Olaf Helmer and Paul Oppenheim. A syntactical definition of probability and of degree of confirmation. The journal of symbolic logic, vol. 10 (1945), pp. 25–60. See errata, ibid., p. iv.

1946 ◽  
Vol 11 (1) ◽  
pp. 17-18
Author(s):  
Max Black
Keyword(s):  
Symbolic Logic ◽  
Definition Of ◽  
Degree Of Confirmation

A Definition of Degree of Confirmation for Very Rich Languages

1956 ◽  
Vol 23 (1) ◽  
pp. 58-62 ◽  
Author(s):  
Hilary Putnam
Keyword(s):  
Definition Of ◽  
Degree Of Confirmation

R. W. Ritchie. A rudimentary definition of addition. The journal of symbolic logic, vol. 30 (1965), pp. 350–354.

1970 ◽  
Vol 35 (3) ◽  
pp. 475-475
Author(s):  
James H. Bennett
Keyword(s):  
Symbolic Logic ◽  
Definition Of

Fair bets and inductive probabilities

1955 ◽  
Vol 20 (3) ◽  
pp. 263-273 ◽  
Author(s):  
John G. Kemeny
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Inductive Probability ◽  
Other Hand ◽  
Frequency Interpretation ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Degree Of Confirmation

The question of what constitutes fairness in betting quotients has been studied by Ramsey, deFinetti, and Shimony. Thanks to their combined efforts we now have a satisfactory definition of fairness.On the other hand, the explication of the concept of degree of confirmation (inductive probability) has progressed rapidly in recent years, thanks primarily to Carnap. This explication has usually proceeded by laying down the axioms for frequency-probabilities, and elaborating on these. While in the case where a frequency interpretation is intended these axioms are clearly justified, in our case they have been laid down without any justification. Carnap's presentation has been criticized for just this reason.The purpose of this paper is to show that the probability axioms are necessary and sufficient conditions to assure that the degrees of confirmation form a set of fair betting quotients. In addition it will be shown that one additional, highly controversial, axiom is precisely the condition needed to assure that not only deFinetti's weaker criterion but Shimony's criterion of fairness is also satisfied.

A note on the Entscheidungsproblem

1936 ◽  
Vol 1 (1) ◽  
pp. 40-41 ◽  
Author(s):  
Alonzo Church
Keyword(s):  
Present Note ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Symbolic Logic ◽  
Recursive Definition ◽  
System L ◽  
Preceding Paragraph ◽  
Individual Variables ◽  
Definition Of ◽  
Positive Integers

In a recent paper the author has proposed a definition of the commonly used term “effectively calculable” and has shown on the basis of this definition that the general case of the Entscheidungsproblem is unsolvable in any system of symbolic logic which is adequate to a certain portion of arithmetic and is ω-consistent. The purpose of the present note is to outline an extension of this result to the engere Funktionenkalkul of Hilbert and Ackermann.In the author's cited paper it is pointed out that there can be associated recursively with every well-formed formula a recursive enumeration of the formulas into which it is convertible. This means the existence of a recursively defined function a of two positive integers such that, if y is the Gödel representation of a well-formed formula Y then a(x, y) is the Gödel representation of the xth formula in the enumeration of the formulas into which Y is convertible.Consider the system L of symbolic logic which arises from the engere Funktionenkalkül by adding to it: as additional undefined symbols, a symbol 1 for the number 1 (regarded as an individual), a symbol = for the propositional function = (equality of individuals), a symbol s for the arithmetic function x+1, a symbol a for the arithmetic function a described in the preceding paragraph, and symbols b1, b2, …, bk for the auxiliary arithmetic functions which are employed in the recursive definition of a; and as additional axioms, the recursion equations for the functions a, b1, b2, …, bk (expressed with free individual variables, the class of individuals being taken as identical with the class of positive integers), and two axioms of equality, x = x, and x = y →[F(x)→F(y)].

scholarly journals On guessing and reasoning: a reply to Paul Bourgine's

1969 ◽  
Vol 2 (1) ◽  
Author(s):  
Priscila L. Farias
Keyword(s):  
Abductive Inference ◽  
Symbolic Logic ◽  
Dear Reader ◽  
Charles Peirce ◽  
Definition Of ◽  
Surprising Fact

Charles Peirce ends the above tentative definition of the grounds for abduction by stating that “the effort should therefore be to make each hypothesis, which is practically no more than a question, as near an even bet as possible” (CP 1.121). The reason why I am quoting this passage of the Collected Papers is that I assume it can justify why should I (with no solid background in symbolic logic or mathematics) dare coming up with a debate on Paul Bourgine’s article “Models of abduction.” Taking the following rendition of an abductive inference as a guide: The surprising fact, C, is observed; But if A were true, C would be a matter of course, Hence, there is reason to suspect that A is true. (CP 5.189) I therefore decided to consider Paul’s article as a “surprising fact,” and to try my best in devising relevant questions about it. I believe should thank my deductive capacities for enabling me to extract hopefully clear and reliable consequences from those questions, but I will only be able to evaluate the pertinence of my efforts after a moment o f induction I expect to share with you, dear reader.

scholarly journals Metalogical Remarks on Induction

Axiomathes ◽  
2021 ◽  
Author(s):  
Jan Woleński
Keyword(s):  
Philosophy Of Science ◽  
Qualitative Approach ◽  
Controversial Issues ◽  
Logical Probability ◽  
Problem Of Induction ◽  
Definition Of ◽  
Degree Of Confirmation ◽  
Quantitative Account

AbstractThe problem of induction belongs to the most controversial issues in philosophy of science. If induction is understood widely, it covers every fallible inference, that is, such that its conclusion is not logically entailed by its premises. This paper analyses so-called reductive induction, that is, reasoning in which premises follow from the conclusion, but the reverse relation does not hold. Two issues are taken into account, namely the definition of reductive inference and its justification. The analysis proposed in the paper employs metalogical tools. The author agrees with the view that a quantitative account of degree of confirmation for universal theories via logical probability is problematic. However, prospect for a qualitative approach look as more promising. Using the construction of maximally consistent sets allows to distinguish good and worthless induction as well as shows how to understand induction in a semantic way. A closer analysis of deductivism in the theory of justification shows that it is a hidden inductivism.

scholarly journals A wonderful analogy between Augustine’s definition of moral-value- functional sense of response-action and Philo’s definition of truth-functional sense of implication in logic

2020 ◽  
Vol 27 (3) ◽  
pp. e94
Author(s):  
Vladimir O. Lobovikov
Keyword(s):  
Interdisciplinary Research ◽  
Precise Definition ◽  
Philosophical Theology ◽  
Symbolic Logic ◽  
Moral Value ◽  
Human Actions ◽  
Moral Response ◽  
Definition Of ◽  
Value Functional

The paper is dvoted to interdisciplinary research at the intersection of symbolic logic, mathematical ethics, and philosophical theology. By comparing definitions of relevant functions, a surprising analogy is discovered between the well-known Philo’s precise definition of implication in logic (classical one) and Augustine’s precise definition of God’s morally good reaction to human actions. The moral-value-table-representation of Augustinian doctrine is compared with moral-value-table-representations of Pelagius’ and Leo Tolstoy’s views of adequate moral-response-actions.

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