scholarly journals Normalisation and subformula property for a system of classical logic with Tarski’s rule

2021 ◽  
Author(s):  
Nils Kürbis
Keyword(s):  
Normal Form ◽  
Classical Logic ◽  
Similar Form ◽  
General Introduction ◽  
Intuitionist Logic ◽  
Introduction Rule ◽  
Definition Of ◽  
Subformula Property

AbstractThis paper considers a formalisation of classical logic using general introduction rules and general elimination rules. It proposes a definition of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system, and gives reduction procedures for them. It is then shown that deductions in the system convert into normal form, i.e. deductions that contain neither maximal formulas nor maximal segments, and that deductions in normal form satisfy the subformula property. Tarski’s Rule is treated as a general introduction rule for implication. The general introduction rule for negation has a similar form. Maximal formulas with implication or negation as main operator require reduction procedures of a more intricate kind not present in normalisation for intuitionist logic.

scholarly journals Normalisation and subformula property for a system of intuitionistic logic with general introduction and elimination rules

Synthese ◽  
2021 ◽  
Author(s):  
Nils Kürbis
Keyword(s):  
Normal Form ◽  
Intuitionistic Logic ◽  
General Introduction ◽  
Main Method ◽  
Philosophical Importance ◽  
Subformula Property

AbstractThis paper studies a formalisation of intuitionistic logic by Negri and von Plato which has general introduction and elimination rules. The philosophical importance of the system is expounded. Definitions of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system are formulated and corresponding reduction procedures for maximal formulas and permutative reduction procedures for maximal segments given. Alternatives to the main method used are also considered. It is shown that deductions in the system convert into normal form and that deductions in normal form have the subformula property.

scholarly journals A Precise Definition of an Inference (by the Example of Natural Deduction Systems for Logics $I_{\langle \alpha,\beta \rangle}$

2017 ◽  
Vol 23 (1) ◽  
pp. 83-104 ◽  
Author(s):  
В.О. Шангин
Keyword(s):  
Classical Logic ◽  
Natural Deduction ◽  
Precise Definition ◽  
The Other ◽  
Deduction System ◽  
Introduction Rule ◽  
Special Cases ◽  
Natural Deduction System ◽  
Alpha Beta ◽  
Definition Of

In the paper, we reconsider a precise definition of a natural deduction inference given by V. Smirnov. In refining the definition, we argue that all the other indirect rules of inference in a system can be considered as special cases of the implication introduction rule in a sense that if one of those rules can be applied then the implication introduction rule can be applied, either, but not vice versa. As an example, we use logics $I_{\langle\alpha, \beta\rangle}, \alpha, \beta \in \{0, 1, 2, 3,\dots \omega\}$, such that $I_{\langle 0, 0\rangle}$is propositional classical logic, presented by V. Popov. He uses these logics, in particular, a Hilbert-style calculus $HI_{\langle\alpha, \beta\rangle}, \alpha, \beta \in \{0, 1, 2, 3,\dots \omega\}$, for each logic in question, in order to construct examples of effects of Glivenko theorem’s generalization. Here we, first, propose a subordinated natural deduction system $NI_{\langle\alpha, \beta\rangle}, \alpha, \beta \in \{0, 1, 2, 3,\dots \omega\}$, for each logic in question, with a precise definition of a $NI_{\langle\alpha, \beta\rangle}$-inference. Moreover, we, comparatively, analyze precise and traditional definitions. Second, we prove that, for each $\alpha, \beta \in \{0, 1, 2, 3,\dots \omega\}$, a Hilbert-style calculus $HI_{\langle\alpha, \beta\rangle}$and a natural deduction system $NI_{\langle\alpha, \beta\rangle}$are equipollent, that is, a formula $A$ is provable in $HI_{\langle\alpha, \beta\rangle}$iff $A$ is provable in $NI_{\langle\alpha, \beta\rangle}$. DOI: 10.21146/2074-1472-2017-23-1-83-104

Control Reversal Effects on Swept-Back Wings

1949 ◽  
Vol 1 (1) ◽  
pp. 3-34
Author(s):  
Haydn Templeton
Keyword(s):  
Mathematical Analysis ◽  
Quantitative Estimation ◽  
Dynamic Condition ◽  
Loss Of Control ◽  
General Introduction ◽  
Mathematical Basis ◽  
Control Power ◽  
Physical Picture ◽  
The Subject ◽  
Definition Of

SummaryAileron reversal effects on swept-back wings in general and elevon reversal effects on tailless swept-back wings in particular are discussed on a non-mathematical basis, attention being confined to the orthodox flap type of control. The main purpose of the paper is to convey in the simplest terms possible a clear physical picture of the conditions producing loss of control power, emphasis being naturally laid upon the part played by structural wing distortion. Certain qualitative features relating to the two phenomena are also discussed. As a general introduction to the discussion on aileron reversal effects, the definition of “aileron power” in relation to the actual dynamic condition of rolling is described at some length. For elevon reversal effects on tailless aircraft the effect of wing flexibility on both “elevon power” and on trim in steady symmetric flight is considered. With the descriptive treatment adopted the analysis is of necessity broad and general but is designed to appeal to those not too familiar with the subject. The results of certain calculations on a hypothetical wing, which may be of interest, are included. A mathematical analysis for the quantitative estimation of both aileron and elevon reversal effects is given in the Appendix.

scholarly journals Tree Trimming: Four Non-Branching Rules for Priest’s Introduction to Non-Classical Logic

2017 ◽  
Vol 12 (2) ◽  
Author(s):  
Marilynn Johnson
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Variable Domain ◽  
Free Logic ◽  
Intuitionist Logic ◽  
Branching Rule ◽  
Branching Rules ◽  
Graham Priest

In An Introduction to Non-Classical Logic: From If to Is Graham Priest (2008) presents branching rules in Free Logic, Variable Domain Modal Logic, and Intuitionist Logic. I propose a simpler, non-branching rule to replace Priest’s rule for universal instantiation in Free Logic, a second, slightly modified version of this rule to replace Priest’s rule for universal instantiation in Variable Domain Modal Logic, and third and fourth rules, further modifying the second rule, to replace Priest’s branching universal and particular instantiation rules in Intuitionist Logic. In each of these logics the proposed rule leads to tableaux with fewer branches. In Intuitionist logic, the proposed rules allow for the resolution of a particular problem Priest grapples with throughout the chapter. In this paper, I demonstrate that the proposed rules can greatly simplify tableaux and argue that they should be used in place of the rules given by Priest.

scholarly journals Proof Compression and NP Versus PSPACE II

2020 ◽  
Author(s):  
Lew Gordeev ◽  
Edward Hermann Haeusler
Keyword(s):  
Sequent Calculus ◽  
Total Weight ◽  
Theoretic Approach ◽  
Minimal Logic ◽  
Open Problems ◽  
Complete Proof ◽  
Short Tree ◽  
Proof Tree ◽  
Definition Of ◽  
Subformula Property

We upgrade [3] to a complete proof of the conjecture NP = PSPACE that is known as one of the fundamental open problems in the mathematical theory of computational complexity; this proof is based on [2]. Since minimal propositional logic is known to be PSPACE complete, while PSPACE to include NP, it suffices to show that every valid purely implicational formula ρ has a proof whose weight (= total number of symbols) and time complexity of the provability involved are both polynomial in the weight of ρ. As in [3], we use proof theoretic approach. Recall that in [3] we considered any valid ρ in question that had (by the definition of validity) a "short" tree-like proof π in the Hudelmaier-style cutfree sequent calculus for minimal logic. The "shortness" means that the height of π and the total weight of different formulas occurring in it are both polynomial in the weight of ρ. However, the size (= total number of nodes), and hence also the weight, of π could be exponential in that of ρ. To overcome this trouble we embedded π into Prawitz's proof system of natural deductions containing single formulas, instead of sequents. As in π, the height and the total weight of different formulas of the resulting tree-like natural deduction ∂1 were polynomial, although the size of ∂1 still could be exponential, in the weight of ρ. In our next, crucial move, ∂1 was deterministically compressed into a "small", although multipremise, dag-like deduction ∂ whose horizontal levels contained only mutually different formulas, which made the whole weight polynomial in that of ρ. However, ∂ required a more complicated verification of the underlying provability of ρ. In this paper we present a nondeterministic compression of ∂ into a desired standard dag-like deduction ∂0 that deterministically proves ρ in time and space polynomial in the weight of ρ. Together with [3] this completes the proof of NP = PSPACE. Natural deductions are essential for our proof. Tree-to-dag horizontal compression of π merging equal sequents, instead of formulas, is (possible but) not sufficient, since the total number of different sequents in π might be exponential in the weight of ρ − even assuming that all formulas occurring in sequents are subformulas of ρ. On the other hand, we need Hudelmaier's cutfree sequent calculus in order to control both the height and total weight of different formulas of the initial tree-like proof π, since standard Prawitz's normalization although providing natural deductions with the subformula property does not preserve polynomial heights. It is not clear yet if we can omit references to π even in the proof of the weaker result NP = coNP.

scholarly journals The scope of the concept of information and the future of information science

2019 ◽  
Vol 43 (1) ◽  
pp. 73-98
Author(s):  
Jiří Tomáš Stodola
Keyword(s):  
Classical Logic ◽  
Conceptual Analysis ◽  
Information Science ◽  
Methodological Problem ◽  
Main Method ◽  
The Future ◽  
Definition Of ◽  
Concept Of Information

The key concept of information science is the concept of information which is tied to a number of complications. The main problem is that there is no definition of this concept. The purpose of this article is an analysis of the concept of information from the position of classical logic. The main method of the article is a conceptual analysis. First, we briefly deal with the overview of the concepts of information, with concepts and their definition as such and with the scope of the concept of information. Then, we provide an analysis of 31 important definitions of the concept of information which were developed within the scope of information science and related fields, and we consider relations between the concept of information and the concepts in other disciplines. Conceptual analysis of the concept of information leads to the conclusion that information is probably a concept that somehow addresses the entire reality, thus that it is a term, which is in the classical logic described as transcendental. This fact, in the view of the fact that information science is a special field, seems to be a serious methodological problem. Problems associated with the broadness of the concept of information have three possible solutions: transformation of information science into the universal science, narrowing the concept of information to a special term, or replacement of the concept of information by a different one. At the end of the article, we briefly point out our solution to the problem.

Agents in Quantum and Neural Uncertainty

2010 ◽  
pp. 50-77 ◽  
Author(s):  
Germano Resconi ◽  
Boris Kovalerchuk
Keyword(s):  
Classical Logic ◽  
Rough Set Theory ◽  
Evidence Theory ◽  
Quantum Superposition ◽  
Unitary Matrices ◽  
Quantum Uncertainty ◽  
Logic Operations ◽  
Definition Of ◽  
High Flexibility ◽  
Sets Theory

This chapter models quantum and neural uncertainty using a concept of the Agent–based Uncertainty Theory (AUT). The AUT is based on complex fusion of crisp (non-fuzzy) conflicting judgments of agents. It provides a uniform representation and an operational empirical interpretation for several uncertainty theories such as rough set theory, fuzzy sets theory, evidence theory, and probability theory. The AUT models conflicting evaluations that are fused in the same evaluation context. This agent approach gives also a novel definition of the quantum uncertainty and quantum computations for quantum gates that are realized by unitary transformations of the state. In the AUT approach, unitary matrices are interpreted as logic operations in logic computations. We show that by using permutation operators any type of complex classical logic expression can be generated. With the quantum gate, we introduce classical logic into the quantum domain. This chapter connects the intrinsic irrationality of the quantum system and the non-classical quantum logic with the agents. We argue that AUT can help to find meaning for quantum superposition of non-consistent states. Next, this chapter shows that the neural fusion at the synapse can be modeled by the AUT in the same fashion. The neuron is modeled as an operator that transforms classical logic expressions into many-valued logic expressions. The motivation for such neural network is to provide high flexibility and logic adaptation of the brain model.

Aims and Scope of This Book

2017 ◽  
Author(s):  
Radim Bělohlávek ◽  
Joseph W. Dauben ◽  
George J. Klir
Keyword(s):  
Fuzzy Logic ◽  
Classical Logic ◽  
Historical Perspective ◽  
Academic Community ◽  
Broad Sense ◽  
Narrow Sense ◽  
General Introduction ◽  
Key Concepts ◽  
Aims And Scope

This chapter is a general introduction to the book and an overview of its content. It describes the aims and scope of the book, and explains why a historical perspective is essential for achieving the aims. It introduces informally the key concepts involved, and the particular challenge fuzzy logic poses to the principle of bivalence in classical logic. It looks at the circumstances that led to the emergence of fuzzy logic in the academic community and as well as at the agendas of two main subareas of fuzzy logic, known as fuzzy logic in the narrow sense and fuzzy logic in the broad sense. The content of each of the subsequent chapters of the book is also briefly described.

Some consistency proofs and a characterization of inconsistency proofs in illative combinatory logic

1987 ◽  
Vol 52 (1) ◽  
pp. 89-110 ◽  
Author(s):  
M. W. Bunder
Keyword(s):  
Normal Form ◽  
Propositional Calculus ◽  
Weak Form ◽  
Combinatory Logic ◽  
Major Premise ◽  
Elimination Rule ◽  
Introduction Rule ◽  
Consistency Proofs ◽  
Arbitrary Term

It is well known that combinatory logic with unrestricted introduction and elimination rules for implication is inconsistent in the strong sense that an arbitrary term Y is provable. The simplest proof of this, now usually called Curry's paradox, involves for an arbitrary term Y, a term X defined by X = Y(CPy).The fact that X = PXY = X ⊃ Y is an essential part of the proof.The paradox can be avoided by placing restrictions on the implication introduction rule or on the axioms from which it can be proved.In this paper we determine the forms that must be taken by inconsistency proofs of systems of propositional calculus based on combinatory logic, with arbitrary restrictions on both the introduction and elimination rules for the connectives. Generally such a proof will involve terms without normal form and cut formulas, i.e. formulas formed by an introduction rule that are immediately removed by an elimination with at most some equality steps intervening. (Such a sequence of steps we call a “cut”.)The above applies not only to the strong form of inconsistency defined above, but also to the weak form of inconsistency defined by: all propositions are provable, if this can be represented in the system.Any inconsistency proof of this kind of system can be reduced to one where the major premise of the elimination rule involved in the cut and its deduction must also appear in the deduction of the minor premise involved in the cut.We can, using this characterization of inconsistency proofs, put appropriate restrictions on certain introduction rules so that the systems, including a full classical propositional one, become provably consistent.

Research in homœopathy

1970 ◽  
Vol 59 (03) ◽  
pp. 129-138
Author(s):  
Anita E. Davies
Keyword(s):  
Point Of View ◽  
General Introduction ◽  
Materia Medica ◽  
Definition Of

SummaryAfter a general introduction and definition of terms, the therapeutic situation is illustrated, followed by a discussion of criteria of success or failure and the factors involved in prescribing the surative remedy—expertise as a doctor, assessment of the symptomatology from the homœopathic point of view, knowledge of the materia medica, methods of prescribing, obstacles to cure, understanding the action of potencies; and throughout suggesting where more individual or group study would increase success, and finally calling for the support of all members of the Faculty.

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