decidability
Quick Summary
Decidability, in logic, refers to whether a question can be answered with a definitive "yes" or "no" through a finite, algorithmic process. This concept is central to metalogic, which examines the properties of formal systems.123
Key discoveries in the 1930s by mathematicians like Alonzo Church and Alan Turing established that not all problems are decidable. They developed the concept of a Turing machine to define mechanical computability. This led to the understanding that certain formal systems, like arithmetic, are undecidable, meaning there's no general algorithm to determine the truth or provability of all statements within them. For instance, Gödel's incompleteness theorems demonstrate that any sufficiently complex formal system will contain undecidable sentences.43
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metalogicDiscoveries about formal mathematical systems…arrived at sharp concepts of decidability. In one sense, decidability is a property of sets (of sentences): that of being subject (or not) to mechanical methods by which to decide in a finite number of steps, for any closed sentence of a given formal system (e.g., of N), whether it…
Read More…arrived at sharp concepts of decidability. In one sense, decidability is a property of sets (of sentences): that of being subject (or not) to mechanical methods by which to decide in a finite number of steps, for any closed sentence of a given formal system (e.g., of N), whether it…
Read More- metalogicThe undecidability theorem and reduction classes
…this class of problems is undecidable is particularly suggestive. Once the concept of mechanical procedure was crystallized, it was relatively easy to find absolutely unsolvable problems—e.g., the halting problem, which asks for each Turing machine the question of whether it will ever stop, beginning with a blank tape. In other…
Read More…this class of problems is undecidable is particularly suggestive. Once the concept of mechanical procedure was crystallized, it was relatively easy to find absolutely unsolvable problems—e.g., the halting problem, which asks for each Turing machine the question of whether it will ever stop, beginning with a blank tape. In other…
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formal logicValidity in LPC…that LPC is not a decidable system. This does not mean that it is never possible to prove that a given wff of LPC is valid—the validity of an unlimited number of such wffs can in fact be demonstrated—but it does mean that in the case of LPC, unlike that…
Read More…that LPC is not a decidable system. This does not mean that it is never possible to prove that a given wff of LPC is valid—the validity of an unlimited number of such wffs can in fact be demonstrated—but it does mean that in the case of LPC, unlike that…
Read More- formal logicAxiomatization of LPC
Rules of uniform substitution for predicate calculi, though formulable, are mostly very complicated, and, to avoid the necessity for these rules, axioms for these systems are therefore usually given by axiom schemata in the sense explained earlier (see above Axiomatization of PC). Given the formation rules and definitions stated in…
Read MoreRules of uniform substitution for predicate calculi, though formulable, are mostly very complicated, and, to avoid the necessity for these rules, axioms for these systems are therefore usually given by axiom schemata in the sense explained earlier (see above Axiomatization of PC). Given the formation rules and definitions stated in…
Read More - formal logicSpecial systems of LPC
…system is that it is decidable. (The introduction of even a single dyadic predicate variable, however, would make the system undecidable, and, in fact, even the system that contains only a single dyadic predicate variable and no other predicate variables at all has been shown to be undecidable.) b.A still…
Read More…system is that it is decidable. (The introduction of even a single dyadic predicate variable, however, would make the system undecidable, and, in fact, even the system that contains only a single dyadic predicate variable and no other predicate variables at all has been shown to be undecidable.) b.A still…
Read More - formal logicValidity in modal logic
…is, like LPC itself, an undecidable system.
Read More…is, like LPC itself, an undecidable system.
Read More - formal logicValidity in PC
…is said to be a decidable one. For other systems it can be proved that no decision procedure is possible; the decision problem for such a system is then said to be unsolvable, and the system is said to be an undecidable one.
Read More…is said to be a decidable one. For other systems it can be proved that no decision procedure is possible; the decision problem for such a system is then said to be unsolvable, and the system is said to be an undecidable one.
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