Abstract
Expressively equivalent logical languages can enunciate logical notions in notationally diversified ways. Frege’s Begriffsschrift, Peirce’s Existential Graphs, and the notations presented by Wittgenstein in the Tractatus all express the sentential fragment of classical logic, each in its own way. In what sense do expressively equivalent notations differ? According to recent interpretations, Begriffsschrift and Existential Graphs differ from other logical notations because they are capable of “multiple readings.” We refute this interpretation by showing that there are at least three different kinds of such multiple readings. While readings of the first kind do not capture any essential difference among notations but only among vocabularies, corresponding to readings of the second and the third kind two general parameters according to which notations may differ are defined: linearity vs. non-linearity, and tabularity vs. non-tabularity. This answers the question of how there can be substantially different but expressively equivalent logical notations.
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Notes
To pre-empt misinterpretations from the get-go: one must not conflate expressive equivalence with notational or syntactic equivalence. Two notations are expressively equivalent if they characterize the same logic (anything expressible in one is also expressible in the other; in model-theoretic terms, have the same models). But two expressively equivalent notations, such as BS and EGs, may clearly differ in terms of not being notationally equivalent. For the sake of clarity and conciseness, the present paper is concerned only with languages expressively equivalent to propositional calculus PC in the sense of being two-element Boolean algebras (Ma and Pietarinen 2018b).
Landini holds that it was Wittgenstein’s mistaken view that his N-operator captures what the ab-notation captures. In our terminology, Landini thinks that Wittgenstein thought his N-notation, which is non-tabular, to be tabular. Whether Landini is right or not, nothing in our arguments depends on this concern.
We do not claim that logical notations can differ according to our parameters only. There can be other parameters, aspects or undiscovered phenomena with respect to which logical notations may differ substantially. What we claim is that our parameters do qualitatively differentiate between logical notations.
We offer a more detailed account of Alpha’s syntax in Section 3.
In a critical review of Macbeth’s book, Sullivan (2009) argues that the fact that a notation is capable of multiple readings says nothing of the notation itself. According to Sullivan, “[d]ifferent “readings” here are simply different translations into English” (2009: 93) or, as in Macbeth’s and Shin’s examples, into a notation richer than the BS or EGs in the amount of the primitive symbols those notations consist of.
More appropriately speaking the Peirce Stroke, as it was Peirce who discovered the functional completeness of the NOR and NAND operators in 1880 (see Peirce 1989: 218–221).
Cf., e.g., Church 1956: 48, 48n111.
Example adapted from Howse et al. 2002. Let us ignore the special cases of non-associative languages, for example, in which some such differences may be non-typographic. Clearly in all cases in which typographical differences aquire relevant new meanings in any language or meta-language of logic, those differences come to life not by changing the typography but by definitions and conventions associated with those changes.
In Hjelmslev’s largely forgotten terminology (Hjelmslev 1961), a “permutation” is a mutation in the ordering of the elements of a sentence, which mutation produces distinct types. When any two elements do not mute, they “substitute,” and substitution produces distinct tokens of the same type.
We refer to Roberts (1973) for the staple presentation of the basics of the system of EGs, and Peirce (1911, 2019) for further details on his own original accounts. It is necessary to appreciate the system of conventions that underlie the way Peirce’s graphical system of logic was set up. For example, given an area of a graph, such as the area of the Sheet of Assertion in the four examples of Illustration 23 a graph can be scribed on any position of that area. As noted above, properties such as commutativity of graphs on the same area at its different positions, as well as the properties of associativity and adjunction, are properties that fall from the properties of the space that the Sheet of Assertion represents and on which the graphs are scribed (see, e.g., Pietarinen and Bellucci 2017). The role of the syntactic rules of transformation governing the system of permissible transformations is irrelevant.
All of these are Peirce’s own definitions and terminology, see especially Peirce 1910. As regards the converse, “exoporeutic” interpretation, the Alpha graph in Illustration 25 would mean ‘It is true that R or it is false that P and Q’.
We mean sentential logic, for full predicate logic is not decidable and not expressible in a truth-tabular form.
A language in which all and only logical equivalents have exactly one and the same expression makes it easy to see that there is a decision procedure for the logic of such language. Propositional logic is decidable (e.g., by truth-tables), full quantification theory only semi-decidable. Landini argues that the undecidability of quantification theory undermined Wittgenstein’s project of finding a notation in which all and only logical equivalents have one and the same representation (2007: 118). As far as the propositional fragment is concerned, decidability is not a problem. Landini is correct that Wittgenstein’s notations (the truth-tabular one we are discussing) and the ab-notation (Landini 2007: 113–114), substantially equivalent to the former, are decision procedures for propositional logic. The failure of the decidability of suitably large fragments of first-order (and higher-order) logics need not have undermined this project of Wittgenstein’s, however. For one, some useful fragments of first-order logic are decidable, and for the full language, the set of inconsistent formulas can be effectively decided. On the other hand, Wittgenstein’s N-operator defines an undecidable theory of predicate logic.
To do so would be to conflate what can be done syntactically with what the semantic constrains are. One cannot conjure a syntactic permutation-invariance up by fiat, because whether one is able to have permutation-invariance in the syntax of the language depends on the semantic facts of the matter.
On the derivation of distributivity, which requires Peirce’s Rule (residuation), see Ma and Pietarinen 2017. We note that the set of rules (L2–L5) is semantically incomplete. To have a complete system, one also needs a rule of erasure as well as the rule of insertion, which are not equivalence rules, plus the axiom that the sheet of assertion represents tautology.
The rule of erasure/insertion of double cut is itself a rule of transformation which Peirce derived from certain more primitive considerations that have to do with a conditionals (the scroll) as the primitive sign and observations that some graphs are blanks (SA). The rule of erasure/insertion is the only rule that is needed to demonstrate how residuation (Peirce’s Rule), in turn, emerges, showing the essentially observational nature of logical rules. The rule of erasure/insertion also achieves the same result as what TA does in linear languages.
Cf. Potter 2008, 161–162.
The present paper is not a review of all possible notational varieties equivalent to sentential logic, let alone the countless other logics. Our claims are limited to what we have justifiably taken to come from the pool of fundamental parameters of notational variability.
Landini (2012) draws attention to the fact that the Grundgezetze have a rule of the ‘amalgamation of the horizontals’ which is absent in the 1879 presentation of the notation, and that this difference is crucial. According to Landini, the amalgamation of horizontals plays an essential role in the proof of a theorem from Basic Law IV, and there seems to be no way of representing this operation without Frege’s BS (2012: 12–13, 52–57). In Landini’s view, Frege’s Grundgesetze notation is a two-dimensional notation because only in this way it can allow for the rule of amalgamation. Whether Landini’s claim is correct does not affect the fact that BS is linear in our sense, although linear on both its dimensions, given that TA is accepted as a rule of logical equivalence, and BS is non-linear if CPC is accepted as a rule of syntactical equivalence.
Notations can be non-tabular along different types of linearity and non-linearity. We do not delve further into such further distinctions in the present paper. We also leave aside the important topic of whether expressively identical notations differ in their simplicity, iconicity, and analyticity (R 300; CP 4.561n), or whether an emphasis on iconicity would count as a criterion for the development of certain graphical notations such as EGs, over others notations such as the algebraic ones. This topic has been covered in the recent literature and it necessitates introducing the quantificational Beta part of the logic of EGs over and above the propositional Alpha for its adequate investigation (Pietarinen 2015a, b). It is only the notational differences prompted by the Alpha part that the present paper is concerned with.
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Acknowledgments
Various parts of the paper were presented at the Chinese Academy of Social Sciences, Logic Section, October 2016, with the title, “Syntax, Semantics and Notationics”; Beijing Normal University May 2017, under the title “Peirce’s Logical Philosophy”; the SeRiC series of seminars at the Department of Philosophy and Communication, University of Bologna, May 2018, with the title “What is a Diagram?”; and the Formal Philosophy Conference in Moscow, October 2018, under the title “Peircenstein.” Our thanks go to the organizers and participants of these seminars and conferences, as well as to the reviewer of Acta Analytica for helpful comments. We are not responsible for any typos or formatting inconsistency that may appear in the paper.
Funding
Research supported by the Estonian Research Council (“The Diagrammatic Mind” (DiaMind), PUT267, 2013–2015, PI Pietarinen; “Abduction in the Age of Fundamental Uncertainty,” PUT 1305, 2016–2018, PI Pietarinen); Academy of Finland (“Logical and Cognitive Investigations of Iconicity,” 1270335, 2013–2017; Chinese Academy of Social Sciences Visiting Fellowship Grant, “Pragmatic Logical Philosophy,” 297291, 2016, PI Pietarinen); HSE University Basic Research Program funded by the Russian Academic Excellence Project “5-100.”
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Bellucci, F., Pietarinen, AV. Notational Differences. Acta Anal 35, 289–314 (2020). https://doi.org/10.1007/s12136-020-00425-1
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DOI: https://doi.org/10.1007/s12136-020-00425-1