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

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 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.

scholarly journals Czy logika formalna ma sens?

2018 ◽  
Vol 13 (3) ◽  
pp. 49-55
Author(s):  
Ludomir Newelski
Keyword(s):  
Critical Thinking ◽  
Formal Logic ◽  
Symbolic Logic ◽  
Logical Inference ◽  
Semantic Inference ◽  
Definition Of ◽  
Rational Reasoning

Does formal logic make sense?This text is a commentary on the book Logika i argumentacja. Praktyczny kurs krytycznego myślenia Logic and Argumentation. A Practical Course In Critical Thinking by Professor Andrzej Kisielewicz. Prof. Kisielewicz argues there, among other things, that formal symbolic logic is inadequate to explain practical rational reasoning. This commentary defends formal logic in this respect. In particular, Prof. Kisielewicz proposes in his book a definition of practical logical inference. According to him, a conclusion follows from a given set of premises if there is no situation, where the premises hold, while the conclusion fails. In this commentary it is pointed out that this is a well-known notion of semantic inference in formal logic. It is also well-known that semantic and syntactic inference in logic are equivalent, i.e. equally strong.

scholarly journals CONSISTENCY OF CIRCUIT EVALUATION, EXTENDED RESOLUTION AND TOTAL NP SEARCH PROBLEMS

2016 ◽  
Vol 4 ◽  
Author(s):  
JAN KRAJÍČEK
Keyword(s):  
Symbolic Logic ◽  
Search Problem ◽  
Bounded Arithmetic ◽  
Search Problems ◽  
Open Problems ◽  
Proof Systems ◽  
Arithmetic Theory ◽  
Propositional Proof Systems ◽  
Propositional Proof ◽  
Definition Of

We consider sets ${\it\Gamma}(n,s,k)$ of narrow clauses expressing that no definition of a size $s$ circuit with $n$ inputs is refutable in resolution R in $k$ steps. We show that every CNF with a short refutation in extended R, ER, can be easily reduced to an instance of ${\it\Gamma}(0,s,k)$ (with $s,k$ depending on the size of the ER-refutation) and, in particular, that ${\it\Gamma}(0,s,k)$ when interpreted as a relativized NP search problem is complete among all such problems provably total in bounded arithmetic theory $V_{1}^{1}$. We use the ideas of implicit proofs from Krajíček [J. Symbolic Logic, 69 (2) (2004), 387–397; J. Symbolic Logic, 70 (2) (2005), 619–630] to define from ${\it\Gamma}(0,s,k)$ a nonrelativized NP search problem $i{\it\Gamma}$ and we show that it is complete among all such problems provably total in bounded arithmetic theory $V_{2}^{1}$. The reductions are definable in theory $S_{2}^{1}$. We indicate how similar results can be proved for some other propositional proof systems and bounded arithmetic theories and how the construction can be used to define specific random unsatisfiable formulas, and we formulate two open problems about them.

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