Predicate logic

What is predicate logic?

Predicate logic is a mathematical model that is used for reasoning with predicates. Predicates are functions that map variables to truth values. They are essentially boolean functions whose value could be true or false, depending on the arguments to the predicate. They are generalizations of propositional variables. A propositional variable is a predicate with no arguments. 

It involves making use of standard forms of logical symbolism that philosophers and mathematicians have been rather familiar with for several decades. Predicate logic is also used in ai.

Most sentences like “Jack is miserly” or “Jonathan passes a plate to Jacob” could be represented in terms of logical formulae where a predicate gets applied to one or multiple arguments (which, when used in predicated logic is similar to, but not identical to, its use to refer to the inputs to a procedure in POP-11).

Even though in its current form predicate logic started getting developed in its current form only in the twentieth century, its roots go back all the way to Aristotle. There are several techniques associated with it that are employed for the analysis of many conceptual structures in our common thought. Since many people understand these analytical techniques rather well, and due to the fact that expressing the formulae of predicate logic in AI languages like LISP or POP-11 is quite easy, predicate logic is a very widely used knowledge representation symbolism within the domain of AI. Predicate logic in AI is mainly used as a foundational framework for representation of knowledge and reasoning.

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Predicate logic in AI even embodies a set of systematic procedures that can prove that specific formulae can or can’t be logically derived from others. These logical inference procedures were pretty much used as the backbone for problem-solving systems in artificial intelligence.

On its own, predicate logic is a rather formal type of representation mechanism. However, there are proponents of predicate logic who believe that it can be used to invent conceptual tools which reproduce much of the subtlety and nuance of ordinary informal thinking.

One popular way of incorporating predicate logic in AI programs involved making use of a machine-based inference procedure known as resolution. This procedure was initially proposed by J. A. Robinson in 1965. It made it rather easy to represent specialized as well as common knowledge by using a set of axioms that are expressed in a special form of predicate calculus formulae and then deriving consequences from the axioms in question.

An artificial intelligence programming language known as Prolog (PROgramming in LOGic) has been developed. It makes use of a resolution inference mechanism along with a restricted form of predicate logic. Its supporters say that it is an effective tool for building knowledge-based systems. A large section of the British Nationalities Act was translated into the logical symbolism of Prolog which made it possible to infer whether a specific person can argue entitlement to a passport under the labyrinthine provisions of the Act.

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What is predicate logic example with solution?

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If we want to assert that all elements of some set (S) satisfy a common property (P).

For example, if S = {John, Jonas, Joe}, we might want to claim that all elements of S are comedians.

We could write;

(John is a comedian)^(Jonas is a comedian)^(Joe is a comedian)  

Alternatively, we can create a template: Comedian() 

Comedian(John)^Comedian(Jonas)^Comedian(Joe)  

Comedian() is called a predicate.

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What are logical expressions in predicate logic?

Logical expressions are also known as boolean expressions. They are the result of applying logical or boolean operators to relational or arithmetic expressions. Logical expressions can be logical variables (the simplest type of logical expression) or relations or they can even be complicated logical tests that involve variables, constants, functions, relational operators, logical operators, and parentheses to control the order of evaluation.

Logical expressions return 1 if the expression is true, 0 if the expression is false, or system-missing if it is missing. A logical expression could pretty much be any expression that returns this value. Any expressions that yield this three-value logic can be considered to be a logical expression.

A logical expression in predicate logic would practically have the same form as a logical expression in propositional logic, along with the addition of atomic formulae (ie., predicates), and the universal and existential quantifiers.

In a case where L1 and L2 are logical expressions, L1 AND L2, L1 OR L2, NOT L1, L1 -> L2, and L1 == L2 will be logical expressions.

In a situation where L1 is a logical expression, (A X) L1 will also be a logical expression. 

If L1 is a logical expression, then (E X) L1 will be a logical expression.

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How is predicate logic used?

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Predicate logic is used to help figure out whether or not a member of a certain set possesses a given property. 

It is used because it allows you to break down simple sentences into smaller parts: predicates and individuals. Predicate logic even makes it possible for you to handle expressions of generalization (quantificational expressions). 

Predicate logic enables you to talk about variables (pronouns). The value for the pronoun is an individual in the domain of the universe that is contextually determined.

It makes it possible for sentences with quantificational expressions to be divided into two interpretive components. 

A simple sentence with a variable or a pronoun (a placeholder) could be evaluated to be true or false with respect to an individual contextually taken as a value for the pronoun. 

The quantificational expression instructs you to limit the domain of individuals who are being considered to a relevant set. It shows you how many different values of the pronoun you need to consider from a specific domain in order to establish the truth for the sentence.

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What are the quantifiers used in predicate logic?

The variable of predicates is quantified and defined by quantifiers. There are two types of quantifiers in predicate logic − Universal Quantifier and Existential Quantifier.

Existential Quantifier:

If p(x) is a proposition over the universe U. Then it is denoted as ∃x p(x) and read as "There exists at least one value in the universe of variable x such that p(x) is true. The quantifier ∃ is called the existential quantifier.

There are several ways to write a proposition, with an existential quantifier, i.e.,

(∃x∈A)p(x) or ∃x∈A such that p (x) or (∃x)p(x) or p(x) is true for some x ∈A.

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Universal Quantifier:

If p(x) is a proposition over the universe U. Then it is denoted as ∀x,p(x) and read as "For every x∈U,p(x) is true." The quantifier ∀ is called the Universal Quantifier.

There are several ways to write a proposition, with a universal quantifier.

∀x∈A,p(x) or p(x), ∀x ∈A Or ∀x,p(x) or p(x) is true for all x ∈A.

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What is first-order predicate logic? And its example? 

First-order logic refers to the symbolized reasoning in which the sentences and statements are divided into a subject & a predicate in the predicate logic model. The predicate modifies or defines the properties of the subject in the logic statement. In first-order logic, a predicate can only refer to a single subject. First-order logic is also called first-order predicate calculus or first-order functional calculus.

A sentence in first-order logic is written in the below format: 

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Px or P(x);

Where P is the predicate; and

x as the subject represented as a variable.

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What is an atomic formula?

An atomic formula is a logical expression. A predicate that has all constant arguments is a ground atomic formula. A proposition would essentially be a predicate that does not have arguments. Because of that, it is a ground atomic formula by default. A predicate that has at least one variable argument would be a nonground atomic formula. A literal would be either an atomic formula or its negation.

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Is predicate logic better than propositional logic?

Predicate logic does happen to be more powerful than propositional logic. One major limitation of propositional logic is that propositional logic applies solely to atomic propositions. It is not possible to talk about properties that apply to categories of objects, or about relationships between those properties.