An extension of the Łukasiewicz logic to the modal logic of quantum mechanics

An attempt is made to include the axioms of Mackey for probabilities of experiments in quantum mechanics into the calculus _x0 of ukasiewicz. The obtained calculusQ contains an additional modal signQ and four modal rules of inference. The propositionQx is read x is confirmed. The most specific rule of inference may be read: for comparable observations implication is equivalent to confirmation of material implication.The semantic truth ofQ is established by the interpretation with the help of physical objects obeying to the rules of quantum mechanics. The embedding of the usual quantum propositional logic inQ is accomplished.Allatum est die 9 Junii 1976     

On superintuitionistic logics as fragments of proof logic extensions

Coming fromI andCl, i.e. from intuitionistic and classical propositional calculi with the substitution rule postulated, and using the sign to add a new connective there have been considered here: Grzegorozyk's logicGrz, the proof logicG and the proof-intuitionistic logicI set up correspondingly by the calculiFor any calculus we denote by the set of all formulae of the calculus and by the lattice of all logics that are the extensions of the logic of the calculus, i.e. sets of formulae containing the axioms of and closed with respect to its rules of inference. In the logiclG the sign is decoded as follows: A = (A & A). The result of placing in the formulaA before each of its subformula is denoted byTrA. The maps are defined (in the definitions of x and the decoding of is meant), by virtue of which the diagram is constructedIn this diagram the maps, x and are isomorphisms, thereforex ^–1 = ; and the maps and are the semilattice epimorphisms that are not commutative with lattice operation +. Besides, the given diagram is commutative, and the next equalities take place: ^–1 = ^–1 and = ^–1 x. The latter implies in particular that any superintuitionistic logic is a superintuitionistic fragment of some proof logic extension.     

Paraconsistent logic and model theory

The object of this paper is to show how one is able to construct a paraconsistent theory of models that reflects much of the classical one. In other words the aim is to demonstrate that there is a very smooth and natural transition from the model theory of classical logic to that of certain categories of paraconsistent logic. To this end we take an extension of da Costa'sC ₁ ⁼ (obtained by adding the axiom A A) and prove for it results which correspond to many major classical model theories, taken from Shoenfield [5]. In particular we prove counterparts of the theorems of o-Tarski and Chang-o-Suszko, Craig-Robinson and the Beth definability theorem.     

Depth relevance of some paraconsistent logics

The paper essentially shows that the paraconsistent logicDR satisfies the depth relevance condition. The systemDR is an extension of the systemDK of [7] and the non-triviality of a dialectical set theory based onDR has been shown in [3]. The depth relevance condition is a strengthened relevance condition, taking the form: If _DR- AB thenA andB share a variable at the same depth, where the depth of an occurrence of a subformulaB in a formulaA is roughly the number of nested 's required to reach the occurrence ofB inA. The method of proof is to show that a model structureM consisting of {M ₀ , M₁, ..., M}, where theM _i s are all characterized by Meyer's 6-valued matrices (c. f, [2]), satisfies the depth relevance condition. Then, it is shown thatM is a model structure for the systemDR.     

A pragmatic interpretation of intuitionistic propositional logic

We construct an extension ^P of the standard language of classical propositional logic by adjoining to the alphabet of a new category of logical-pragmatic signs. The well formed formulas of are calledradical formulas (rfs) of ^P;rfs preceded by theassertion sign constituteelementary assertive formulas of ^P, which can be connected together by means of thepragmatic connectives N, K, A, C, E, so as to obtain the set of all theassertive formulas (afs). Everyrf of ^P is endowed with atruth value defined classically, and everyaf is endowed with ajustification value, defined in terms of the intuitive notion of proof and depending on the truth values of its radical subformulas. In this framework, we define the notion ofpragmatic validity in ^P and yield a list of criteria of pragmatic validity which hold under the assumption that only classical metalinguistic procedures of proof be accepted. We translate the classical propositional calculus (CPC) and the intuitionistic propositional calculus (IPC) into the assertive part of ^P and show that this translation allows us to interpret Intuitionistic Logic as an axiomatic theory of the constructive proof concept rather than an alternative to Classical Logic. Finally, we show that our framework provides a suitable background for discussing classical problems in the philosophy of logic.This paper is an enlarged and entirely revised version of the paper by Dalla Pozza (1991) worked out in the framework of C.N.R. project n. 89.02281.08, and published in Italian. The basic ideas in it have been propounded since 1986 by Dalla Pozza in a series of seminars given at the University of Lecce and in other Italian Universities. C. Garola collected the scattered parts of the work, helped in solving some conceptual difficulties and refining the formalism, yielded the proofs of some propositions (in particular, in Section 3) and provided physical examples (see in particular Remark 2.3.1).     

Least-squares theory based on general distributional assumptions with an application to the incomplete observations problem

The linear regression modely=x+ is reanalyzed. Taking the modest position that x is an approximation of the best predictor ofy we derive the asymptotic distribution ofb andR ², under mild assumptions.The method of derivation yields an easy answer to the estimation of from a data set which contains incomplete observations, where the incompleteness is random.     

Mathematical quantum theory I: Random ultrafilters as hidden variables

The basic purpose of this essay, the first of an intended pair, is to interpret standard von Neumann quantum theory in a framework of iterated measure algebraic truth for mathematical (and thus mathematical-physical) assertions — a framework, that is, in which the truth-values for such assertions are elements of iterated boolean measure-algebras (cf. Sections 2.2.9, 5.2.1–5.2.6 and 5.3 below).The essay itself employs constructions of Takeuti's boolean-valued analysis (whose origins lay in work of Scott, Solovay, Krauss and others) to provide a metamathematical interpretation of ideas sometimes considered disparate, heuristic, or simply ill-defined: the collapse of the wave function, for example; Everett's many worlds'-construal of quantum measurement; and a natural product space of contextual (nonlocal) hidden variables.More precisely, these constructions permit us to write down a category-theoretically natural correlation between ideal outcomes of quantum measurements u of a universal wave function, and possible worlds of an Everett-Wheeler-like many-worlds-theory.The universal wave function, first, is simply a pure state of the Hilbert space (L ₂([0, 1]) ^M in a model M an appropriate mathematical-physical theory T, where T includes enough set-theory to derive all the analysis needed for von Neumann-algebraic formulations of quantum theory.The worlds of this framework can then be given a genuine model-theoretic construal: they are random models M(u) determined by M-random elements u of the unit interval [0, 1], where M is again a fixed model of T.Each choice of a fixed basis for a Hilbert space H in a model of M of T then assigns ideal spectral values for observables A on H (random ultrafilters on the range of A regarded as a projection-valued measure) to such M-random reals u. If is the universal Lebesgue measure-algebra on [0, 1], these assignments are interrelated by the spectral functional calculus with value 1 in the boolean extension (V( )) ^M , and therefore in each M(u).Finally, each such M-random u also generates a corresponding extension M(u) of M, in which ideal outcomes of measurements of all observables A in states are determined by the assignments just mentioned from the random spectral values u for the universal position-observable on L ₂([0, 1]) in M.At the suggestion of the essay's referee, I plan to draw on its ideas in the projected sequel to examine more recent modal and decoherence-interpretations of quantum theory, as well as Schrödinger's traditional construal of time-evolution. A preliminary account of the latter — an obvious prerequisite for any serious many-worlds-theory, given that Everett's original intention was to integrate time-evolution and wave-function collapse — is sketched briefly in Section 5.3. The basic idea is to apply results from the theory of iterated measure-algebras to reinterpret time-ordered processes of measurements (determined, for example, by a given Hamiltonian observable H in M) as individual measurements in somewhat more complexly defined extensions M(u) of M.In plainer English: if one takes a little care to distinguish boolean- from measure-algebraic tensor-products of the universal measure-algebra L, one can reinterpret formal time-evolution so that it becomes internal to the universal random models M(u).     

Suspensions and detentions in an urban,low-income school: punishment or reward?

Disciplinary records for 3rd through 8th grade students (n = 314) in an inner-city, public school were examined for one school year to assess students' variation in response to discipline. Rates of disciplinary referrals were compared for students who received no detentions or suspensions throughout the year (never group n = 117), students who received one or more detention or suspension in the fall but not in the spring (fall group n = 62), and students who received one or more detention or suspension in the fall and one or more detention or suspension in the spring (fall + spring group n = 75). Results indicated that during the fall, the fall group had nearly equivalent rates of referrals to the fall + spring group; however, the fall group exhibited significantly lower rates of referrals during winter and spring that were nearly equivalent to the never group, as would be expected for a punishment procedure. In contrast, the fall + spring group evidenced increases in referrals across the year, suggesting the possibility that detentions and suspensions were functioning as rewards for this group. The fall + spring group was rated by teachers and peers at mid-year as highly aggressive, lacking social skills, and high on hyperactivity, whereas the fall group and the never group were statistically equivalent on teacher and peer ratings. Implications for mental health programs for urban schools are discussed, especially the need for alternatives to detention and suspension for the subset of students who account for the majority of school discipline.     

A semantical investigation into Leśniewski's axiom of his ontology

A structure A for the language L, which is the first-order language (without equality) whose only nonlogical symbol is the binary predicate symbol , is called a quasi -struoture iff (a) the universe A of A consists of sets and (b) a b is true in A ([p) a = {p } & p b] for every a and b in A, where a(b) is the name of a (b). A quasi -structure A is called an -structure iff (c) {p } A whenever p a A. Then a closed formula in L is derivable from Leniewski's axiom x, y[x y u (u x) u; v(u, v x u v) u(u x u y)] (from the axiom x, y(x y x x) x, y, z(x y z y x z)) iff is true in every -structure (in every quasi -structure).     

Cut-free tableau calculi for some propositional normal modal logics

We give sound and complete tableau and sequent calculi for the prepositional normal modal logics S4.04, K4B and G ⁰(these logics are the smallest normal modal logics containing K and the schemata A A, A A and A ( A); A A and AA; A A and ((A A) A) A resp.) with the following properties: the calculi for S4.04 and G ⁰are cut-free and have the interpolation property, the calculus for K4B contains a restricted version of the cut-rule, the so-called analytical cut-rule.In addition we show that G ⁰is not compact (and therefore not canonical), and we proof with the tableau-method that G ⁰is characterised by the class of all finite, (transitive) trees of degenerate or simple clusters of worlds; therefore G ⁰is decidable and also characterised by the class of all frames for G ⁰.Research supported by Fonds zur Förderung der wissenschaftlichen Forschung, project number P8495-PHY.Presented by W. Rautenberg     

Syntactical investigations intoBI logic andBB′I logic

In this note, we will study four implicational logicsB, BI, BB and BBI. In [5], Martin and Meyer proved that a formula is provable inBB if and only if is provable inBBI and is not of the form of » . Though it gave a positive solution to theP - W problem, their method was semantical and not easy to grasp. We shall give a syntactical proof of the syntactical relation betweenBB andBBI logics. It also includes a syntactical proof of Powers and Dwyer's theorem that is proved semantically in [5]. Moreover, we shall establish the same relation betweenB andBI logics asBB andBBI logics. This relation seems to say thatB logic is meaningful, and so we think thatB logic is the weakest among meaningful logics. Therefore, by Theorem 1.1, our Gentzentype system forBI logic may be regarded as the most basic among all meaningful logics. It should be mentioned here that the first syntactical proof ofP - W problem is given by Misao Nagayama [6].Presented byHiroakira Ono     

Three prepositional calculi of probability

Attempts are made to transform the basis of elementary probability theory into the logical calculus.We obtain the propositional calculus NP by a naive approach. As rules of transformation, NP has rules of the classical propositional logic (for events), rules of the ukasiewicz logic ₀ (for probabilities) and axioms of probability theory, in the form of rules of inference. We prove equivalence of NP with a fragmentary probability theory, in which one may only add and subtract probabilities.The second calculus MP is a usual modal propositional calculus. It has the modal rules x x, x y x y, x x, x y (y x), (y x), in addition to the rules of classical propositional logic. One may read x as x is probable. Imbeddings of NP and of ₀ into MP are given.The third calculus P is a modal extension of ₀. It may be obtained by adding the rule ((xy)y) xy to the modal logic of quantum mechanics Q [5]. One may read x in P as x is observed. An imbedding of NP into P is given.     

Action incompleteness

The author has previously introduced an operator into dynamic logic which takes formulae to terms; the suggested reading of A was the bringing about of A or the seeing to it that A. After criticism from S. K. Thomason and T. J. Surendonk the author now presents an improved version of his theory. The crucial feature is the introduction of an operatorOK taking terms to formulae; the suggested reading of OK is always terminates.     

Paraconsistent algebras

The prepositional calculiC _n , 1 n introduced by N.C.A. da Costa constitute special kinds of paraconsistent logics. A question which remained open for some time concerned whether it was possible to obtain a Lindenbaum's algebra forC _n . C. Mortensen settled the problem, proving that no equivalence relation forC _n . determines a non-trivial quotient algebra.The concept of da Costa algebra, which reflects most of the logical properties ofC _n , as well as the concept of paraconsistent closure system, are introduced in this paper.We show that every da Costa algebra is isomorphic with a paraconsistent algebra of sets, and that the closure system of all filters of a da Costa algebra is paraconsistent.     

A further note on freeman's measure of association

This note extends and elaborates Hubert's attempt to provide an interpretation of Freeman's measure of association,. The measure is used in a a contingency table when observations are ordered on one variable and unordered on the other. No attempt is made explore the distribution of.     

A finite analog to the löwenheim-skolem theorem

The traditional model theory of first-order logic assumes that the interpretation of a formula can be given without reference to its deductive context. This paper investigates an interpretation which depends on a formula's location within a derivation. The key step is to drop the assumption that all quantified variables must have the same range and to require only that the ranges of variables in a derivation must be related in such way as to preserve the soundness of the inference rules. With each (consistent) derivation there is associated a Buridan-Volpin (orBV) structure [M, {r(x)}] which is simply a Tarski structureM for the language and a map giving the ranger(x) of each variablex in the derivation. IfLK* is (approximately) the classical sequent calculusLK of Gentzen from which the structural contraction rules have been dropped, then our main result reads: If a set of first-ordered formulas has a Tarski modelM, then from any normal derivationD inLK* of can be constructed aBV modelM _D=[M, {r(x)}] of where each ranger(x) is finite.Presented byMelvin Fitting;     

A note on syntactical and semantical functions

We say that a semantical function is correlated with a syntactical function F iff for any structure A and any sentence we have A F A .It is proved that for a syntactical function F there is a semantical function correlated with F iff F preserves propositional connectives up to logical equivalence. For a semantical function there is a syntactical function F correlated with iff for any finitely axiomatizable class X the class ^–1X is also finitely axiomatizable (i.e. iff is continuous in model class topology).     

Omega-consistency and the diamond

G is the result of adjoining the schema (qAA)qA to K; the axioms of G* are the theorems of G and the instances of the schema qAA and the sole rule of G* is modus ponens. A sentence is -provable if it is provable in P(eano) A(rithmetic) by one application of the -rule; equivalently, if its negation is -inconsistent in PA. Let -Bew(x) be the natural formalization of the notion of -provability. For any modal sentence A and function mapping sentence letters to sentences of PA, inductively define A by: p = (p) (p a sentence letter); = ; (AB)su}= (A B); and (qA)= -Bew(A )(S) is the numeral for the Gödel number of the sentence S). Then, applying techniques of Solovay (Israel Journal of Mathematics 25, pp. 287–304), we prove that for every modal sentence A, _G A iff for all , _PA A ; and for every modal sentence A, _G* A iff for all , A is true.I should like to thank David Auerbach and Rohit Parikh.     

On defining necessity in terms of entailment

In their book Entailment, Anderson and Belnap investigate the consequences of defining Lp (it is necessary that p) in system E as (pp)p. Since not all theorems are equivalent in E, this raises the question of whether there are reasonable alternative definitions of necessity in E. In this paper, it is shown that a definition of necessity in E satisfies the conditions { _E Lpp, _EL(pq)(LpLq), _E pLp} if and only if its has the form C ₁.C₂ .... C_np, where each C _iis equivalent in E to either pp or ((pp)p)p.     

On the definability of the quantifier “there exist uncountably many”

In paper [5] it was shown that a great part of model theory of logic with the generalized quantifier Q x = there exist uncountably many x is reducible to the model theory of first order logic with an extra binary relation symbol. In this paper we consider when the quantifier Q x can be syntactically defined in a first order theory T. That problem was raised by Kosta Doen when he asked if the quantifier Q x can be eliminated in Peano arithmetic. We answer that question fully in this paper.I would like to thank Kosta Doen and Zoran Markovi who made valuable suggestions and remarks on a draft of this paper.