First order logic is complete , which means (I think) given a set of sentences A and a sentence B, then either B or ~B can be arrived at through the rules of inference being applied to A. If B is arrived at, then A implies B in every interpretation. ... So FOL is decidable.
Who proved the completeness of first-order logic?
This result, known as the Completeness Theorem for first-order logic, was proved by Kurt G๖del in 1929. According to the Completeness Theorem provability and semantic truth are indeed two very different aspects of the same phenomena. In order to prove the Completeness Theorem, we first need a formal notion of proof.
Is first-order logic incomplete?
First order arithmetic is incomplete . ... Second order arithmetic is more expressive – except when it’s not – and is also incomplete and also complete, except when it means something different. Oh, and full second order-logic might not really be a logic at all.
Is there second order logic?
In logic and mathematics second-order logic is an extension of first-order logic , which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.
Is first-order logic complete and consistent?
According to Wikipedia, first order logic is complete . What is the proof of this? (Also, in the same paragraph, it says that its undecidable. Couldn’t you just enumerate all possible proofs and disproofs to decide it though?)
What is the difference between first-order and second-order logic?
First-order logic uses only variables that range over individuals (elements of the domain of discourse); second-order logic has these variables as well as additional variables that range over sets of individuals.
Is propositional logic complete?
Truth-functional propositional logic and first-order predicate logic are semantically complete , but not syntactically complete (for example, the propositional logic statement consisting of a single propositional variable A is not a theorem, and neither is its negation).
Why is second-order logic incomplete?
Theorem: 2nd order logic is incomplete: 1) The set T of theorems of 2nd order logic is effectively enumerable . 2) The set V of valid sentences of 2nd order logic is not effectively enumerable. ... 4) So there must be a valid sentence of 2nd order logic that is not a theorem of 2nd order logic.
Is second-order logic decidable?
Logical systems extending first-order logic, such as second-order logic and type theory, are also undecidable . The validities of monadic predicate calculus with identity are decidable, however.
What are first and second-order questions?
First-order questions or claims are within a discipline or AOK . Analysis uses the methods of the discipline or AOK. ■ Second-order questions or claims are about the discipline or AOK (its methods for constructing knowledge).
Is First-Order Logic useful?
First-order logic can be useful in the creation of computer programs . It is also of interest to researchers in artificial intelligence ( AI ). There are more powerful forms of logic, but first-order logic is adequate for most everyday reasoning.
What is First-Order Logic with example?
Definition A first-order predicate logic sentence G over S is a tautology if F |= G holds for every S-structure F. Examples of tautologies (a) ∀x.P(x) → ∃x.P(x); (b) ∀x.P(x) → P(c); (c) P(c) → ∃x.P(x); (d) ∀x(P (x) ↔ ¬¬P (x)); (e) ∀x(¬(P1(x) ∧ P2(x)) ↔ (¬P1(x) ∨ ¬P2(x))).
What is the point of First-Order Logic?
First-order logic is also known as Predicate logic or First-order predicate logic. First-order logic is a powerful language that develops information about the objects in a more easy way and can also express the relationship between those objects .
What is the difference between first order logic and propositional logic?
Propositional Logic converts a complete sentence into a symbol and makes it logical whereas in First-Order Logic relation of a particular sentence will be made that involves relations, constants, functions, and constants .
What does Second Order mean in math?
Second-order arithmetic, an axiomatization allowing quantification of sets of numbers . Second-order differential equation, a differential equation in which the highest derivative is the second. Second-order logic, an extension of predicate logic.
What is first order and second order?
A first-order reaction rate depends on the concentration of one of the reactants. A second-order reaction rate is proportional to the square of the concentration of a reactant or the product of the concentration of two reactants.
