# Interpreting a statement in first order logic

• Mr Davis 97
In summary, if ##a## and ##b## are real numbers with ##a \ne 0##, then there exists a solution to ##ax+b=0##.
Mr Davis 97

## Homework Statement

Rewrite the following statements in symbolic form:

a) If ##a## and ##b## are real numbers with ##a \ne 0##, then ##ax+b=0## has a solution.
b) If ##a## and ##b## are real numbers with ##a \ne 0##, then ##ax+b=0## has a unique solution.

## The Attempt at a Solution

Attempts at solution:

Let ##P(x,a,b)## be the statement that ##ax+b=0## is true.
a) ##\forall a \in \mathbb{R} - \{0\} \forall b \in \mathbb{R} \exists x \in \mathbb{R} P(x,a,b)##
b) ##\forall a \in \mathbb{R} - \{0\} \forall b \in \mathbb{R} \exists x \in \mathbb{R} (P(x,a,b) \wedge \forall y (P(y,a,b) \implies y=x))##

Is that at all right? Is there an easier way? It all seems very cumbersome.

Mr Davis 97 said:

## Homework Statement

Rewrite the following statements in symbolic form:

a) If ##a## and ##b## are real numbers with ##a \ne 0##, then ##ax+b=0## has a solution.
b) If ##a## and ##b## are real numbers with ##a \ne 0##, then ##ax+b=0## has a unique solution.

## The Attempt at a Solution

Attempts at solution:

Let ##P(x,a,b)## be the statement that ##ax+b=0## is true.
a) ##\forall a \in \mathbb{R} - \{0\} \forall b \in \mathbb{R} \exists x \in \mathbb{R} P(x,a,b)##
b) ##\forall a \in \mathbb{R} - \{0\} \forall b \in \mathbb{R} \exists x \in \mathbb{R} (P(x,a,b) \wedge \forall y (P(y,a,b) \implies y=x))##

Is that at all right? Is there an easier way? It all seems very cumbersome.
Looks good to me, although I'm no logic expert, and I would add an ##\in \mathbb{R}## to the ##y##.
To make it less cumbersome a professor of mine used ##\exists !## to express uniqueness, but this isn't an official symbol. Some even used a symbol for "without loss of generality".

fresh_42 said:
Looks good to me, although I'm no logic expert, and I would add an ##\in \mathbb{R}## to the ##y##.
To make it less cumbersome a professor of mine used ##\exists !## to express uniqueness, but this isn't an official symbol. Some even used a symbol for "without loss of generality".

Didn't know that this isn't an official symbol. I use it all the time.

Math_QED said:
Didn't know that this isn't an official symbol. I use it all the time.
I don't know for certain, I only checked it in Wikipedia. But it makes sense, as it combines two statements in one. It is convenient, but not pure in a logical sense.

## 1. What is first order logic and why is it important in interpreting statements?

First order logic, also known as predicate logic, is a formal language used to represent and reason about relationships, properties, and objects. It is important in interpreting statements because it provides a precise and unambiguous way to express complex ideas and arguments, making it a powerful tool for formalizing knowledge and reasoning.

## 2. How do you interpret a statement in first order logic?

Interpreting a statement in first order logic involves breaking it down into its logical components, such as quantifiers, variables, and predicates, and defining their meanings in a specific domain. The statement is then evaluated based on the defined meanings to determine its truth value.

## 3. What are the common symbols and notations used in first order logic?

The most common symbols and notations used in first order logic are quantifiers (∀ and ∃), logical connectives (¬, ∧, ∨, →, and ↔), variables (x, y, z), predicates (P, Q, R), and equality (=). These symbols allow us to express complex logical statements in a concise and consistent manner.

## 4. What are the limitations of first order logic?

First order logic is limited in its ability to handle certain types of knowledge, such as uncertainty and vagueness. It also struggles with handling self-reference and expressing some types of common sense knowledge. Additionally, it is not capable of reasoning about its own consistency.

## 5. How is first order logic used in artificial intelligence and computer science?

First order logic is used in artificial intelligence and computer science as a formal language for knowledge representation and automated reasoning. It serves as the foundation for many logical and probabilistic reasoning systems, including expert systems, natural language processing, and automated theorem proving.

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