What is first-order logic examples?

What is first-order logic examples?

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))).

How do you represent first-order logic?

First-Order logic:

1. Objects: A, B, people, numbers, colors, wars, theories, squares, pits, wumpus….
2. Relations: It can be unary relation such as: red, round, is adjacent, or n-any relation such as: the sister of, brother of, has color, comes between.
3. Function: Father of, best friend, third inning of, end of….

What is a valid formula of first-order logic?

Every first-order formula is equivalent to a NNF formula. It can be computer by extending the propositional NNF normalisation with specific laws to handle quantifiers. Example: to compute the NNF of ∀x. (∀y.P(x,y) ∨ Q(x)) → ∃z.P(x,z).

What are the kind of symbols used in first-order logic?

The syntax of first-order logic is defined relative to a signature. A signature σ consists of a set of constant symbols, a set of function symbols and a set of predicate symbols. Each function and predicate symbol has an arity k > 0. We will often refer to predicates as relations.

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.

Which is an example of a valid formula?

Finally, an example of a valid formula is p ∨ ¬p. A valid formula, often also called a theorem, corresponds to a correct logical argument, an argument that is true regardless of the values of its atoms. For example p ⇒ p is valid. No matter what p is, p ⇒ p always holds.

What is a satisfiable formula?

A formula is satisfiable if there exists an interpretation (model) that makes the formula true. A formula is valid if all interpretations make the formula true. In general, the question whether a sentence of first-order logic is satisfiable is not decidable.

Is first-order logic complete?

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.

Is truth functional logic decidable?

A logical system is decidable if there is an effective method for determining whether arbitrary formulas are theorems of the logical system. For example, propositional logic is decidable, because the truth-table method can be used to determine whether an arbitrary propositional formula is logically valid.

What is the difference between first 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.

What is valid formula?

A valid formula, often also called a theorem, corresponds to a correct logical argument, an argument that is true regardless of the values of its atoms. For example p ⇒ p is valid. No matter what p is, p ⇒ p always holds.

Which is the formal representation of first order logic?

First-order logic is also called Predicate logic and First-order predicate calculus (FOPL). It is a formal representation of logic in the form of quantifiers. In predicate logic, the input is taken as an entity, and the output it gives is either true or false.

When to use a constant in first order logic?

If (Ax)P(x)is true, then P(c)is true, where cis a constant in the domain of x. For example, from (Ax)eats(Ziggy, x)we can infer eats(Ziggy, IceCream). The variable symbol can be replaced by any ground term, i.e., any constant symbol or function symbol applied to ground terms only.

How are sentences built up in first order logic?

Someone is liked by everyone: (Ey)(Ax)likes(x,y) Sentences are built up from terms and atoms: A term(denoting a real-world individual) is a constant symbol, a variable symbol, or an n-place function of n terms.

Which is an example of disjunction in first order logic?

E.g., (Ax) dolphin(x) => mammal(x) Existential quantification corresponds to disjunction (“or”) in that (Ex)P(x)means that Pholds for some value of xin the domain associated with that variable.