Another variant of natural deduction

1956 ◽  
Vol 21 (1) ◽  
pp. 52-55 ◽  
Author(s):  
Irving M. Copi
Keyword(s):  
Natural Deduction ◽  
Propositional Function ◽  
Image Position

Since 1934 various different techniques for natural deduction have been developed by Gentzen, Jaśkowski, Rosser, Quine, and others (see [1], pp. 147–167; [2], especially footnotes 1, 3, and 4; and [3], pp. 75-83, 96-107, and 289-294). It has been pointed out to me by Professor Donald Kalish of U.C.L.A. that the restrictions placed upon Universal Generalization (UG) and Existential Instantiation (EI) in [3] force one to construct a less natural proof than seems desirable for such arguments as(cf. [3], p. 139, and [1], pp. 175f.) It is my purpose in this note to formulate an alternative restriction on UG which will permit a more natural proof for such arguments, and to prove the consistency of the altered rule.In the notation of [3], “The expression ‘Φμ’ will denote any propositional function in which there is at least one free occurrence of the variable denoted by ‘μ’. The expression ‘Φν’ will denote the result of replacing all free occurrences of μ in Φμ by ν, with the added proviso that when ν is a variable it must occur free in Φν at all places at which μ occurs free in Φμ.” ([3], p. 100.) The statement of EI is relatively unrestricted; beingprovided that ν is a variable which occurs free in no earlier step.” ([3], p. 104.)

A note on some intermediate propositional calculi

1984 ◽  
Vol 49 (2) ◽  
pp. 329-333 ◽  
Author(s):  
Branislav R. Boričić
Keyword(s):  
Positive Solution ◽  
Natural Deduction ◽  
Propositional Calculus ◽  
Conservative Extension ◽  
Image Position ◽  
Classical Propositional Calculus ◽  
Propositional Calculi

This note is written in reply to López-Escobar's paper [L-E] where a “sequence” of intermediate propositional systems NLCn (n ≥ 1) and corresponding implicative propositional systems NLICn (n ≥ 1) is given. We will show that the “sequence” NLCn contains three different systems only. These are the classical propositional calculus NLC1, Dummett's system NLC2 and the system NLC3. Accordingly (see [C], [Hs2], [Hs3], [B 1], [B2], [Hs4], [L-E]), the problem posed in the paper [L-E] can be formulated as follows: is NLC3a conservative extension of NLIC3? Having in mind investigations of intermediate propositional calculi that give more general results of this type (see V. I. Homič [H1], [H2], C. G. McKay [Mc], T. Hosoi [Hs 1]), in this note, using a result of Homič (Theorem 2, [H1]), we will give a positive solution to this problem.NLICnand NLCn. If X and Y are propositional logical systems, by X ⊆ Y we mean that the set of all provable formulas of X is included in that of Y. And X = Y means that X ⊆ Y and Y ⊆ X. A(P1/B1, …, Pn/Bn) is the formula (or the sequent) obtained from the formula (or the sequent) A by substituting simultaneously B1, …, Bn for the distinct propositional variables P1, …, Pn in A.Let Cn(n ≥ 1) be the string of the following sequents:Having in mind that the calculi of sequents can be understood as meta-calculi for the deducibility relation in the corresponding systems of natural deduction (see [P]), the systems of natural deductions NLCn and NLICn (n ≥ 1), introduced in [L-E], can be identified with the calculi of sequents obtained by adding the sequents Cn as axioms to a sequential formulation of the Heyting propositional calculus and to a system of positive implication, respectively (see [C], [Ch], [K], [P]).

A sequent calculus for type assignment

1977 ◽  
Vol 42 (1) ◽  
pp. 11-28 ◽  
Author(s):  
Jonathan P. Seldin
Keyword(s):  
Normal Form ◽  
Main Part ◽  
Natural Deduction ◽  
Sequent Calculus ◽  
Cut Elimination ◽  
Type Assignment ◽  
Image Position ◽  
Atomic Type

The sequent calculus formulation (L-formulation) of the theory of functionality without the rules allowing for conversion of subjects of [3, §14E6] fails because the (cut) elimination theorem (ET) fails. This can be most easily seen by the fact that it is easy to prove in the systemandbut not (as is obvious if α is an atomic type [an F-simple])The error in the “proof” of ET in [14, §3E6], [3, §14E6], and [7, §9C] occurs in Stage 3, where it is implicitly assumed that if [x]X ≡ [x] Y then X ≡ Y. In the above counterexample, we have [x]x ≡ ∣ ≡ [x](∣x) but x ≢ ∣x. Since the formulation of [2, §9F] is not really satisfactory (for reasons stated in [3, §14E]), a new seguent calculus formulation is needed for the case in which the rules for subject conversions are not present. The main part of this paper is devoted to presenting such a formulation and proving it equivalent to the natural deduction formulation (T-formulation). The paper will conclude in §6 with some remarks on the result that every ob (term) with a type (functional character) has a normal form.The conventions and definitions of [3], especially of §12D and Chapter 14, will be used throughout the paper.

A normalization theorem for set theory

1988 ◽  
Vol 53 (3) ◽  
pp. 673-695 ◽  
Author(s):  
Sidney C. Bailin
Keyword(s):  
Set Theory ◽  
Natural Deduction ◽  
Inference Rule ◽  
Major Premise ◽  
Elimination Rule ◽  
Introduction Rule ◽  
Normalization Theorem ◽  
Image Position ◽  
Well Foundedness

In this paper we present a normalization theorem for a natural deduction formulation of Zermelo set theory. Our result gets around M. Crabbe's counterexample to normalizability (Hallnäs [3]) by adding an inference rule of the formand requiring that this rule be used wherever it is applicable. Alternatively, we can regard the result as pertaining to a modified notion of normalization, in which an inferenceis never considered reducible if A is T Є T, even if R is an elimination rule and the major premise of R is the conclusion of an introduction rule. A third alternative is to regard (1) as a derived rule: using the general well-foundedness rulewe can derive (1). If we regard (2) as neutral with respect to the normality of derivations (i.e., (2) counts as neither an introduction nor an elimination rule), then the resulting proofs are normal.

Inferential equivalence and natural deduction

1957 ◽  
Vol 22 (3) ◽  
pp. 237-240
Author(s):  
H. Hiż
Keyword(s):  
Natural Deduction ◽  
Identity Relation ◽  
Sentential Calculus ◽  
Image Position

There are axiomatizations of the sentential calculus which, in effect, assert the inferential equivalence (mutual deducibility) of classes of sentences. A well known axiomatization of this sort consists ofAnother one, closely connected with A1–A3, is comprised ofB1–B3 jointly assert the inferential equivalence of a sentence of the form (p ⊃ q) ⊃ r and two sentences of the forms q ⊃ r and ∼p ⊃ r respectively. This axiomatization requires the rule of substitution and the rule of detachment for conditionals.The existence of such axiomatizations suggests a possibility of formulating natural deduction by means of inferential equivalence instead of the usual one-sided inference, customarily represented by ‘⊦’. ‘H’ will stand here for inferential equivalence. It may be looked upon as a combination of signs ‘⊦’ and ‘⊣’. ‘H’ is thought of as a metasystematic constant functor of two arguments each of which is a class of names of sentences. For purposes of this paper the first argument may be just a single name of a sentence, the second a pair (not necessarily ordered) of names of sentences. The role ‘H’ plays is determined by **1 –*5 below. Besides that ‘H’ remains here uninterpreted. In particular, it is not directly postulated that it is an identity relation.

Correction to A note on the Entscheidungsproblem

1936 ◽  
Vol 1 (3) ◽  
pp. 101-102 ◽  
Author(s):  
Alonzo Church
Keyword(s):  
Propositional Function ◽  
Free Variables ◽  
System L ◽  
Universal Quantifiers ◽  
Individual Variables ◽  
The Individual ◽  
Image Position

In A note on the Entscheidungsproblem the author gave a proof of the unsolvability of the general case of the Entscheidungsproblem of the engere Funktionenkalkül. This proof, however, contains an error, in order to correct which it is necessary to modify the “additional axioms” of the system L so that they contain no free variables (either free individual variables or free propositional function variables).The additional axioms of L other than x=y→[F(x)→F(y)] contain no free propositional function variables, and hence it is sufficient to replace each one by an expression obtained from it by quantifying all the individual variables by means of universal quantifiers initially placed—thus, in particular, x = x is replaced by (x)[x=x]. Moreover the axiom x=y→[F(x)→F(y)] may be replaced by the following set of axioms:and similar axioms for each of the functions b1, b2, …, bk.

Comments on a variant form of natural deduction

1965 ◽  
Vol 30 (2) ◽  
pp. 119-121 ◽  
Author(s):  
William Tuthill Parry
Keyword(s):  
Natural Deduction ◽  
Journal Article ◽  
Symbolic Logic ◽  
Variant Form ◽  
Propositional Function ◽  
The Third ◽  
Formal Statement

This note shows that the system of natural deduction proposed by Copi in this Journal (1956), made by varying one restriction on Universal Generalization (UG) of the system of his Symbolic logic (1954), is incorrect. The original Symbolic logic system, also incorrect, was corrected in the third printing (1958) by modification of another restriction on UG; but combining this modification with that of the Journal article does not give a correct system.1. In Symbolic logic — cited as SL (or as SL54 to indicate the first printing) — Copi precedes his formal statement of the quantification rules by a set of conventions applying to all the rules.The expression ‘Φμ’ will denote any propositional function in which there is at least one free occurrence of the variable denoted by ‘μ’. The expression ‘Φν’ will denote the result of replacing all free occurrences of μ in Φμ by ν, with the added provision that [a] when ν is a variable it must occur free in Φν at all places at which μ occurs free in Φμ.The italicized provision we call Restriction a. In the third printing of Symbolic logic — cited as SL58 — Restriction a appears explicitly as part of each of the four quantification rules.

Inner models for set theory—Part II

1952 ◽  
Vol 17 (4) ◽  
pp. 225-237 ◽  
Author(s):  
J. C. Shepherdson
Keyword(s):  
Set Theory ◽  
Complete Model ◽  
Universal Class ◽  
Propositional Function ◽  
Class V ◽  
Completeness Condition ◽  
Inner Models ◽  
Condition C ◽  
Definition Of ◽  
Image Position

In this paper we continue the study of inner models of the type studied inInner models for set theory—Part I.The present paper is concerned exclusively with a particular kind of model, the ‘super-complete models’ defined in section 2.4 of I (page 186). The condition (c) of 2.4 and the completeness condition 1.42 imply that such a model is uniquely determined when its universal class Vmis given. Writing condition (c) and the completeness conditions 1.41, 1.42 in terms of Vm, we may state the definition in the form:3.1. Dfn.A classVmis said to determine a super-complete model if the model whose basic notions are defined by,satisfies axiomsA, B, C.N. B. This definition is not necessarily metamathematical in nature. If desired, it could be written out quite formally as the definition of a notion ‘SCM(U)’ (‘Udetermines a super-complete model’) thus:whereψ(U) is the propositional function expressing in terms ofUthe fact that the model determined byUaccording to 3.1 satisfies the relativization of axioms A, B, C. E.g. corresponding to axiom A1m, i.e.,,ψ(U) contains the equivalent term. All the relativized axioms can be similarly expressed in this way by first writing out the relativized form (after having replaced all defined symbols which occur by the corresponding formulae in primitive notation) and then replacing ‘(Am)ϕ(Am) bywhich is in turn replaced by, and similarly replacing ‘(xm)ϕ(xm)’ by ‘(xm)ϕ(xm)’ by ‘(X)(X ϵ U ▪ ⊃ ▪ ϕ(X)), andThusψ(U) is obtained in primitive notation.

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

A Magnetic Spectrometer for a Scanning Transmission Electron Microscope

1974 ◽  
Vol 32 ◽  
pp. 436-437
Author(s):  
A. Kosiara ◽  
J. W. Wiggins ◽  
M. Beer
Keyword(s):  
Scanning Transmission Electron Microscope ◽  
Magnetic Spectrometer ◽  
Fringe Field ◽  
Curvature Radius ◽  
Magnetic Pole ◽  
Quadrupole Field ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
Under Construction ◽  
Image Position

A magnetic spectrometer to be attached to the Johns Hopkins S. T. E. M. is under construction. Its main purpose will be to investigate electron interactions with biological molecules in the energy range of 40 KeV to 100 KeV. The spectrometer is of the type described by Kerwin and by Crewe Its magnetic pole boundary is given by the equationwhere R is the electron curvature radius. In our case, R = 15 cm. The electron beam will be deflected by an angle of 90°. The distance between the electron source and the pole boundary will be 30 cm. A linear fringe field will be generated by a quadrupole field arrangement. This is accomplished by a grounded mirror plate and a 45° taper of the magnetic pole.

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