Tarski ’ s Claim thirty years later

Tarski’s Claim (TC = Theorem 8 in [13]) follows from simple considerations in (type-free) lambda-calculus. The present note records essentially a proof of Lemma 1.1 in [16], i.e. TCL = the type-free lambdacalculus variant of TC, as well as a few historical comments appearing there. Additional remarks are meant to insure the fact that TCL can be transferred verbatim to typed lambda-calculus [TCLT]. (TCLT is just a notational variant of the derivation of TC in ordinary Lukasiewicz / traditional style.) The Addendum contains a transcription, in type-free lambda-calculus terms, of a Lukasiewicz / traditional style derivation of TC (notationally equivalent to TCLT), due to John Halleck [6] (September, 2010). Tarski’s Claim. Alfred Tarski claimed in 1925, without proof, the following (meta-) statement [hereafter TC]: Let L be a propositional logic in a propositional language containing at least implication (→). If L is finitely axiomatizable with modus ponens for → (and substitution) then L is also axiomatizable with a single axiom, and modus ponens (and substitution), provided it contains [K] p→(q→p) – [irrelevance] [D] p→(q→((p→(q→r))→r)) – [pairing] as theorems (‘theses’). Apparently, the Claim above was first mentioned in print in 1929, by Stanis law Leśniewski, Tarski’s PhD advisor, in [10], §§1–11. Tarski published his Claim slightly later, still without a proof, as Theorem 8, in [13]. Beyond Leśniewski, the original proof of TC was known to other Polish logicians, during the early twenties (as, e.g., to Jan Lukasiewicz), as well as to some other people, mathematicians and / or philosophers, at a later time. Among them one could mention, for instance, Carew A. Meredith (who attended Lukasiewicz’s lectures held at the Royal Irish Academy from 1947 on, cf. [14], page 514) and his occasional collaborator, Arthur N. Prior (cf., e.g., [15], §10, page 181). In retrospect, it is also likely that one could have included Bo leslaw Sobociński in the list. On the other hand, by the end of the seventies, any information as regards the ‘original method’ of proof seems to have been irretrievably lost. So, David Meredith (Carew’s cousin), in correspondence: 1Cf. pp. 58–59, in the German original; the full paper, containing also §12, dated 1938, appears, in English translation, in his Collected Works [11], II, pp. 410–605; cf., specifically, [11], II, §10, page 467.