Abstract
At first the title of this paper will provoke both logicians and Leibnitians to shake their heads: while talking about calculi of strict implication one rather associates it with the names of C 1. Lewis, W. Ackermann, or some other contemporary logician, but hardly with the 17th century philosopher G. W. Leibniz1. Yet already some 30 years ago in an essay on “Leibniz’s interpretation of his logical calculi” the wellknown Leibniz scholar, Nicholas Rescher (1954), had put forward the then bold thesis: in an interpretation that Leibniz himself had suggested, one of these calculi would become “a precursor of C. I. Lewis’ systems of strict implication” (p.10), Unfortunately, Rescher did not give a factual justification of this prophetic view. Had it been done, it would have thrown some light on the real significance of Leibniz’s logic. One reason for this omission consists, perhaps, in the fact that Rescher incorrectly interpreted the important logical constant ‘est Ens’, or, synonymously, ‘est Res’ or ‘est Possibile’ as logical necessity instead of logical possibility.
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Notes
The “indefinita” function as disguised conceptual-quantifiers for Leibniz; for a detailed account of the Leibnitian logic of the quantifiers, cf. our paper “‘Unbestimmte Begriffe bei Leibniz” Studia Leibnitiana, XVI (1984), 1–26.
The first one appeared in Studia Leibnitiana XVIII (1986), 1–37, the second is forthcoming in the contributions to the symposium Leibniz: Questions de Logique, Brtlssel, Louvain-la-Neuve 1985.
In the Definitiones from around 1679 (AV, 1, 146–7) one can find a marginal sketch of a characteristic in which especially ‘est’ is abbreviated by ‘e’ Leibniz did not use this symbol, however, in any of the known drafts of a calculus.
Compare M. Dummet, Review of Rescher (1954) in Journal of Symbolic Logic 21 (1956), p.198; also cf. Castaneda (1976), ref.3, p.484, where Leibniz is blamed for the following: “… having analyzed ‘some A’s are B’s’ as ‘AB exists’ [he] does not go on to interpret this as ‘AB contains existence’ which would be symbolized as ‘AB ≡ AB (Existence)’. He takes this step tin GP VII, 213] but he leaves the concept existence or entity somewhat isolated.” Fortunately Leibniz did not further pursue this mistaken approach otherwise, and the “serious troubles” that Castaneda deduces later on (especially pp.489 ff) do not refute the system of Leibnitian logic but only the miscarried reconstruction of it by Castaneda.
That ‘Ens’ should not be viewed as a conceptual-constant has first been noticed by L. Couturat: cf. his La Logique de Leibniz, Paris 1901 (Reprint Hildesheim 1966), p.353, ref.2. A very detailed discussion of this point may also be found in
R. Kauppi, Über die Leibnizche Logik, Helsinki 1960, especially pp.215–222. Finally, it should be pointed out that analogous interpretation of the truth concept as a 1st order conceptual-constant as probed in §108 GI leads to the same difficulties. With ‘V’ abbreviating Verum’ Leibniz attempts to interpret the proposition ‘A est verum’ — in which ‘A’ stands for a concept such as ‘Homo’ -predicatively and thus obtains in accordance with K6 the equation A = AV: “A = A verum seu A est verum”. However, because of ABeA, this interpretation would entail that if A is true, then every conjunction AB is true as well, especially AĀ would be true. But that is absurd.
Cf. also R. Kauppi’s (1960) similar interpretation of these principles; however, she expresses a qualification: “In dieser Form sind sie nicht ausdrücklich von LEIBNIZ aufgestellt worden” (p.182, footnote 2).
In De Vero et Falso, Aff irmatione et Negatione, et de Contradictoriis (AV, 1, 86–8) it is said analogously “12) Si posîtis enuntiationibus sequatur nova et haec sit falsa, etiam aliqua ex illis erit falsa”. By the way, Leibniz remarks: “Hoc est axioma” instead of “Hoc est regula”!
AV, 1, 184–90; the following quotation does not appear in the partial edition in Grua, 322–4, but is found in F. Schmidt (ed.), G. W. Leibniz — Fragmente zur Logik (Berlin-East 1960), 474–8. Later references to this fragment refer to the Edition in AV.
C. I. Lewis & H. G. Langford, Symbolic Logic, New York2 1959, S.124.
Analysis Particularum, ed. by F. Schupp in Studia Leib-nitiana Sonderheft 8 (1979), 138–53; quotation p.145.
Cf. Kauppi (1960), especially ch.IV, §3; also Kauppi, “Zur Analyse der hypothetischen Aussage bei Leibniz”, in A. Heinekamp & F. Schupp (ed.), Die Intensionale Logik bei Leibniz und in der Gegenwart, Wiesbaden 1979, 1–9;
H. Burkhardt, Logik und Semiotik in der Philosophie von Leibniz, München 1980, passim (cf. under the heading ‘hypothetisch’);
H. Ishiguro in M. Hooker (ed.), Leibniz: Critical and Interpretative Essays, Minneapolis 1982, 90–102 is indeed concerned with the theme “Leibniz on hypothetical truth” but she ignores the reduction of hypothetical to categorical propositions that is advocated by Leibniz with verve; cf. finally F. Schupp’s (1982) commentary to GI, o.c., especially 164–5, where further literature is given.
O.c., p. 110; cf. similarly the versions in Specimen Certitudinis, o.c., p.372.
O.c., p.82; Leibniz says that the former proposition “resolvi potesti in hac duas” — i.e., into the latter propositions.
This need not mean that 11.7 were derivable from MPS and the other laws of strict implication. If one drops 11.7 without substitution, weaker systems result that are designated as respective “‘nought systems’“(S1°, S2°, S30,…) in the terminology of J. J. Zeman, Modal Logic—The Lewis Modal Systems (Oxford 1973).
As to the “lack” of the law of associativity cf. K. Durr, Neue Beleuchtung einer Theorie von Leibniz (Darmstadt 1930), pp.53 ff; Rescher (1954) p.11; Kauppi (1960), p.173; and FL Burk-hardt (1980), p.353 with suggestions to further readings (ref.399).
Cf. Zeman (1973), pp. 176–7.
There the “rule of necessitation” is valid only for the truth-functional tautologies and for the sentences of the form ¬P¬β ⇒ γ; cf. Zeman (1973), pp.104 ff. and 184 ff.
H. Poser, Zur Theorie der Modalbegriffe bei G. W. Leibniz, Wiesbaden 1969, p.60.
S4, unlike T, contains only finitely many non-equivalent “modalities”; cf. Zeman (1973), 179–81.
Cf. his paper “Leibniz’s Theories of Contingency”, reprinted in M Hooker (ed.), Leibniz: Critical and Interpretative Essays, 243–83. On p.275 Adams remarks: “… the characteristic axiom of S5… is not valid on the emonstrability conception of necessity. For a proposition may be indemonstrable without being demonstrably indemonstrable.”
For a short outline of deontic-logical systems cf., for instance, F. von Kutschers, Einführung in die Logik der Normen, Werte und Entscheidungen (Freiburg 1973), ch.1. Systems of epistemic logic are described in detail in our Glauben, Wissen und Wahrscheinlichkeit (Wien 1980).
Cf. A. R. Anderson & N. D. Belnap, Entailment I (Princeton 1975), 12 and 22.3.
Cf. Zeman (1973) p.86; as a corollary of this proposition one obtains the further meta-theorem, that S1° — thus also Leibniz’s L1S — covers the whole of propositional-logic.
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© 1987 Martinus Nijhoff Publishers, Dordrecht
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Lenzen, W. (1987). Leibniz’s Calculus of Strict Implication. In: Srzednicki, J. (eds) Initiatives in Logic. Reason and Argument, vol 2. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3673-7_1
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