⇔ , ! can be omitted. Below, we will also use ! even if it is not inevitable.
- 6.
Compare it with [\(\mathbf{True}^{\mathbf{P}\boldsymbol{\pi }}_{w}\) p] ⇔ d f [p w = 1 ]. To the two notions of truth there correspond two notions of falsity. Cf. [27] for analysis of truth in TIL.
- 7.
Adopted from [23, 24]. The definitions of All and Some are borrowed from [32] where Tichý showed also concise proofs of various relations between constructions involving them. Further remarks: definienda suggested in brackets are usual in the topic of generalized quantifiers; common construal of generalized quantifiers as (binary) relations between classes presupposes Schönfinkel’s reduction (‘currying’) which is not generally valid in the logic utilizing partial functions.
- 8.
Cf. [16].
- 9.
Recall that this concept can be utilized for an apt definition of contradictoriness between propositional constructions.
- 10.
- 11.
Abbreviating thus “contrariety and subcontrariety”.
- 12.
Subaltern, Contrary, Subcontrary /(o o ω o ω ) (the relations between propositions).
- 13.
\(\neg \boldsymbol{E}\) is a natural abbreviation of \(\lambda w.\neg [\mathbf{True}^{\mathbf{T\boldsymbol{\pi }}_{w}}\ \boldsymbol{E}]\). Analogously for \(\neg \boldsymbol{A}\), \(\neg \boldsymbol{I}\), \(\neg \boldsymbol{O}\).
- 14.
We will treat the compound symbols of quantifiers as simple. Their obvious definitions see below.
- 15.
A rare example of their distinguishing as Apuleian Square and the (Gottschalk’s) Logical Quatern can be found in [30, p. 294].
- 16.
- 17.
(Schematic) statements such as \(\lambda w.\boldsymbol{\square }\boldsymbol{P}\) are analytic—they are constructions constructing constant propositions, which is apparent from their ‘bodies’ (e.g. \(\boldsymbol{\square }\boldsymbol{P}\)) which contain no free possible world variables.
- 18.
Analogously as above, statements such as λ w[Requisite \(\boldsymbol{G\,F}\)] are analytic—they are constructions constructing constant propositions, which is apparent from their ‘bodies’ which (normally) contain no free possible world variables.
- 19.
For an exhaustive study of such notions see [22] from which we borrow our definitions.
- 20.
After finishing my paper, M. Duží—who is also using Tichý’s logic—reminded me that she proposed the essentials of the modal modern reading of the Square in her presentation [11]. She also proposed to call properties concepts, which leads to the quadruple of statements about their two basic relations; schematically: “The concept F is subsumed by/compatible with the concept G/non-G”.
- 21.
Cf. [22], where void properties are defined and related to accidental, essential and partly essential properties. However, there is only a partial correspondence of the quadruple of those properties with the Square studied below because requisites and potentialities are not essential or accidental properties, but kinds of properties essential for or accidental for (cf. Sect. 5.3).
- 22.
Cf. [28].
- 23.
We keep here the original notation though it clashes with our previous use of “U”.
- 24.
The work on the paper has been partly supported by the grant of the Czech Science Foundation (GACR) ‘Semantic Notions, Paradoxes and Hyperintensional Logic Based on Modern Theory of Types,’ registration no. GA16-19395S.
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Raclavský, J. (2017). Two Standard and Two Modal Squares of Opposition. In: Béziau, JY., Basti, G. (eds) The Square of Opposition: A Cornerstone of Thought. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-45062-9_8
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