Classical propositional logic (and other propositional logics) are generally presented with atomic formulas, due to a philosophical idea of Wittgenstein (1921), baptized by Bertrand Russell “Logical Atomism” (1924). This philosophical view is controversial and from a mathematical point of view it is possible to construct propositional logic without atoms. This is what we will show here in the case of classical propositional logic. Surprisingly enough this has not yet been studied in details. On the one hand Gödel very succinctly talked about that in a informal way (1929/1930). On the other hand Suszko developed what he called “abstract logic”, a general theory of propositional logics without the atomic assumption, but did not study in details particular cases.

We will here present a precise mathematical definition of classical propositional logic without atoms, present a semantics for it, a sequent calculus and prove the completeness theorem using a very general abstract version of this theorem (Beziau 2001).

References

• J.-Y. Béziau, “Sequents and bivaluations”, Logique et Analyse, 44 (2001), pp.373-394.
• K. Gödel, “Eine Eigenschaft der Realisierungen des Aussagenkalküls”, Ergebnisse eines mathematischen Kolloquiums, 2 (1929/30), pp.20-21.
• B. Russell, “Logical Atomism”, in J. H. Muirhead (ed.), Contemporary British Philosophers, London: Allen and Unwin, 1924, pp.356–383.
• R. Suszko (with S. L. Bloom and D. J. Brown) “A note on abstract logics”, Bulletin de l’Académie Polonaise des Sciences, 18 (1970), pp. 109-110.
• L. Wittgenstein, “Logisch-Philosophische Abhandlung”, Annalen der Naturphilosophie, 14 (1921).