When C. I. Lewis developed his theory of strict implication, he left open the question which of his various systems should be regarded as being closest to our intuitions—though he was inclined to favor the system S2. There are to be found in the literature numerous discussions of this question; most of these have condemned S2 as being too strong, and have proposed ways of weakening it.

In the present paper I shall attempt to throw some light on this question by setting up a syntactical definition of “possibility.” I shall show that every system of modal logic constructed on the basis of this definition is at least as strong as the Lewis system S4.

As the intuitive basis for the syntactical definition of possibility, I take the position that to say a sentence is possible means that there exists a true sentence of the same form. Thus, for example, it would be said that the sentence, “Lions are indigenous to Alaska,” is possible, because of the fact that the sentence, “Lions are indigenous to Africa,” has the same form and is true.

We consider fine hierarchies in recursion theory, descriptive set theory, logic and complexity theory. The main results state that the sets of values of different Boolean terms coincide with the levels of suitable fine hierarchies. This gives new short descriptions of these hierarchies and shows that collections of sets of values of Boolean terms are almost well ordered by inclusion. For the sake of completeness we mention also some earlier results demonstrating the usefulness of fine hierarchies.

In the paper we study the class of weakly algebraizable logics, characterized by the monotonicity and injectivity of the Leibniz operator on the theories of the logic. This class forms a new level in the non-linear hierarchy of protoalgebraic logics.

The present paper contains an emendation to the theory of formal deducibility which was presented in lectures at the University of Notre Dame in the spring of 1948 [4]. The method of proof developed in these lectures for the elimination theorem, which corresponds to the “Hauptsatz” of Gentzen's thesis [6], failed when modal connectives were present. This deficiency was remedied soon thereafter; but it was not possible to include the new proof in those lectures. This new proof is presented below.

Before proceeding to this proof, the elimination theorem will be formulated, and its proof outlined, in an abstract form. Certain general conditions on the rules are formulated in §1; then, in §2, it is shown that the first two stages of the elimination theorem go through for any rules satisfying the conditions. The modifications necessary to extend this proof to cases where modal rules are present will then be taken care of in the later sections. A section on the extension of the elimination theorem to the singular form of LC is added at the end.

This is the first part of a two part work on the monadic theory of short orders (embedding neither ω1 nor ω1*. This part provides the technical groundwork for decidability results. Other applications are possible.

Recent discussion serves to bring out, amply and convincingly, the utility of observing the ordinary correct use of words and phrases for the purpose of clearing up philosophical problems. In this paper, I shall endeavour to show, by means of an example, that the reverse method may have its interest, too. An attempt will be made to cultivate a minor deviation from the accepted ways of using certain words and phrases in idiomatic English as well as in the formalized “languages” of the logicians. The words and phrases in question are those for the formalization of which a logician employs (free or bound) variables. Cases in point are the words customarily called quantifiers. The deviation I have in mind affects the relation of these words to the notion of identity. The deviation is illustrated by the following sentences:

(1a) Any two points of a straight line completely determine that line;

(2a) He is John's brother if he has the same parents as John;

(3a) Mazzini did more for the emancipation of his country than any living man of his time.

These examples may be contrasted with the following closely related sentences:

(1b) Any two distinct points of a straight line completely determine that line;

(2b) He is John's brother if he has the same parents as John and if he is not John himself;

(3b) Mazzini did more for the emancipation of his country than any other living man of his time.

One of the most important and distinctive tools in recursion theory has been the priority method whereby a recursively enumerable (r.e.) set A is constructed by stages to satisfy a sequence of conditions {Rn }n∈ω called requirements. If n < m, requirement Rn is given priority over Rm and action taken for Rm at some stage s may at a later stage t > s be undone for the sake of Rn thereby injuring Rm at stage t. The first priority method was invented by Friedberg [2] and Muchnik [11] to solve Post's problem and is characterized by the fact that each requirement is injured at most finitely often.

Shoenfield [20, Lemma 1], and then independently Sacks [17] and Yates [25] invented a much more powerful method in which a requirement may be injured infinitely often, and the method was applied and refined by Sacks [15], [16], [17], [18], [19] and Yates [25], [26] to obtain many deep results on r.e. sets and their degrees. In spite of numerous simplifications and variations this infinite injury method has never been as well understood as the finite injury method because of its apparently greater complexity.

The purpose of this paper is to reduce the Sacks method to two easily understood lemmas whose proofs are very similar to the finite injury case. Using these lemmas we can derive all the results of Sacks on r.e. degrees, and some by Yates and Robinson as well, in a manner accessible to the nonspecialist. The heart of the method is an ingenious observation of Lachlan [7] which is combined with a further simplification of our own.

In this note it is shown that the property of orthomodularity of the lattice of orthoclosed subspaces of a pre-Hilbert space is not determined by any first-order properties of the relation ⊥ of orthogonality between vectors in . Implications for the study of quantum logic are discussed at the end of the paper.

The key to this result is the following:

Ifis a separable Hilbert space, andis an infinite-dimensional pre-Hilbert subspace of, then (, ⊥) and (, ⊥) are elementarily equivalent in the first-order languageL2of a single binary relation.

Choosing to be a pre-Hilbert space whose lattice of orthoclosed subspaces is not orthomodular, we obtain our desired conclusion. In this regard we may note the demonstration by Amemiya and Araki [1] that orthomodularity of the lattice of orthoclosed subspaces is necessary and sufficient for a pre-Hilbert space to be metrically complete, and hence be a Hilbert space. Metric completeness being a notoriously nonelementary property, our result is only to be expected (note also the parallel with the elementary L2-equivalence of the natural order (Q, <) of the rationals and its metric completion to the reals (R, <)).

To derive (1), something stronger is proved, viz. that (, ⊥) is an elementary substructure of (, ⊥).

A subset A ⊆ M of a totally ordered structure M is said to be convex, if for any a, b ∈ A: [a < b → ∀t (a < tb → t ∈ A)]. A complete theory of first order is weakly o-minimal (M. Dickmann [D]) if any model M is totally ordered by some ∅-definable formula and any subset of M which is definable with parameters from M is a finite union of convex sets. We prove here that for any model M of a weakly o-minimal theory T. any expansion M+ of M by a family of unary predicates has a weakly o-minimal theory iff the set of all realizations of each predicate is a union of a finite number of convex sets (Theorem 63). that solves the Problem of Cherlin-Macpherson-Marker-Steinhorn [MMS] for the class of weakly o-minimal theories.

Given separate though similar proofs of the finite model property for individual modal calculi such as S5, S4, S2, and the Feys-von Wright system T, the problem arises of generalising the arguments and establishing the property for modal calculi en masse. In other words, we would like to be able to show in one fell swoop that any modal calculus satisfying certain general syntactic conditions has the finite model property.

Schnorr randomness is a notion of algorithmic randomness for real numbers closely related to Martin-Löf randomness. After its initial development in the 1970s the notion received considerably less attention than Martin-Löf randomness, but recently interest has increased in a range of randomness concepts. In this article, we explore the properties of Schnorr random reals, and in particular the c.e. Schnorr random reals. We show that there are c.e. reals that are Schnorr random but not Martin-Löf random, and provide a new characterization of Schnorr random real numbers in terms of prefix-free machines. We prove that unlike Martin-Löf random c.e. reals, not all Schnorr random c.e. reals are Turing complete, though all are in high Turing degrees. We use the machine characterization to define a notion of “Schnorr reducibility” which allows us to calibrate the Schnorr complexity of reals. We define the class of “Schnorr trivial” reals, which are ones whose initial segment complexity is identical with the computable reals, and demonstrate that this class has non-computable members.

The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We show that every mapping is sequentially continuous if and only if it is sequentially nondiscontinuous and strongly extensional, and that “every mapping is strongly extensional”, “every sequentially nondiscontinuous mapping is sequentially continuous”, and a weak version of Markov's principle are equivalent. Also, assuming a consequence of Church's thesis, we prove a version of the Kreisel-Lacombe-Shoenfield-Tseĭtin theorem.

The purpose of this work is to present Gentzen-style formulations of S5 and S4 based on sequents of higher levels. Sequents of level 1 are like ordinary sequents, sequents of level 2 have collections of sequents of level 1 on the left and right of the turnstile, etc. Rules for modal constants involve sequents of level 2, whereas rules for customary logical constants of first-order logic with identity involve only sequents of level 1. A restriction on Thinning on the right of level 2, which when applied to Thinning on the right of level 1 produces intuitionistic out of classical logic (without changing anything else), produces S4 out of S5 (without changing anything else).

This characterization of modal constants with sequents of level 2 is unique in the following sense. If constants which differ only graphically are given a formally identical characterization, they can be shown inter-replaceable (not only uniformly) with the original constants salva provability. Customary characterizations of modal constants with sequents of level 1, as well as characterizations in Hilbert-style axiomatizations, are not unique in this sense. This parallels the case with implication, which is not uniquely characterized in Hilbert-style axiomatizations, but can be uniquely characterized with sequents of level 1.

These results bear upon theories of philosophical logic which attempt to characterize logical constants syntactically. They also provide an illustration of how alternative logics differ only in their structural rules, whereas their rules for logical constants are identical.

This paper presents a new technique for handling modal logics with transitive frames, i.e. extensions of the modal system K4. In effect, the technique is based on the following fundamental result, to be obtained below in §3.

Given a formula φ, we can effectively construct finite frames 1, …, n which completely characterize the set of all transitive general frames refuting φ. More exactly, an arbitrary general frame refutes φ iff contains a (not necessarily generated) subframe such that (1) i, for some i ϵ {1, …, n}, is a p-morphic image of (after Fine [1985] we say is subreducible to i), (2) is cofinal in , and (3) every point in that is not in does not get into “closed domains” which are uniquely determined in i, by φ.

This purely technical result has, as it turns out, rather unexpected and profound consequences. For instance, it follows at once that if φ determines no closed domains in the frames 1, …, n associated with it, then the normal extension of K4 generated by φ has the finite model property and so is decidable. Moreover, every normal logic axiomatizable by any (even infinite) set of such formulas φ also has the finite model property. This observation would not possibly merit any special attention, were it not for the fact that the class of such logics contains almost all the standard systems within the field of K4 (at least all those mentioned by Segerberg [1971] or Bull and Segerberg [1984]), all logics containing S4.3, all subframe logics of Fine [1985], and a continuum of other logics as well.

The so-called Curry-Howard interpretation (Curry [1934], Curry and Feys [1958], Howard [1969], Tait [1965]) is known to provide a rather neat term-functional account of intuitionistic implication. Could one refine the interpretation to obtain an almost as good account of other neighbouring implications, including the so-called ‘resource’ implications (e.g. linear, relevant, etc.)?

We answer this question positively by demonstrating that just by working with side conditions on the rule of assertability conditions for the connective representing implication (‘→’) one can characterise those ‘resource’ logics. The idea stems from the realisation that whereas the elimination rule for conditionals (of which implication is a particular case) remains virtually unchanged no matter what kind of conditional one has (i.e. linear, relevant, intuitionistic, classical, etc., all have modus ponens), the corresponding introduction rule carries an element of vagueness which can be explored in the characterisation of several sorts of conditionals. The rule of →-introduction is classified as an ‘improper’ inference rule, to use a terminology from Prawitz [1965]. Now, the so-called improper rules leave room for manoeuvre as to how a particular logic can be obtained just by imposing conditions on the discharge of assumptions that would correspond to the particular logical discipline one is adopting (linear, relevant, ticket entailment, intuitionistic, classical, etc.). The side conditions can be ‘naturally’ imposed, given that a degree of ‘vagueness’ is introduced by the presentation of those improper inference rules, such as the rule of →-introduction:

which says: starting from assumption ‘A’, and arriving at ‘B’ via an unspecified number of steps, one can discharge the assumption and conclude that ‘A’ implies ‘B’.

This paper concerns certain relationships between the ordering of degrees of unsolvability and the jump operation. It is shown that every minimal degree a < 0′ satisfies a″ = 0″. To restate this result in more suggestive language and compare it with related results, we shall use notation based on the now standard terminology of “high” and “low” degrees. Let Hn be the class of degrees a < 0′ such that a(n) = 0(n+1), and let Ln be the class of degrees a ≤ 0′ such that an = 0n. (Observe that Hi ⊆ Lj and Li ⊆ Lj, whenever i ≤ j, and Hi ∩ Lj = ∅ for all i andj.) The result mentioned above may now be restated in the form that every minimal degree a ≤ 0′ is in L2. This extends an earlier result of S. B. Cooper ([1], see also [4]) that no minimal degree a < 0′ is in H1. In the other direction, Sasso, Epstein, and Cooper ([10], [15]) have shown that there is a minimal degree a < 0′ which is not in L1. Also, C.E.M. Yates [14, Corollary 11.14], showed the existence of a minimal degree a < 0′ in L1. Thus each minimal degree a < 0′ lies in exactly one of the two classes L1L2 – L1 and each of the classes contains minimal degrees.

Our results are not restricted to the degrees below 0′. We show in fact that every minimal degree a satisfies a″ = (a ∪ 0′)′. To restate this result and discuss extensions of it, we extend the “high-low” classification of degrees from the degrees below 0′ to degrees in general. There are a number of fairly plausible ways of doing this, but we choose the one way we know of doing so which leads to interesting results. Let GHn be the class of degrees a such that a(n) = (a ∪ 0′)(n), and let GLn be the class of degrees a such that a(n) = (a ∪ 0′)(n−1).

We show basic facts about dp-minimal ordered structures. The main results are: dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has nonempty interior, and any theory of pure tree is dp-minimal.

The quantified relevant logic RQ is given a new semantics in which a formula ∀xA is true when there is some true proposition that implies all x-instantiations of A. Formulae are modelled as functions from variable-assignments to propositions, where a proposition is a set of worlds in a relevant model structure. A completeness proof is given for a basic quantificational system QR from which RQ is obtained by adding the axiom EC of ‘extensional confinement’: ∀x(A ⋁ B) → (A ⋁ ∀xB), with x not free in A. Validity of EC requires an additional model condition involving the boolean difference of propositions. A QR-model falsifying EC is constructed by forming the disjoint union of two natural arithmetical structures in which negation is interpreted by the minus operation.

Introduction. In setting up his definition of analyticity, Carnap uses various non-constructive rules, of which we shall be concerned with only one. This one we shall call simply “Carnap's rule.” It can be roughly stated thus: If f(0), f(1), f(2), … are all provable, then (x)f(x) shall be provable.

In this paper will be briefly considered the logics got by starting with the system of Principia plus Peano's axioms and allowing one, two, …, ω, ω+l, and so on up to, but not including ω uses of Carnap's rule (interspersed with uses of the ordinary rules), as well as the system which includes Principia and Peano's axioms and is closed under application of Carnap's rule and the ordinary rules. Under suitable assumptions as to consistency, it is shown that in each of these logics there occur undecidable propositions and that a formula which states the consistency of the logic exists in the logic but is not provable in the logic. An interesting side result is that the logic got by allowing co applications of Carnap's rule is not closed under Carnap's rule.

We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that dependence is equivalent to bounded non-forking assuming NTP2.