Argument

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JHY

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Is the following argument valid and sound ?
Is premise 2 true or false ? Is the conclusion true or false ?

Screenshot (1305).png
 
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I do not like the question.

This is a valid argument

Major premise: If 5x = 20, then x = 4.
Minor premise: 5x = 20.
Conclusion: x = 4.

If both premises are true, the conclusion is true. Modus ponens.

This is also a valid argument.

Major premise: If 5x = 20, then x = 4.
Minor premise: x [imath]\ne[/imath] 4.
Conclusion: 5x [imath]\ne[/imath] 20.

If both premises are true, the conclusion is true. Modus tollens.

The reason I do not like the question is that the major premise is obviously true. So it is easy to get mixed up about truth and validity. An invalid argument may be true, and a valid argument may be false.

Modus ponens

[math]p \implies q \text { and } p \implies q.[/math]
Modus tollens

[math]p \implies q \text { and not } q \implies \text {not } p.[/math]
Both are logically valid. Whether they are true depends on the truth of the premises.
 
Dr.Peterson said:
The argument (form) is valid; the argument (as a whole) is sound only if the premises are both true:

Soundness - Wikipedia

en.wikipedia.org en.wikipedia.org

What are your thoughts about whether the second premise is true?
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My thought is that : The second premise can be true, since these is no rule saying that x must be a particular value. Am I right ?
 
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JHY said:
The second premise can be true, since these is no rule saying that x must be a particular value. Am I right ?
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It CAN be true; but that is not the same as IS.

What have you learned about variables in logical arguments? (We have no idea yet what course you are taking, so you need to tell us. I'd give a very different answer according to how formal a type of logic you are learning.)
 
Dr.Peterson said:
It CAN be true; but that is not the same as IS.

What have you learned about variables in logical arguments? (We have no idea yet what course you are taking, so you need to tell us. I'd give a very different answer according to how formal a type of logic you are learning.)
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I have learned just the very basic : 【 The three forms of argument ( Form I : All A are B --> C is A --> C is B, Form II : If p then q --> p is true --> q is true, Form III : If p then q --> Not q is true --> Not p is true ) and the definition of soundness and validity of an argument ]
I think variables in logical argument is not in that syllabus, I think I have no idea about it.
 
JHY said:
I have learned just the very basic : 【 The three forms of argument ( Form I : All A are B --> C is A --> C is B, Form II : If p then q --> p is true --> q is true, Form III : If p then q --> Not q is true --> Not p is true ) and the definition of soundness and validity of an argument ]
I think variables in logical argument is not in that syllabus, I think I have no idea about it.
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You clearly haven't been taught symbolic logic (since if you had, you wouldn't use arrows to separate parts of an argument), or traditional names for arguments (modus ponens, and so on). In fact, the first two forms are sort of equivalent, but are different styles (categorical vs. propositional).

But if they are using variables in an example, I'd expect them to have given you at least some terminology for a statement that is neither true nor false, but conditionally true. I'd want to look at what you were taught in order to find the terminology you should use. Is there any more context that you can show?

In any case, [imath]x\ne 4[/imath] has a truth value that depends on the value of x, so you can't call it true.
 
Dr.Peterson said:
You clearly haven't been taught symbolic logic (since if you had, you wouldn't use arrows to separate parts of an argument), or traditional names for arguments (modus ponens, and so on). In fact, the first two forms are sort of equivalent, but are different styles (categorical vs. propositional).

But if they are using variables in an example, I'd expect them to have given you at least some terminology for a statement that is neither true nor false, but conditionally true. I'd want to look at what you were taught in order to find the terminology you should use. Is there any more context that you can show?

In any case, [imath]x\ne 4[/imath] has a truth value that depends on the value of x, so you can't call it true.
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All I have learned about statement is that
1. " Statement is a sentence that is either true or false but not both simultaneously "
2. " command and question are not statement because they are neither true nor false. Therefore they are not statements"
3. " Sentence such as x + 2 = 5 can either be true or false depending on the value of x. Therefore it is not a statement. "
 
JHY said:
All I have learned about statement is that
1. " Statement is a sentence that is either true or false but not both simultaneously "
2. " command and question are not statement because they are neither true nor false. Therefore they are not statements"
3. " Sentence such as x + 2 = 5 can either be true or false depending on the value of x. Therefore it is not a statement. "
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You are "quoting" above - Please provide the reference . Is that from your textbook ? What is the name of the book? Author ? Publisher?
 
JHY said:
3. " Sentence such as x + 2 = 5 can either be true or false depending on the value of x. Therefore it is not a statement. "
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This is one of the things I was expecting, as it is commonly taught at the level I think your course is (such as a Survey of Mathematics course I have taught). The book I used says, "A sentence that can be judged either true of false is called a statement." The authors carefully avoid using anything like [imath]x\ne 2[/imath] in any arguments.

What confused me is that if your book does use such a "statement", but also says that it is not a statement (as yours explicitly does!), then the question is nonsense. You can't use a non-statement as a premise in an argument!

In reality, however, we can use variables in a statement; [imath]x\ne 2[/imath] does have a truth value, but only for specific values of the variable. This is how logic is often used in practice; for instance, we can say "if [imath]x=2[/imath] then [imath]x+1=3[/imath]". This is a true statement!

A Discrete Math book I have says the same sort of thing your book says, but in more detail:

A statement is a sentence that is true or false but not both. ... The truth or falsity of "He is a college student" depends on the reference for the pronoun he. For some values of he the sentence is true; for others it is false. If the sentence were preceded by other sentences that made the pronoun's reference clear, then the sentence would be a statement. Considered on its own, however, the sentence is neither true nor false, and so it is not a statement.​

They give a similar example with variables. And they definitely use such statements later in the book; but they are (generally) careful to first say, "x and y are particular real numbers" so that their statements can be called statements.

But the answer to your question is still what I said: not knowing the value of x, premise 2 may be either true or false, and therefore the conclusion may be either true or false, though the argument is valid. If I were helping you in person, I would be looking through your book to see if they said, for example, that x refers to a specific value, or if they give other examples using variables. As it appears only from what you have quoted, they are inconsistent, and therefore you are naturally confused.
 
Dr.Peterson said:
This is one of the things I was expecting, as it is commonly taught at the level I think your course is (such as a Survey of Mathematics course I have taught). The book I used says, "A sentence that can be judged either true of false is called a statement." The authors carefully avoid using anything like [imath]x\ne 2[/imath] in any arguments.

What confused me is that if your book does use such a "statement", but also says that it is not a statement (as yours explicitly does!), then the question is nonsense. You can't use a non-statement as a premise in an argument!

In reality, however, we can use variables in a statement; [imath]x\ne 2[/imath] does have a truth value, but only for specific values of the variable. This is how logic is often used in practice; for instance, we can say "if [imath]x=2[/imath] then [imath]x+1=3[/imath]". This is a true statement!

A Discrete Math book I have says the same sort of thing your book says, but in more detail:

A statement is a sentence that is true or false but not both. ... The truth or falsity of "He is a college student" depends on the reference for the pronoun he. For some values of he the sentence is true; for others it is false. If the sentence were preceded by other sentences that made the pronoun's reference clear, then the sentence would be a statement. Considered on its own, however, the sentence is neither true nor false, and so it is not a statement.​

They give a similar example with variables. And they definitely use such statements later in the book; but they are (generally) careful to first say, "x and y are particular real numbers" so that their statements can be called statements.

But the answer to your question is still what I said: not knowing the value of x, premise 2 may be either true or false, and therefore the conclusion may be either true or false, though the argument is valid. If I were helping you in person, I would be looking through your book to see if they said, for example, that x refers to a specific value, or if they give other examples using variables. As it appears only from what you have quoted, they are inconsistent, and therefore you are naturally confused.
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Thank you very much for explaining. Now I understand the problem. It clears my doubt.
 
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