7. define logic. why is logic important?
8. define premise
9. what is the law of syllogism? give example
10. what is the law of contrapositive? give example
11. what is the law of detachment give example

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Accepted Answer

### Question 7
# Brief Explanations:
- **Define Logic**: Logic is the study of reasoning, specifically the principles and methods of valid inference and argumentation. It deals with distinguishing correct (valid) reasoning from incorrect (invalid) reasoning.
- **Importance of Logic**:
    - **Critical Thinking**: It helps in developing critical thinking skills, enabling individuals to analyze arguments, identify fallacies, and make sound judgments.
    - **Problem - Solving**: In various fields like mathematics, science, and philosophy, logical reasoning is essential for formulating hypotheses, proving theorems, and solving complex problems.
    - **Communication**: It enhances the clarity and coherence of communication by providing a framework for constructing valid arguments and understanding the arguments of others.
    - **Decision - Making**: Assists in making rational decisions by evaluating different options based on logical principles.
# Answer:
- **Definition of Logic**: Logic is the study of principles of valid reasoning, inference, and argumentation, focusing on distinguishing valid from invalid reasoning.
- **Importance of Logic**: It fosters critical thinking, aids in problem - solving across disciplines, improves communication by enabling clear argumentation, and supports rational decision - making.

### Question 8
# Brief Explanations:
In logic and argumentation, a premise is a statement or proposition that serves as the basis or starting point of an argument. It is a claim that is put forward to provide support or evidence for a conclusion. For example, in the argument "All men are mortal (premise 1), Socrates is a man (premise 2), therefore Socrates is mortal (conclusion)", the two statements "All men are mortal" and "Socrates is a man" are premises.
# Answer:
A premise is a statement or proposition in an argument that provides support or evidence for the conclusion of that argument.

### Question 9
# Brief Explanations:
- **Law of Syllogism**: The Law of Syllogism is a rule of deductive reasoning. If we have two conditional statements, where the conclusion of the first conditional is the hypothesis of the second conditional, then we can derive a new conditional statement. In symbolic form, if we have $p
ightarrow q$ (if $p$, then $q$) and $q
ightarrow r$ (if $q$, then $r$), then we can conclude that $p
ightarrow r$ (if $p$, then $r$).
- **Example**:
    - Let $p =$ "It is raining", $q=$ "The ground is wet", and $r=$ "The grass will grow".
    - The first conditional: If it is raining ($p$), then the ground is wet ($q$) ($p
ightarrow q$).
    - The second conditional: If the ground is wet ($q$), then the grass will grow ($r$) ($q
ightarrow r$).
    - By the Law of Syllogism, we can conclude: If it is raining ($p$), then the grass will grow ($r$) ($p
ightarrow r$).
# Answer:
- **Law of Syllogism**: If we have two conditional statements $p
ightarrow q$ and $q
ightarrow r$, we can infer the conditional statement $p
ightarrow r$.
- **Example**: Let $p =$ "A figure is a square", $q=$ "A figure has four equal sides", $r=$ "A figure is a rhombus". The first conditional: If a figure is a square ($p$), then it has four equal sides ($q$) ($p
ightarrow q$). The second conditional: If a figure has four equal sides ($q$), then it is a rhombus ($r$) ($q
ightarrow r$). By the Law of Syllogism, we conclude: If a figure is a square ($p$), then it is a rhombus ($r$) ($p
ightarrow r$).

### Question 10
# Brief Explanations:
- **Law of Contrapositive**: The Law of Contrapositive states that a conditional statement $p
ightarrow q$ is logically equivalent to its contrapositive $
eg q
ightarrow
eg p$. In other words, if the statement "If $p$, then $q$" is true, then the statement "If not $q$, then not $p$" is also true, and vice - versa.
- **Example**:
    - Let the conditional statement $p
ightarrow q$ be "If a number is divisible by 4 ($p$), then it is even ($q$)".
    - The contrapositive of this statement is $
eg q
ightarrow
eg p$, which is "If a number is not even ($
eg q$), then it is not divisible by 4 ($
eg p$)".
    - We know that the original statement is true (since any number divisible by 4 is even). The contrapositive is also true because if a number is odd (not even), it cannot be divisible by 4.
# Answer:
- **Law of Contrapositive**: A conditional statement $p
ightarrow q$ is logically equivalent to its contrapositive $
eg q
ightarrow
eg p$, meaning $p
ightarrow q$ and $
eg q
ightarrow
eg p$ have the same truth value.
- **Example**: Consider the conditional "If a shape is a square ($p$), then it has four right angles ($q$)". Its contrapositive is "If a shape does not have four right angles ($
eg q$), then it is not a square ($
eg p$)". Both the original statement and its contrapositive are true.

### Question 11
# Brief Explanations:
- **Law of Detachment (Modus Ponens)**: The Law of Detachment is a valid form of deductive reasoning. If we have a conditional statement $p
ightarrow q$ (if $p$, then $q$) and we know that $p$ is true (the hypothesis of the conditional is true), then we can conclude that $q$ is true (the conclusion of the conditional is true). In symbolic form, if $p
ightarrow q$ is true and $p$ is true, then $q$ is true.
- **Example**:
    - Let the conditional statement $p
ightarrow q$ be "If it is sunny ($p$), then I will go for a walk ($q$)".
    - Suppose we know that "It is sunny" ($p$ is true).
    - By the Law of Detachment, we can conclude that "I will go for a walk" ($q$ is true).
# Answer:
- **Law of Detachment**: If a conditional statement $p
ightarrow q$ is true and its hypothesis $p$ is true, then its conclusion $q$ is true (symbolically: if $p
ightarrow q$ and $p$ are true, then $q$ is true).
- **Example**: Let the conditional be "If a student studies hard ($p$), then he will pass the exam ($q$)". If we know that "A student studies hard" ($p$ is true), then by the Law of Detachment, we can conclude that "He will pass the exam" ($q$ is true).

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