Ordinary Logic Research Articles

Compatible complement in Piron’s system and ordinary modal logic

Previous critiques of the relevant of Piron’s questions-propositions system as a generated of quantum mechanics by interpretation are reinforced by confrontation with the de Morgan laws and with the ordinary modal logic.

Narrative and Sexual Excess

The principle protagonist of In a Year of Thirteen Moons is a transsexual, someone who has crossed an absolute border, transgressed the fundamental divide of gender upon which, according to psychoanalytic theory, all society the systems which compose it, including language itselfis based. Such a thoroughgoing anomaly would seem to demand an extraordinary restructuring of the narrative form which tells it. Instead, Thirteen Moons exhibits a balanced, stubbornly rectilinear form apparently indifferent to the transgression which, by an ordinary logic, should constitute the collapse of its classical symmetry. This essay examines some of the strategies by which, like the secondary revision of the dream text, the film binds the threatening and potentially disruptive sexuality to a plot all too familiar, a scenario all too recurrent. In S/Z, Roland Barthes isolates for discussion five codes that structure the narrative text: the hermeneutic, which sets up, holds in suspense, and discloses an enigma; the proairetic, which is the code of actions and gestures; the cultural or referential, which indicates the system of science, of knowledge within which the text operates; the semic, which controls the connotative level or the character of the text; and the symbolic, which sets up a system of antitheses and thus allows for classifications and legal substitutions. The classical realist text is constructed out of a hierarchical arrangement of these codes, whereby the hermeneutic and proairetic dominate the other codes by imposing on them an irreversible logico-temporal order. To use Barthes's word, the hermeneutic and the proairetic vectorize the text. In an elaboration of the relationship among the codes of the realist text, Peter Wollen has stated that the hermeneutic and proairetic codes are correlated with the symbolic order and encode the textual body with a gender identity, while the semic and symbolic codes (he does not mention the cultural) are correlated with the semiotic and release a kind of random erotics, a promiscuous force of uncathected energy.1 The dominant, sequential codes anchor the nar-

A hintikka possible worlds model for certainty levels in medical decision making

Symbolic logic has long been an appealing tool in efforts to formalize medical decision making [2], [9], [10]. Most concepts of diagnosis and plans for patient management can be formulated as a collection of assertions which are either true or false or indeterminate for a given patient. For example: "the patient has hyponatremia"; "the patient has Addison's crisis"; "the patient should receive steroids". The relationships among assertions such as these superficially resemble the connectives of symbolic logic: nega tion, conjunction, alternation, material implication, and material equivalence. For many applications, however, the binary, true-false character of ordinary symbolic logic renders it incapable of reproducing decisions based upon probability and cost considerations. For example, we might diagnose maturity onset diabetes mellitus in preference to glucagonoma in an obese 56-year-old male with hyper glycemia, because the former is far more probable. Likewise, we might recommend sputum cytology determinations in preference to an exploratory thoracotomy for working up a possible lung cancer case, because the former is less costly (both monetarily and in terms of patient suffering). Applications of decision theory to medical decision making have often proved difficult to work with, in part because of the unavailability of a sufficiently complete and reliable set of probabilities and costs. It is our view that symbolic logic most closely resembles the intuitive process of medical decision making, and thus that potentially the most productive approach to this prob lem is to develop an extension of ordinary symbolic logic capable of handling costs and probabilities. The "certainty levels" calculus is a system of symbolic logic which has been used to study cellular response in injury [12], [19], the pathogenesis of congenital heart disease [13], [14], and the reporting

The modal logic of quantum logic

Modal logic is concerned with the concepts of necessity and possibility and a certain class of object propositions. In this paper we develop the basic concepts of a modal logic which is related to propositions about quantum physical objects. Since the object logic of quantum mechanical propositions is given by the calculi of quantum logic, the structure investigated in this paper will be called the modal logic of quantum logic. The object language and logic of quantum physical propositions is developed here within the dialogic approach to quantum logic [1]. On the basis of these object-linguistic structures we investigate the language of meta-propositions which state the material or formal truth of objectpropositions. Applying again the dialogic technique to meta-propositions the important notion of a formally true meta-proposition can be defined. Using this concept it turns out that the formal logic of meta-propositions is equivalent to the corresponding structure in ordinary logic, i.e., to the effective (intuitionistic) logic. The modalities "necessary" and "possible" are introduced here in the framework of a meta-linguistic interpretation, which considers the modalities as statements about the object-propositions under discussion [9]. On this basis it is found that meta-propositions which state the material truth of a special class of object propositions may be considered as quantum logical modalities (Section 2). A detailed investigation then shows that these quantum logical modalities are intimately related to the important quantum mechanical concepts of commensurability and objectivity. An illustration of the quantum logical modalities by relations between projection operators in Hilbert space concludes this part. In the third section we introduce the concept of a formally true modality which leads to the modal logic of quantum logic. It is shown that this logic of quantum logical modalities contains all the quantum logical restrictions which come from the possible incommensurability of quantum physical

Is nondistributivity for microsystems empirically founded?

Some authors have proposed nondistributive logic as a way out of the difficulties usually met in trying to describe typical quantum phenomena (e.g. the double-slit experiment). We show, however, that, if one takes serously the wave-corpuscle dualism, which was after all the central fact around which quantum theory was developed, ordinary (distributive) logic can fully account for the empirical observations. We, furthermore, point out that there are unavoidable physical difficulties connected with the adoption of a nondistributive corpuscolar approach.

Boolean Features of Sign-Flipping and Signed-Carry Logic in Base -2

A direct self-contained proof of sign-flipping logic for base -2 binary numbers is derived and the same approach is extended to the derivation of a particular adder logic. Sign-flipping equations are contrasted with the logic for assimilated complement in classical binary. Logic equations for signed-carry addition and subtraction are then derived for various codes of interstage carry. Reduction of these operations to ordinary addition and subtraction logic, as well as reduction of base -2 subtraction to base -2 addition and use of Boolean complements is analyzed. Lastly, addition logic equations are derived for sign flipping as a special case of base -2 addition logic.

A 53 MHz Randomly Triggered Synchronous Predetermined Scaler for Accelerator Diagnostics

A 53 MHz random triggered recycling, predetermined start-stop scaler is described. The scaler has a range from 2-to-9999 counts, sensitivity of 40 mV rms and provides both NIM and TTL standard logic outputs. The scaler is composed mainly of standard ECL integrated circuits including universal counters, configured in a synchronous parallel carry connection, and ordinary logic gates and flip-flops. Special circuitry has been designed to prevent start-up counting errors due to the phase randomness between the applied start triggering and input signal transitions. Following a start trigger, the scaler cycles until disabled, delivering an output pulse each time the predetermined count is reached. The time jitter of the output pulses with respect to the periodic input signal is less than 0.1 ns. High resolution examination of the behavior of individual charge bunches within the accelerator is made possible when the scaler is set to count the machine's harmonic number. If the scaler is set one digit away from the harmonic number, the output pulses advance one bunch for each turn around the machine, allowing a sequential display of the contents of each bucket in the machine.

Intensional Language and Ordinary Logic

Intensional Language and Ordinary Logic

3. Personality Factors in Thinking

Many authors would agree that thinking is never separate from needs, motives, and emotions. But in light of their theoretical positions, different investigators have solved this problem in quite different ways. Some, such as Bleuler (1912), have contrasted "autistic" thinking with "realistic" thinking; others, such as T. Rib0 (1906), have posited a special "logic of feelings" that is in contrast to ordinary logic. The connection between thinking and affect has been viewed as the coexistence of two special forms of thinking, one affective and the other logical, there being two interconnected but separate spheres - thinking and feeling.

Ordinary Logic and Logic in Teaching

Review Article Review of Robert H. Ennis, Ordinary Logic.

"Ordinary Logic," by R. H. Ennis

"Ordinary Logic," by R. H. Ennis

Fuzzy Logic and its Application to Switching Systems

Fuzzy logic deals with propositions which may be ascribed values between falsehood and truth (0 and 1) subjectively in either a continuous or a discrete fashion. This is in contrast to ordinary logic (two-valued or k-valued logic) in which a given proposition is ascribed values objectively using either deterministic or probabilistic approaches.

The two-valued orientation of current equilibrium theory

As represented in papers by Cartwright and Harary, Abelson and Rosenberg, and Lambert, current formalizations of interpersonal equilibrium theory characterize balanced systems of individual orientation as two-valued. Within this very restricted characterization of balance, we show that ordinary logic must yield the same conclusions as “psycho-logic.” We cite some dissatisfaction with the restricted applicability of this sort of equilibrium theory. Finally, we urge that current equilibrium theory be considered a special case of a more general theory encompassing response spaces with a dimensionality greater than one, and we sketch an approach to building such a theory.

Algebra of Observation

Birkof£-von Neumann's contention') that observational propositions in Quantum Mechanics do not obey the Boolean logic, but the modular logic was suggested by the structure of the Hilbert space, and only indirectly related to the nature of observation in physics. Husimi2 ) was the first to undertake the task of establishing directly experiment:J.l foundation of quantum logic. The present paper has the same aim as Husimi's early paper,') but adopts a different approach. A feature of the present paper is its methodology of extracting an algebraic structure from a pro­ babilistic structure. A concrete observational interpretation is given for the first time to the conjunction of two incompatible propositions. A side benefit of this work is clarification of the origin of the Booleanity of the ordinary logic, which has to be sought outside the implicational lattice.

Independence proofs in predicate logic with infinitely long expressions

The systems of predicate logic considered here are extensions of ordinary first-order predicate logic. They were obtained by first admitting infinite conjunctions, disjunctions, and quantifications into formulas, and then adjusting the axiom schemata and rules of inference to accommodate them. In this paper, only systems with formulas of denumerable length and “formal proofs” with denumerably many steps will be discussed, though such restrictions are hardly essential. The systems with denumerable conjunctions and disjunctions, but finite quantifications, are semantically complete; i.e., valid formulas are provable. However, systems in which denumerably infinite quantifications are also allowed, are semantically incomplete.

Symmetrie switching functions: Matrix logic V

Symmetric logical functions are classified in ordinary and threshold function. The first class is defined as being operative for only k of p inputs being active. Threshold functions are operative for at least k of p inputs. Ordinary symmetric and threshold functions are mutually related and one class may generate functions of the other class. Transistor circuits are already extensively used to perform threshold functions in form of resistor-coupled nonlinear amplifiers. Recently a new logical element has been conceived which performs ordinary symmetric logic. To evaluate the merits of these logical elements efficiently a method has been developed which may be adapted to logical design by computer routines. Solution of problems involving up to ten variables is feasible without the aid of computers.

Elimination of extra-logical postulates

Under certain circumstances a postulate can be eliminated in favor of a mere definition, or convention of notational abbreviation. Suppose e.g. that a given system presupposes the machinery of ordinary logic and contains in addition a single extra-logical primitive: the relation Pt, “is a spatial part of.” Suppose this relation is governed by a single extra-logical postulate, to the effect that Pt is transitive:Now instead of Pt, the relation O of spatial overlapping might be taken as primitive; Pt could then be defined in terms of O as follows:The transitivity of Pt then follows by purely logical principles from the definition. The statement (1) with its ‘Pt’ clauses expanded according to (2) is a purely logical theorem, demonstrable independently of any stipulations or assumptions concerning the properties of O. Thus, through a change involving neither increase nor decrease of primitive ideas, the need of adopting (1) as a postulate is removed.Or again, consider a system comprising ordinary logic and the sole extra-logical primitive S, the relation of simultaneity, governed by a sole postulate to the effect that S is symmetrical:Here instead of S the relation N, “is no later than” might be taken as primitive; S could then be defined in terms of N as follows:Postulation of (3) then becomes unnecessary, for (3) is an abbreviation, according to (4), of a theorem which is demonstrable within pure logic.

Modal functions in two-valued logic

In this paper it will be shown that the fundamental modal functions (“… is necessary,” “… is possible,” etc.) may be represented in the ordinary two-valued logic of propositions, relatively to a subset S of the set K of all propositions of the logic. The only restriction on S is that no member of S can occur in any other member of S. The representation concerns the set SK consisting of all members of K in which occur members of S. Every member of SK may be regarded as a truth-function of the members of S. If S consists of only one member, P, and if f(P) and g(P) are any members of SK, then we define ∣f(P)∣ as (f(P)·f(~P)). The proposition “f(P) is S-necessary” may then be expressed by ∣f(P)· Furthermore we may express “f(P) is S-impossible” by ∣~f(P)∣, abbreviated to f(P)*; we may express “f(P) is S-possible” by abbreviated to f(P); and we may express “f(P) S-strictly implies g(P)” by (f(P)·~g(P))*, abbreviated to f(P)⊰g(P). Thus is obtained a novel analysis of modality, contrasting, on the one hand, with Carnap's syntactic theory of modality, and, on the other hand, with Lewis's view of modal concepts as underivable from other logical concepts. We would maintain that modal concepts are not absolute, but always relative to some S, generally unspecified.In the formal exposition which follows, we shall assume tacitly but not explicitly the important distinction between the symbols we are discussing and symbols denoting or designating the former.

III. On the theory of consistence of logical class-frequencies, and its geometrical representation

1. In the ordinary treatment of logic the field of discussion is strictly limited to premises of a non-numerical character, numerically definite data being rigidly excluded. The statistician obtains no help from ordinary logic towards solving even the simplest problems, e. g ., the deduction of inferences from data of the type “ x percent, of A’s are B, y percent, of A’s are C,” or the inferring of association between B and C from known associations of A with B and with C. It is now more than half a century since De Morgan, in the chapter “On the Numerically Definite Syllogism,” of his ‘Formal Logic’ (1847), laid the foundations of a theory of strictly quantitative logic. Substituting the modified notation of Jevons, employed by me in a recent paper, for De Morgan’s own notation, his Theorem may be expressed in the form “if (AB) + (AC) > (A), (BC) must be at least equal to the difference (AB) + (AC) —(A).” For if, e. g ., we imagine (A) boxes, into which we deal a certain number (viz., (AB)) of cards marked (B), one into each box, leaving only (A) — (AB) unoccupied boxes, and then proceed to deal into the remaining boxes cards marked C, one into each; some of these must fall into boxes already occupied by a B if their number exceed (A) — (AB). From similar simple reasoning De Morgan derived the complete conditions of consistence for (AB), (AC), and (BC). He does not, however, consider the case of more than three attributes, and the whole discussion is rendered very lengthy owing to his standpoint being still that of the older logicians.