This paper shows that, for the Hertz–Gentzen Systems of 1933, extended by a classical rule T1 and using certain axioms, all derivations are analytic: every cut formula occurs as a subformula in the cut’s conclusion. Since the Stoic cut rules are instances of Gentzen’s Cut rule of 1933, from this we infer the decidability of the propositional logic of the Stoics. We infer the correctness for this logic of a “relevance criterion” and of two “balance criteria”, and hence that a (...) particular derivable sequent has no derivation that is “normal” in the sense that the first premiss of each cut is cut-free. We also infer that Cut is not admissible in the Stoic system, based on the standard Stoic axioms, the T1 rule and the instances of Cut with just two antecedent formulae in the first premiss. OPEN ACCESS. (shrink)

This book traces the development of formal logic from its origins inancient Greece to the present day. The authors first discuss the work oflogicians from Aristotle to Frege, showing how they were influenced by thephilosophical or mathematical ideas of their time. They then examinedevelopments in the present century.

This book aids in the rehabilitation of the wrongfully deprecated work of William Parry, and is the only full-length investigation into Parry-type propositional logics. A central tenet of the monograph is that the sheer diversity of the contexts in which the mereological analogy emerges – its effervescence with respect to fields ranging from metaphysics to computer programming – provides compelling evidence that the study of logics of analytic implication can be instrumental in identifying connections between topics that would otherwise remain (...) hidden. More concretely, the book identifies and discusses a host of cases in which analytic implication can play an important role in revealing distinct problems to be facets of a larger, cross-disciplinary problem. It introduces an element of constancy and cohesion that has previously been absent in a regrettably fractured field, shoring up those who are sympathetic to the worth of mereological analogy. Moreover, it generates new interest in the field by illustrating a wide range of interesting features present in such logics – and highlighting these features to appeal to researchers in many fields. (shrink)

This paper develops an interpretation of the fourth account of conditionals in Sextus Empiricus's Outlines of Pyrrhonism that conceptually links it with contemporary ?relevance? interpretations of entailment. It is argued that the third account of conditionals, which analyzes the truth of a conditional in terms of the joint impossibility of antecedent and denial of consequent, should not be interpreted in terms of a relative incompatibility of antecedent and denial of consequent because of Stoic acceptance of the truth of some conditionals (...) of the form p ? ?p and its converse. Rather, it is suggested, ancient attempts to avoid the so-called paradoxes of implication involve the fourth account of conditionals. I hypothesize that this account is related to Stoic attempts to define truth conditions for conditionals in terms of a theory of the concludency (validity) of arguments in opposition to the more common procedure (represented by the first three accounts of conditionals) of specifying truth conditions for conditionals ?semantically? and using those truth conditions in the development of a theory of argument validity. (shrink)

Dans les Esquisses pyrrhoniennes, Sextus Empiricus entreprend de réfuter le sunèmmenon, qui correspond en logique à notre « implication ». Sa réfutation procède circulairement, chacun des quatre critères proposés se voyant réfuté par le critère qui lui succède, de sorte que la dernière réfutation marque implicitement le retour à la première. Une telle situation est exemplaire parce qu’elle trouve un étrange écho dans la controverse qui oppose les logiciens du xx e siècle : savoir si la conséquence logique peut être (...) réduite à la validité du conditionnel matériel ou si elle n’exige pas plutôt ce que les logiciens appellent la « pertinence ». Envisagée dans cette perspective, la batterie des réfutations sceptiques se réduit au rôle d’adjuvant dans une controverse qui la dépasse, comme si la logique elle-même devenait sceptique lorsqu’elle se heurte à une aporie constitutive. (shrink)

The majority of formal accounts attribute to Stoic logicians the classical truth-functional understanding of the material conditional and exclusive disjunction.These interpretations were disputed,...

In this paper I would like to argue that Stoic logic is a kind ofrelevant logic rather than the classical logic. To realize this purpose I willtry to keep as close as possible to Stoic calculus as expressed with the helpof their arguments.

Controversy has surrounded the interpretation of the so-called 'Diodorean' and 'Chrysippean' conditionals of the Stoics. I critically evaluate and reject two interpretations of each of them: as expressing natural laws, and as strict conditionals. In doing so I engage with the work of authors such as Frede, Gould, Hurst, the Kneales, Mates, and Prior. I conclude by offering my own proposal for where these Stoic conditionals should be located on a 'ladder' of logical strength.

We give a simple sequent calculus presentation of R.B. Angell’s logic of analytic containment, recently championed by Kit Fine as a plausible logic of partial content.

I develop a basic theory of content within the framework of truthmaker semantics and, in the second part, consider some of the applications to subject matter, common content, logical subtraction and ground.

I provide a truthmaker semantics for Angell’s system of analytic implication and establish completeness.

The concept of entailment is often connected with deducibility: A is said to entail B iff B is logically deducible from A.1 It has also been connected to the concept of containment in Kant’s sense of analytic containment: A entails B only if the meaning of B is contained in the meaning of A. But the concepts of deducibility and containment are two distinct concepts, and the failure to distinguish them leads to faulty attempts to merge them in formal systems. (...) One such attempt is Anderson and Belnap’s system, E, in which a Fitch-type theory of natural deduction is modified to incorporate a certain sense of “containment”2. Another is Parry’s system, AI, of “analytic implication” which began with a more restricted sense of containment but has usually been presented as a theory of deducibility. (shrink)

ABSTRACT: An introduction to Stoic logic. Stoic logic can in many respects be regarded as a fore-runner of modern propositional logic. I discuss: 1. the Stoic notion of sayables or meanings (lekta); the Stoic assertibles (axiomata) and their similarities and differences to modern propositions; the time-dependency of their truth; 2.-3. assertibles with demonstratives and quantified assertibles and their truth-conditions; truth-functionality of negations and conjunctions; non-truth-functionality of disjunctions and conditionals; language regimentation and ‘bracketing’ devices; Stoic basic principles of propositional logic; 4. (...) Stoic modal logic; 5. Stoic theory of arguments: two premisses requirement; validity and soundness; 6. Stoic syllogistic or theory of formally valid arguments: a reconstruction of the Stoic deductive system, which consisted of accounts of five types of indemonstrable syllogisms, which function as nullary argumental rules that identify indemonstrables or axioms of the system, and four deductive rules (themata) by which certain complex arguments can be reduced to indemonstrables and thus shown to be formally valid themselves; 7. arguments that were considered as non-syllogistically valid (subsyllogistic and unmethodically concluding arguments). Their validity was explained by recourse to formally valid arguments. (shrink)

ABSTRACT: For the Stoics, a syllogism is a formally valid argument; the primary function of their syllogistic is to establish such formal validity. Stoic syllogistic is a system of formal logic that relies on two types of argumental rules: (i) 5 rules (the accounts of the indemonstrables) which determine whether any given argument is an indemonstrable argument, i.e. an elementary syllogism the validity of which is not in need of further demonstration; (ii) one unary and three binary argumental rules which (...) establish the formal validity of non-indemonstrable arguments by analysing them in one or more steps into one or more indemonstrable arguments (cut type rules and antilogism). The function of these rules is to reduce given non-indemonstrable arguments to indemonstrable syllogisms. Moreover, the Stoic method of deduction differs from standard modern ones in that the direction is reversed (similar to tableau methods). The Stoic system may hence be called an argumental reductive system of deduction. In this paper, a reconstruction of this system of logic is presented, and similarities to relevance logic are pointed out. (shrink)

In this article, we consider variations of Nuel Belnap’s ‘artificial reasoner’. In particular, we examine cases in which the artificial reasoner is faulty, e.g. situations in which the reasoner is unable to calculate the value of a formula due to an inability to retrieve the values of its atoms. In the first half of the article, we consider two ways of modelling such circumstances and prove the deductive systems arising from these two types of models to be equivalent to Graham (...) Priest’s first-degree entailment with an ‘emptiness’ value and Richard Angell’s analytic containment, making computational interpretations of these systems possible. The Belnap-type semantics for AC bring FDE φ and AC in line with other containment logics in their neighborhood. The second half of the article examines formal questions, such as whether AC admits an analysis along the lines of that given to the related system of William Parry’s system of analytic implication, as suggested by Kurt Gödel and confirmed by Kit Fine. Furthermore, a natural means of extending these systems to languages with an intensional implication connective is investigated. (shrink)