A question about logical equivalence and bound variables within Predicate Logic

How do you identify free variables in predicate logic?

A variable is free in a formula if it occurs at least once in the formula without being introduced by one of the phrases “for some x” or “for all x.” Henceforth, a formula S in which x occurs as a free variable will be called “a condition…

How do you prove logical equivalence with quantifiers?

Statements involving predicates and quantifiers are logically equivalent if and only if they have the same truth value for every predicate substituted into these statements and for every domain of discourse used for the variables in the expressions. The notation S ≡ T indicates that S and T are logically equivalent.

What is predicate logic explain with example?

Predicate Logic – Definition

A predicate with variables can be made a proposition by either assigning a value to the variable or by quantifying the variable. The following are some examples of predicates − Let E(x, y) denote “x = y” Let X(a, b, c) denote “a + b + c = 0” Let M(x, y) denote “x is married to y”

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Why do we need predicate logic?

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

What is a bound variable in logic?

A bound variable, in contrast, is a variable that has been bound to a specific value or range of values in the domain of discourse or universe. This may be achieved through the use of logical quantifiers, variable-binding operators, or an explicit statement of allowed values for the variable (such as, “…

What is the difference between a free variable and a bound variable?

A free variable is a variable that has no limitations, while a bound variable, on the other hand, is a variable with limitations. To determine whether your variable is free or bound, use these two criteria. Bound variables have limitations; free variables don’t. Bound variables can be swapped; free variables can’t.

How do you do logical equivalence?

Two logical statements are logically equivalent if they always produce the same truth value. Consequently, p≡q is same as saying p⇔q is a tautology. Beside distributive and De Morgan’s laws, remember these two equivalences as well; they are very helpful when dealing with implications. p⇒q≡¯q⇒¯pandp⇒q≡¯p∨q.

Is it raining proposition or not?

logic! A proposition is a statement that is either true or false. Propositional logic talks about Boolean combinations of propositions and inferences we can make about them. E.g., If it is raining, then it is cloudy.

What is the difference between universal quantifier and existential quantifier?

The universal quantifier, meaning “for all”, “for every”, “for each”, etc. The existential quantifier, meaning “for some”, “there exists”, “there is one”, etc. A statement of the form: x, if P(x) then Q(x). A statement of the form: x such that, if P(x) then Q(x).

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What is the semantics of predicate logic?

The semantics of Predicate Logic does two things. It assigns a meaning to the individuals, predicates, and variables in the syntax. It also systematically determines the meaning of a proposition from the meaning of its constituent parts and the order in which those parts combine (Principle of Compositionality).

What is the difference between propositional logic and predicate logic?

Propositional logic is the logic that deals with a collection of declarative statements which have a truth value, true or false. Predicate logic is an expression consisting of variables with a specified domain. It consists of objects, relations and functions between the objects.

What is the syntax of predicate logic?

In Predicate Logic, each variable combines with and is bound by a single quantifier. Predicate Logic has two such quantifiers: ∀ (the universal quantifier) and ∃ (the existential quantifier).

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).

How do you represent simple facts in predicate logic?

Representing Simple Facts in Logic Consider the following set of sentences.

  1. Marcus was a man.
  2. Marcus was a Pompeian.
  3. All Pompeians were Romans.
  4. Caesar was a ruler.
  5. All Romans were either loyal to Caesar or hated him.
  6. Every one is loyal to someone.
  7. People only try to assassinate rulers they are not loyal to.

How do you say but in predicate logic?

When translating from English sentences into logical form, “but” generally means the same as “and”, and the phrase “neither A nor B” is translated as “not A and not B”.

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What propositions are logically equivalent?

The propositions are equal or logically equivalent if they always have the same truth value. That is, p and q are logically equivalent if p is true whenever q is true, and vice versa, and if p is false whenever q is false, and vice versa. If p and q are logically equivalent, we write p = q.

How many possible truth values does the proposition have?

two possible values

A proposition has only two possible values: it is either true or false. We often abbreviate these values as T and F, respectively. Given a proposition p, we form another proposition by changing its truth value.