|Decidability: a question, for example, whether a property applies to an object or not, is decidable if a result can be achieved within a finite time. For this decision process, an algorithm is chosen as a basis. See also halting problem, algorithms, procedures, decision theory._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
|Berka I 329
Decision problem/Logic/Berka: appeared historically for the first time in Leibniz with the idea of a purely arithmetical "ars iudicandi".
Behmann: (1922)(1): "The main problem of modern logic".
I. It is to be decided with exactly stated means, whether a relevant formula of a (logical) calculus is valid.
II. If it is not universal, it is to be decided whether it is valid in none of the areas or whether it is valid in an area. If it is valid in any area, one must determine which cardinal number this area has.
III. It is to be decided whether a relevant formula is valid in all areas with a finite number of elements or not."
Berka: this is a basically semantic formulation of the E problem.
E Problem/syntactical: it is to be decided with the help of exactly defined processes that have to fulfill certain conditions whether a relevant formula of a calculus is provable or refutable.
Statement Calculus/E-Problem: by Lukasiewicz (1921)(3), Post (1921)(4), Wittgenstein (1921)(5) positively solved.
1. H. Behmann, Beiträge zur Algebra der Logik, insbesondere zum Entscheidungsproblem, Math. Ann. 86 (1922), 163-229
2. R. Ackermann, Solvable Classes of the Decision Problem, Amsterdam (3. ed.) 1968
3. J. Lukasiewicz, Logica dwuwartosciowa, PF 23 (1921), 189-205
4. E. L. Post, Introduction to a general theory of elemantary propositions, American Journal of Mathematics 43 (1921) , 163-185
5. L. Wittgenstein, Logisch-Philosophische Abhandlung, Ann. Naturphil. 14 (1921), 185-262_____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. The note [Author1]Vs[Author2] or [Author]Vs[term] is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
G. W. Leibniz
Philosophical Texts (Oxford Philosophical Texts) Oxford 1998
Logik Texte Berlin 1983