# How is First Order Logic complete but not decidable?

First-order logic is complete because all entailed statements are provable, but is undecidable because there is no algorithm for deciding whether a given sentence is or is not logically entailed.

## Is first-order logic decidable?

First-order logic is not decidable in general; in particular, the set of logical validities in any signature that includes equality and at least one other predicate with two or more arguments is not decidable. Logical systems extending first-order logic, such as second-order logic and type theory, are also undecidable.

## Is first order satisfiability decidable?

For first-order logic (FOL), satisfiability is undecidable. More specifically, it is a co-RE-complete problem and therefore not semidecidable.

## Is first-order logic semi decidable?

Theorem 1. Validity of first-order formulas is semi-decidable. Proof. Recall that a semi-decision procedure for validity should halt and return “valid” when given a valid formula as input, but otherwise may compute forever.

## Is first-order logic sound and complete?

There are many deductive systems for first-order logic which are both sound (i.e., all provable statements are true in all models) and complete (i.e. all statements which are true in all models are provable).

## Why is first-order logic complete?

First-order logic is complete because all entailed statements are provable, but is undecidable because there is no algorithm for deciding whether a given sentence is or is not logically entailed.

## What is the difference between completeness and Decidability?

Decidable A theory T is decidable if there exists an effective procedure to determine whether T⊢φ where φ is any sentence of the language. Completeness A theory T is syntactically complete if for every sentence of the language φ it is true that T⊢φ or T⊢¬φ.

## Is first-order logic incomplete?

First order arithmetic is incomplete. Except that it’s also complete. 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.

## What is completeness logic?

completeness, Concept of the adequacy of a formal system that is employed both in proof theory and in model theory (see logic). In proof theory, a formal system is said to be syntactically complete if and only if every closed sentence in the system is such that either it or its negation is provable in the system.

## Does completeness imply soundness?

Soundness means that you cannot prove anything that’s wrong. Completeness means that you can prove anything that’s right. In both cases, we are talking about a some fixed system of rules for proof (the one used to define the relation ⊢ ).

## Why is predicate logic not Decidable?

It’s hard to say what the “cause” is – mathematical phenomena have proofs, not causes. But the key reason for the undecidability is that predicate logic is too powerful; it’s powerful enough to describe the algorithm you might try to use, so it can circumvent it.

## 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. 3) Thus, by Lemma One, V is not a subset of T.

## Is predicate 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).

## Who determined through their completeness theorem that first-order logic is complete?

Kurt G๖del

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.

## Is second order logic complete?

Several deductive systems can be used for second-order logic, although none can be complete for the standard semantics (see below). Each of these systems is sound, which means any sentence they can be used to prove is logically valid in the appropriate semantics.

## Is classical logic complete?

On Wikipedia this property is also called syntactical completeness. Propositional classical logic is Post-complete. First-order classical logic and propositional intuitionistic logic are not Post-complete. For some references, you can have a look here and here (and at their bibliography).

## Who is the father of classical logic?

Aristotle is a towering figure in ancient Greek philosophy, who made important contributions to logic, criticism, rhetoric, physics, biology, psychology, mathematics, metaphysics, ethics, and politics. He was a student of Plato for twenty years but is famous for rejecting Plato’s theory of forms.

## Who is the father of traditional logic?

Aristotle

philosopher Immanuel Kant called Aristotle, the ancient Greek philosopher, the “father of logic.” If we are thinking only of traditional, or formal, logic (which is the only kind of logic we study in this book), this is true.