Any intermediate propositional logic (i.e., a logic including intuitionistic logic and contained in classical logic) can be extended to a calculus with epsilon- and tau-operators and critical formulas. For classical logic, this results in Hilbert’s ε-calculus. The first and second ε-theorems for classical logic establish conservativity of the ε-calculus over its classical base logic. It is well known that the second ε-theorem fails for the intuitionistic ε-calculus, as prenexation is impossible. The paper investigates the effect of adding critical ε- and τ -formulas and using the translation of quantifiers into ε- and τ -terms to intermediate logics. It is shown that conservativity over the propositional base logic also holds for such intermediate ετ -calculi. The “extended” first ε-theorem holds if the base logic is finite-valued Gödel-Dummett logic, fails otherwise, but holds for certain provable formulas in infinite-valued Gödel logic. The second ε-theorem also holds for finite-valued first-order Gödel logics. The methods used to prove the extended first ε-theorem for infinite-valued Gödel logic suggest applications to theories of arithmetic.

]]>The use of the symbol ∨ for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol ∨ in its historical and logical context. Some sources say that disjunction in its use as connecting propositions or formulas was introduced by Peano; others suggest that it originated as an abbreviation of the Latin word for “or”, *vel*. We show that the origin of the symbol ∨ for disjunction can be traced to Whitehead and Russell’s pre-*Principia* work in formal logic. Because of *Principia*’s influence, its notation was widely adopted by philosophers working in logic (the logical empiricists in the 1920s and 1930s, especially Carnap and early Quine). Hilbert’s adoption of ∨ in his *Grundzüge der theoretischen Logik* guaranteed its widespread use by mathematical logicians. The origins of other logical symbols are also discussed.

We investigate a recent proposal for modal hypersequent calculi. The interpretation of relational hypersequents incorporates an accessibility relation along the hypersequent. These systems give the same interpretation of hypersequents as Lellman’s linear nested sequents, but were developed independently by Restall for S5 and extended to other normal modal logics by Parisi. The resulting systems obey Došen’s principle: the modal rules are the same across different modal logics. Different modal systems only differ in the presence or absence of external structural rules. With the exception of S5, the systems are modular in the sense that different structural rules capture different properties of the accessibility relation. We provide the first direct semantical cut-free completeness proofs for K, T, and D, and show how this method fails in the case of B and S4.

]]>Any set of truth-functional connectives has sequent calculus rules that can be generated systematically from the truth tables of the connectives. Such a sequent calculus gives rise to a multi-conclusion natural deduction system and to a version of Parigot’s free deduction. The elimination rules are “general,” but can be systematically simplified. Cut-elimination and normalization hold. Restriction to a single formula in the succedent yields intuitionistic versions of these systems. The rules also yield generalized lambda calculi providing proof terms for natural deduction proofs as in the Curry-Howard isomorphism. Addition of an indirect proof rule yields classical single-conclusion versions of these systems. Gentzen’s standard systems arise as special cases.

]]>Published in the UK on August 17, 2021. Available in hardcover and paperback.

*An Introduction to Proof Theory provides* an accessible introduction to the theory of proofs, with details worked out and many examples and exercises. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic (and arithmetic), natural deduction and the normalization theorems (for both NJ and NK), the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these results. The second half examines Gentzen’s consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal proof theory are developed from scratch. The proof methods needed, especially proof by induction, are introduced in stages throughout the text.

This article makes available some early letters chronicling the relationship between the biologist Joseph H. Woodger and the logician Alfred Tarski. Using twenty-five unpublished letters from Tarski to Woodger preserved in the Woodger Papers at University College, London, I reconstruct their relationship for the period 1935–1950. The scientific aspects of the correspondence concern, among other things, Tarski’s reports on the work he is doing, his interests, and his — sometimes critical but always well-meaning — reactions to Woodger’s attempts at axiomatizing and formalizing biology using the system of Principia Mathematica. Perhaps the most interesting letter from a philosophical point of view is a very informative letter on nominalism dated November 21, 1948. But just as fascinating are the personal elements, the dramatic period leading to the second world war, their reaction to the war events, Tarski’s anguish for his family stranded back in Poland, the financial worries, and his first reports on life in the East Coast and, as of 1942, at the University of California, Berkeley.

There is much that is interesting in this correspondence, but what struck me most was letter 8, dated May 22, 1939. This is exactly the time when Tarski was considering offers by Quine and others to come to America. Tarski was hesitant, something we now, in hindsight, find puzzling. The Fefermans, in their biography of Tarski, also found this puzzling. Their explanation was that Tarski’s “surpreme absence of self-doubt” was the determining factor. Leśniewski had just died and, the Fefermans conjectured, Tarski was certain that as “clearly the pre-eminent logician in Poland”, he would be appointed as Leśniewski’s successor in Warsaw (p. 106–107).

Letter 8 paints a completely different picture, which any current precariously employed scientist can appreciate. Quine suggested that Tarski should travel to the US in the Fall of 1939 and stay for a year. There was no offer of employment, no grant money, nothing, just vague assurances that invitations and perhaps honoraria would be forthcoming. In Poland, Tarski made ends meet as a high-school teacher and adjunct lecturer at Warsaw University. This is Tarski’s evaluation of his situation:

It seems to me that it would be criminally reckless if I decided to follow Quine’s “suggestions” and your advice. You don’t take into account a circumstance. I have two parents, a wife and two children. And these are all people, and no marmots, no bears, etc.: they are not able to sink into sleep for a few months. If I decide to go to America and spend the next school year there (regardless of whether I get a job there), I have to inform the school that I am giving up my hours next year; as a result, a third of my income will be taken away as early as 1 August (i.e. the money I receive at school). I have to do the same at the university—I have to inform them that I do not intend to give lectures in the coming year; as a result, on 1 September, the second third of my income will be eliminated. The last third (i.e. the money I get in university for my adjunct position) will be eliminated on October 1st if I don’t take up my duty at that time. My savings are enough for the passport, the visa, the trip to America and back and still for a one-month stay in America. So what will my family live on in September? My wife earns about 6.50£ a month, but that’s too little for one person, let alone for five.

He goes on to contrast Quine’s offer to him with the much more generous and definite offers of funding to Carnap and Woodger himself. Later letters detail his repeated attempts at rescuing his family from Nazi-occupied Poland. Read the whole thing here.

]]>First, of course there’s Frege’s basic law V, which was shown inconsistent by Russell (Russell’s paradox). The inconsistency gave us Russell’s theory of types and Whitehead & Russell’s *Principia mathematica,* and, much later, the neo-logicist systems of second-order arithmetic that replace Basic Law V with Hume’s principle.

An early version of the logic underlying *Principia*, the so-called substitutional theory, turned out to be inconsistent. Russell found the inconsistency himself; see this paper by Bernie Linsky (thanks Landon Elkind for pointing me to this).

Hilbert and Ackermann also had notorious problems with the correct formulation of their substitution rules, although their errors in successive versions of *Grundzüge der theoretischen Logik* made the system only unsound and not outright inconsistent (as far as I know)—see Pager’s 1960 paper and references therein.

In the early 1930s, Alonzo Church (“A set of postulates for the foundations of logic“) and Haskell Curry (“Grundlagen der kombinatorischen Logik“) proposed new logical systems for the foundations of mathematics. Both were shown to be inconsistent by Kleene and Rosser in 1935 (in the case of Curry, only the equality axioms proposed in 1934 lead to contradiction). The consistent subsystem that Church extracted is—you guessed it—the lambda calculus (see Cantini’s SEP entry for details). On the other hand, Curry’s analysis of the inconsistency gave us Curry’s paradox.

Quine’s system of New Foundations was originally introduced in his *Mathematical Logic* in 1940. The system as presented in the first (1940) edition allows the derivation of the Burali-Forti paradox. This was again proved by Rosser, who thus leads the scoreboard in number of systems shown inconsistent.

Rosser actually went on to use Quine’s NF in his own textbook *Logic for Mathematicians*, published in 1953. In his development of mathematics in NF, he discussed and used the axiom of choice. Unfortunately, that same year Specker showed that NF also *disproves* the axiom of choice (thanks to Martin Davis for pointing this out). In his review of Specker’s paper, Rosser suggested that a restriction of AC is all that’s needed for the results developed in his book. There, he only proposed *countable* choice as an axiom (Axiom 15, p. 512) which (as far as is known today) is consistent (thanks to Randall Holmes).

A version of the theory of constructions developed in the 1960s by Georg Kreisel and Nicolas Goodman was inconsistent as shown in Goodman’s 1968 dissertation (see this paper by Dean & Kurokawa).

Per Martin-Löf’s original (1971) system of constructive type theory was also inconsistent—shown by Jean-Yves Girard in his 1972 dissertation (Girard’s paradox), and this led to important developments in type theory.

Another famous example from set theory is the inconsistency of Reinhardt cardinals, proved by Kunen. This has perhaps a different flavor than the other examples: presumably (I don’t have access to Reinhardt’s thesis) he proposed the axiom (that there is a non-trivial elementary embedding of the universe into itself) originally as an object of investigation, rather than as something taken to be obviously true (as Frege took Basic Law V). (Thanks Toby Meadows and David Schrittesser for reminding me of this nevertheless. Factoid: Reinhardt is my academic uncle.)

Inconsistency is a problem in classical logic because from an inconsistency you can prove anything: thus classically, inconsistent theories are trivial. One way to get around this is to change the logic in such a way that inconsistencies by themselves don’t make the theory “explode”. Newton da Costa pioneered such logics and in the 1960s he developed paraconsistent set theories on their basis. The hope was that they would prove contradictions such as Russell’s paradox without thereby proving absolutely everything. Unfortunately, some of those nevertheless turned out to be trivial (shown in 1985 by Arruda; h/t Luis Estrada-González).

Panu Raatikainen reminded me that the list is sometimes referred to as Martin Davis’ honor roll (after this FOM post).

]]>MUltlog dates back to 1996 and was written by Gernot Salzer. The code still works, and it turned out that it’s not hard to install it on Linux — all you need is TeXLive and SWI Prolog installed, which is provided packaged in standard distributions like Ubuntu or Debian (as is TCL/TK). I’ve taken Gernot’s code, put it on Github, and gussied it up a little bit. The output of Multlog is a paper that describes the calculi for the system that you input. For instance, below you’ll find the output if you feed it paraconsistent weak Kleene logic.

**UPDATE: **As of v1.14, the distribution and the new website includes a number of examples, including such favorites as FDE, LP, and classical logic with some fun connectives & quantifiers.

Here are some other finite-valued logics which were introduced somewhat recently (see www.logic.at/multlog#examples for these and more).

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