On the inconsistency of systems similar to

1978 ◽  
Vol 43 (1) ◽  
pp. 1-2 ◽  
Author(s):  
M. W. Bunder ◽  
R. K. Meyer
Keyword(s):  
General System ◽  
Modus Ponens ◽  
Combinatory Logic ◽  
Deduction Theorem ◽  
The Absolute ◽  
Image Position ◽  
Weakened Form

This note shows that the inconsistency proof of the system of illative combinatory logic given in [1] can be simplified as well as extended to the absolute inconsistency of a more general system.One extension of the result in [1] lies in the fact that the following weakened form of the deduction theorem for implication will lead to the inconsistency:Also the inconsistency follows almost as easily foras it does for ⊢ H2X for arbitrary X, so we will consider the more general case.The only properties we require other than (DT), (1) and Rule Eq for equality are modus ponens,andLet G = [x] Hn−1x⊃: . … H2x⊃ :Hx⊃ . x ⊃ A, where A is arbitrary. Then if Y is the paradoxical combinator and X = YG, X = GX.Now X ⊂ X, i.e.,

The inconsistency of

1976 ◽  
Vol 41 (2) ◽  
pp. 467-468
Author(s):  
M. W. Bunder
Keyword(s):  
Fixed Point ◽  
Modus Ponens ◽  
The Other ◽  
Deduction Theorem ◽  
Image Position

In [4] Curry raised the possibility that his system proposed in ξ15C of [3] might be inconsistent. In this paper this inconsistency is proved using a method also employed in [1].From Curry's axiom ⊦LH, it follows thatholds for arbitrary X.The other results from that are required areModus Ponens, and the Deduction Theorem for implication:Assuming ⊦HA, we define as in [1]:and letwhere Y is the paradoxical (or fixed point) combinator.We have X = G2X, so, by (2), H X ⊦ X ⊃ G2X which is HX ⊦ X ⊃. H2X ⊃ G1X. Clearly HX ⊦ H(G1X) and, by (5), HX ⊦ H3X, so that, by (3), H X ⊦ H2X ⊃. X ⊃ G1X and by (5) and Modus Ponens HX ⊦ X ⊃ G1X. This is HX ⊦ X ⊃: HX ⊃. X ⊃ A which by (3) and Modus Ponens giveswhich gives, by (4), HX ⊦ X ⊃ A. Now, by (1) and (DT),which is ⊦G1X. But we have ⊦H2X so, by the (DT), ⊦H2X ⊃ G1X which is ⊦G2X. Thus we have proved ⊦X.

Consistency notions in illative combinatory logic

1977 ◽  
Vol 42 (4) ◽  
pp. 527-529 ◽  
Author(s):  
M. W. Bunder
Keyword(s):  
Propositional Calculus ◽  
General System ◽  
Combinatory Logic ◽  
Definition Of ◽  
Image Position

In this paper we show that certain notions of consistency currently in use in illative combinatory logic are not always necessary nor always sufficient for the acceptability of the system.The three notions of consistency we will consider are:(A) Not all terms of the system are assertible.(B) Not all propositions of the system are assertible.(C) The system is Q-consistent, i.e. if Q (for equality) is in or is added to the system as a primitive with axioms such asPost defined a system of propositional calculus to be inconsistent if all propositions were assertible. In a more general system where there may be terms which are not propositions (although all propositions are terms), we need to distinguish notions (A) and (B) as two possible consistency notions “in the sense of Post”.In [3] Curry and Feys assume that Q is in the system defined as λxλy.Ξ(C*x)(C*y) or as a primitive, but Seldin in [5] also talks about Q-consistency of a system (F22) to which Q has to be added. We therefore consider the definition of Q-consistency that includes all three possibilities.In [3] Curry and Feys state that Q-consistency is essential for acceptability. This is because there the purpose was to have an assertional system of illative combinatory logic in which combinatory equality was represented by the assertion of QXY. However if Q is to represent some other form of equality, Q-consistency may become inessential for acceptability or even incompatible with it.

Post completeness in modal logic

1972 ◽  
Vol 37 (4) ◽  
pp. 711-715 ◽  
Author(s):  
Krister Segerberg
Keyword(s):  
Modal Logic ◽  
Upper Bound ◽  
Modus Ponens ◽  
Proper Subset ◽  
Proper Extension ◽  
Image Position

Let ⊥, →, and □ be primitive, and let us have a countable supply of propositional letters. By a (modal) logic we understand a proper subset of the set of all formulas containing every tautology and being closed under modus ponens and substitution. A logic is regular if it contains every instance of □A ∧ □B ↔ □(A ∧ B) and is closed under the ruleA regular logic is normal if it contains □⊤. The smallest regular logic we denote by C (the same as Lemmon's C2), the smallest normal one by K. If L and L' are logics and L ⊆ L′, then L is a sublogic of L', and L' is an extension of L; properly so if L ≠ L'. A logic is quasi-regular (respectively, quasi-normal) if it is an extension of C (respectively, K).A logic is Post complete if it has no proper extension. The Post number, denoted by p(L), is the number of Post complete extensions of L. Thanks to Lindenbaum, we know thatThere is an obvious upper bound, too:Furthermore,.

Irula riddles

1979 ◽  
Vol 42 (2) ◽  
pp. 361-368
Author(s):  
K. V. Zvelebil
Keyword(s):  
Social Organization ◽  
South India ◽  
Oral Traditions ◽  
Verbal Art ◽  
Tribal Group ◽  
Economic Sphere ◽  
Literary Cultures ◽  
The Absolute ◽  
Complex Forms ◽  
Image Position

The Irulas 2 are a tribal complex of four tribes inhabiting the lower slopes of the northern, eastern and southern parts of the Nilgiri mountains of South India. They speak a tribal language of their own—the ërla na:ya—in four dialects; it belongs, historically, to the Tamil-Malayalam group of South Dravidian.3 Two of the tribes intermarry, so that the Irula complex forms a tribal group of three endogamous units. Linguistically and from the point of social organization, the Irula situation may be thus symbolized asThe creativity of Irula-speaking tribes finds expression mostly in music,4 dance, and above all, in verbal art.5 They have a wealth of oral traditions characteristic for most pre-literary cultures; though modernization—thus far mostly in the socio-economic sphere—has had its impact on the Irula-speaking tribes, the absolute majority of the Irulas are still illiterate. Hence storytelling, oral rendering of myths, legends and genealogies, and other forms of verbal art are still very much alive.

On the Absolute Nörlund Summability of a Fourier Series

1973 ◽  
Vol 16 (4) ◽  
pp. 599-602
Author(s):  
D. S. Goel ◽  
B. N. Sahney
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Partial Sums ◽  
The Absolute ◽  
Image Position ◽  
Absolute Nörlund Summability

Let be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us write(1.1)If(1.2)as n→∞, we say that the series is summable by the Nörlund method (N,pn) to σ. The series is said to be absolutely summable (N,pn) or summable |N,pn| if σn is of bounded variation, i.e.,(1.3)

scholarly journals Pointwise characteristic factors for Wiener–Wintner double recurrence theorem

2015 ◽  
Vol 36 (4) ◽  
pp. 1037-1066 ◽  
Author(s):  
IDRIS ASSANI ◽  
DAVID DUNCAN ◽  
RYO MOORE
Keyword(s):  
Full Measure ◽  
Orthogonal Complement ◽  
Ergodic System ◽  
Absolute Value ◽  
Recurrence Theorem ◽  
The Absolute ◽  
Image Position ◽  
Null Set

In this paper we extend Bourgain’s double recurrence result to the Wiener–Wintner averages. Let $(X,{\mathcal{F}},{\it\mu},T)$ be a standard ergodic system. We will show that for any $f_{1},f_{2}\in L^{\infty }(X)$, the double recurrence Wiener–Wintner average $$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)e^{2{\it\pi}int}\end{eqnarray}$$ converges off a single null set of $X$ independent of $t$ as $N\rightarrow \infty$. Furthermore, we will show a uniform Wiener–Wintner double recurrence result: if either $f_{1}$ or $f_{2}$ belongs to the orthogonal complement of the Conze–Lesigne factor, then there exists a set of full measure such that the supremum on $t$ of the absolute value of the averages above converges to $0$.

scholarly journals Absolute Calibration of Stellar Temperatures

1973 ◽  
Vol 54 ◽  
pp. 153-155
Author(s):  
H. Kienle ◽  
D. Labs
Keyword(s):  
Spectral Energy Distribution ◽  
Black Body ◽  
Absolute Temperature ◽  
Spectral Energy ◽  
Absolute Calibration ◽  
Standard Star ◽  
The Absolute ◽  
Stellar Temperatures ◽  
Image Position ◽  
Radiation Laboratory

The scale of effective temperatures Teff is based on observed absolute radiation temperatures Tr, which are defined by Planck's radiation law where TAu designs the absolute temperature of the gold point. A relative scale of radiation temperatures can be derived from spectrophotometric comparisons with a standard star. The absolute calibration of the standard star (α Lyr or Sun) demands a careful comparison with a standard radiation source of well known spectral energy distribution (Black Body or Synchrotron). With ground-based observations atmospheric extinction is to be taken into account; with extraterrestrial observations detectors may be used which are absolutely calibrated in a radiation laboratory under space conditions.

Pure three-valued Łukasiewiczian implication

1966 ◽  
Vol 31 (3) ◽  
pp. 399-405 ◽  
Author(s):  
Storrs McCall ◽  
R. K. Meyer
Keyword(s):  
Modus Ponens ◽  
The Matrix ◽  
Image Position

The matrix defining Łukasiewicz's three-valued logic, constructed in 1920 and described at length in [1], is the following: This matrix was axiomatized in 1931 by Wajsberg (see [6]), who showed that the following axioms together with the rules of substitution and modus ponens were sufficient:

Independence of Rose's axioms for m-valued implication

1969 ◽  
Vol 34 (2) ◽  
pp. 283-284 ◽  
Author(s):  
Ernest Edmonds
Keyword(s):  
Modus Ponens ◽  
Image Position

Rose has shown in [2] that the following axioms are sufficient, with modus ponens, for m-valued Łukasiewiczian implication.

scholarly journals Note on the Standard of Relative Viscosity, and on “Negative Viscosity.”

1906 ◽  
Vol 25 (1) ◽  
pp. 227-230 ◽  
Author(s):  
W. W. Taylor
Keyword(s):  
Capillary Tube ◽  
Relative Viscosity ◽  
Absolute Viscosity ◽  
Flow Through ◽  
The Absolute ◽  
Negative Viscosity ◽  
Image Position

The absolute viscosity calculated from the formula(where p = the pressure, t the time, r the radius, l the length of capillary, and v the volume of liquid), which connects the viscosity of a liquid with the rate of flow through a long capillary tube, is not often made use of, mainly on account of the difficulty of accurately determining some of the constants (r in particular).

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