A basic logic

1942 ◽  
Vol 7 (3) ◽  
pp. 105-114 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Finite Number ◽  
Strong Evidence ◽  
Decision Problem ◽  
Simple System ◽  
Basic Logic ◽  
Recursive Methods ◽  
Universal Quantification

This paper is concerned with finding a fairly simple system of logic which is “basic” in the sense that every system of logic is definable in it. If a “system”is regarded as being a class of propositions, rather than a class of sentences, then every class of propositions which is a system should be definable in the basic logic. The system of logic proposed in this paper will not be proved to be a basic logic, but strong evidence that it is basic will be given at the end of the paper. Evidence will also be given that the class of propositions which are not provable in the system is not definable in any system of logic. It will be established that the decision problem is unsolvable for the system. Notable characteristics of the system are its lack of negation and universal quantification, and its similarity to systems proposed by Church, Curry, and Rosser.By a “system” will be meant a class of propositions the membership of which can be specified by ordinary recursive methods, so that if the membership of a given proposition in the class can be established at all, such membership can be established in a finite number of steps. (This is not the same as demanding that a criterion must exist for determining, in a finite number of steps, whether or not some given proposition is a member of the class. Such a demand would require a solution of the decision problem for every system.)

A definition of existence in terms of abstraction and disjunction

1957 ◽  
Vol 22 (4) ◽  
pp. 343-344
Author(s):  
Frederic B. Fitch
Keyword(s):  
Finite Number ◽  
Infinite Number ◽  
Recursive Relation ◽  
Basic Logic ◽  
Class A ◽  
Definition Of ◽  
Image Position

Greater economy can be effected in the primitive rules for the system K of basic logic by defining the existence operator ‘E’ in terms of two-place abstraction and the disjunction operator ‘V’. This amounts to defining ‘E’ in terms of ‘ε’, ‘έ’, ‘o, ‘ό’, ‘W’ and ‘V’, since the first five of these six operators are used for defining two-place abstraction.We assume that the class Y of atomic U-expressions has only a single member ‘σ’. Similar methods can be used if Y had some other finite number of members, or even an infinite number of members provided that they are ordered into a sequence by a recursive relation represented in K. In order to define ‘E’ we begin by defining an operator ‘D’ such thatHere ‘a’ may be thought of as an existence operator that provides existence quantification over some finite class of entities denoted by a class A of U-expressions. In other words, suppose that ‘a’ is such that ‘ab’ is in K if and only if, for some ‘e’ in A, ‘be’ is in K. Then ‘Dab’ is in K if and only if, for some ‘e and ‘f’ in A, ‘be’ or ‘b(ef)’ is in K; and ‘a’, ‘Da’, ‘D(Da)’, and so on, can be regarded as existence operators that provide for existence quantification over successively wider and wider finite classes. In particular, if ‘a’ is ‘εσ’, then A would be the class Y having ‘σ’ as its only member, and we can define the unrestricted existence operator ‘E’ in such a way that ‘Eb’ is in K if and only if some one of ‘εσb’, ‘D(εσ)b’, ‘D(D(εσ))b’, and so on, is in K.

scholarly journals Large prospective losses lead to sub-optimal sensorimotor decisions in humans

2018 ◽  
Author(s):  
Tyler J. Adkins ◽  
Richard L. Lewis ◽  
Taraz G. Lee
Keyword(s):  
Bounded Rationality ◽  
Recent Work ◽  
Human Behavior ◽  
Strong Evidence ◽  
Decision Problem ◽  
Computational Models ◽  
Cognitive Biases ◽  
Expected Gain ◽  
Potential Losses ◽  
Biases And Heuristics

AbstractThe rationality of human behavior has been a major problem in philosophy for centuries. The pioneering work of Kahneman and Tversky provides strong evidence that people are not rational. Recent work in psychophysics argues that incentivized sensorimotor decisions (such as deciding where to reach to get a reward) maximizes expected gain, suggesting that it may be impervious to cognitive biases and heuristics. We rigorously tested this hypothesis using multiple experiments and multiple computational models. We obtained strong evidence that people deviated from the objectively rational strategy when potential losses were large. They instead appeared to follow a strategy in which they simplify the decision problem and satisfice rather than optimize. This work is consistent with the framework known as bounded rationality, according to which people behave rationally given their computational limitations.

A further consistent extension of basic logic

1950 ◽  
Vol 14 (4) ◽  
pp. 209-218 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Special Kind ◽  
Basic Logic ◽  
Real Numbers ◽  
Consistent Theory ◽  
The Real ◽  
Recursively Enumerable ◽  
Consistent Extension ◽  
Universal Quantification ◽  
Excluded Middle ◽  
Additional Assumptions

1.1. In two previous papers a consistent theory of real numbers has been outlined by the author, using a system K′. This latter system is an extension of a system K, which is “basic” in the sense that every finitary (recursively enumerable) subclass of its well-formed expressions is in a certain sense represented in it. The system L described below is a further extension of K. The system K′ lacks two important features possessed by L: a symbol for a special kind of implication (or “conditionality”) and a symbol for the modal concept “necessity.” The presence of the implication symbol, and the additional assumptions that go with it, make available in L various kinds of restricted universal quantification not available in K′, for example, universal quantification restricted to the real numbers of the author's theory of real numbers.1.2. If ‘~[a & ~a]’ is a theorem of L, then the proposition expressed by ‘a’ may be said to L-satisfy the principle of excluded middle. I t is always the case that ‘a’ L-satisfies the principle of excluded middle (or rather that the proposition expressed by ‘a’ does so) if and only if ‘a’ or ‘~a’ is a theorem of L. An example of a proposition that does not L-satisfy the principle of excluded middle is that expressed by ‘’, namely the proposition that asserts that the class of classes that are not members of themselves is a member of itself.

scholarly journals A review on statistical Bayes decision theory in the case of more than one individual

1983 ◽  
Vol 2 (4) ◽  
pp. 206-214
Author(s):  
D. J. De Waal
Keyword(s):  
Decision Theory ◽  
Finite Number ◽  
Decision Problem ◽  
Decision Problems ◽  
Nash Solution ◽  
Bayes Decision Theory ◽  
Binomial Parameter ◽  
The Nash Solution

A review on statistical Bayes decision theory in the case of more than one individual is given. Although several solutions are pointed out, special attention is given to the Nash solution as a good candidate to use in practical decision problems. Attention is given to a decision problem where the best decision out of a finite number of decisions has to be taken and there is also an estimation of a binomial parameter by several Bayesians.

A complete theory of natural, rational, and real numbers

1950 ◽  
Vol 15 (3) ◽  
pp. 185-196 ◽  
Author(s):  
John R. Myhill
Keyword(s):  
Number Theory ◽  
Complete Theory ◽  
Future Research ◽  
Present System ◽  
Basic Logic ◽  
Real Numbers ◽  
Mathematical System ◽  
Classical Mathematics ◽  
Universal Quantification ◽  
Range Of Values

The purpose of the present paper is to construct a fragment of number theory not subject to Gödel incompletability. Originally the system was designed as a metalanguage for classical mathematics (see section 10); but it now appears to the author worthwhile to present it as a mathematical system in its own right, to serve however rather as an instrument of computation than of proof. Its resources in the latter respect seem very extensive, sufficient apparently for the systematic tabulation of every function used in any but the most recondite physics. The author intends to pursue this topic in a later paper; the present one will simply present the system, along with proofs of consistency and completeness and a few metatheorems which will be used as lemmas for future research.Completeness is achieved by sacrificing the notions of negation and universal quantification customary in number-theoretic systems; the losses consequent upon this are made good in part by the use of the ancestral as a primitive idea. The general outlines of the system follow closely the pattern of Fitch's “basic logic”; however the latter system uses combinatory operators in place of the variables used in the present paper, and if variables are introduced into Fitch's system by definition their range of values will be found to be much more extensive than that of my variables. The present system K is thus a weaker form of Fitch's system.It is apparently not known whether or not Fitch's system is complete.

Criteria and Methods for Reliable Three-Dimensional Image Reconstruction

1974 ◽  
Vol 32 ◽  
pp. 330-331
Author(s):  
R. A. Crowther
Keyword(s):  
Image Reconstruction ◽  
Finite Number ◽  
Density Distribution ◽  
Electron Micrographs ◽  
Mathematical Problem ◽  
Three Dimensional ◽  
Ideal Case ◽  
Dimensional Image ◽  
General Object ◽  
The Ideal

The reconstruction of a three-dimensional image of a specimen from a set of electron micrographs reduces, under certain assumptions about the imaging process in the microscope, to the mathematical problem of reconstructing a density distribution from a set of its plane projections.In the absence of noise we can formulate a purely geometrical criterion, which, for a general object, fixes the resolution attainable from a given finite number of views in terms of the size of the object. For simplicity we take the ideal case of projections collected by a series of m equally spaced tilts about a single axis.

Concurrent Electrodermal and Eyeblink Conditioning With Masked and Unmasked Stimuli

2021 ◽  
Vol 35 (1) ◽  
pp. 35-42
Author(s):  
José Luis Marcos ◽  
Azahara Marcos
Keyword(s):  
Unconditioned Stimulus ◽  
Strong Evidence ◽  
Eyeblink Conditioning ◽  
Noise Burst ◽  
Conditioning Phase ◽  
Contingency Awareness ◽  
Blank Screen ◽  
Neutral Face ◽  
Fearful Face

Abstract. The aim of this study was to determine if contingency awareness between the conditioned (CS) and unconditioned stimulus (US) is necessary for concurrent electrodermal and eyeblink conditioning to masked stimuli. An angry woman’s face (CS+) and a fearful face (CS−) were presented for 23 milliseconds (ms) and followed by a neutral face as a mask. A 98 dB noise burst (US) was administered 477 ms after CS+ offset to elicit both electrodermal and eyeblink responses. For the unmasking conditioning a 176 ms blank screen was inserted between the CS and the mask. Contingency awareness was assessed using trial-by-trial ratings of US-expectancy in a post-conditioning phase. The results showed acquisition of differential electrodermal and eyeblink conditioning in aware, but not in unaware participants. Acquisition of differential eyeblink conditioning required more trials than electrodermal conditioning. These results provided strong evidence of the causal role of contingency awareness on differential eyeblink and electrodermal conditioning.

Sign in / Sign up

Export Citation Format

Share Document