arguments

[chrislav]

The definition of a valid argument is the following:
the conclusion of an argument is true whenever the premises of the argument is true:
Is the following argument valid?
premise (1) : if 3+4=8,then 5+3=2
premise (2) 3+4 is not 8

Concusion: 5+3 is not 2

You can consider the question as a challenging question

 

[AvgStudent]

I'll give this a try
Since [imath]3+4 \neq 8[/imath], then you start out with a false assumption. Anything derived from a false assumption can be either true or false.
Therefore, the conclusion is that [imath]5+3[/imath] may or may not equal 2.

 

[pka]

The definition of a valid argument is the following:
the conclusion of an argument is true whenever the premises of the argument is true:
Is the following argument valid?
premise (1) : if 3+4=8,then 5+3=2
premise (2) 3+4 is not 8

Concusion: 5+3 is not 2

You can consider the question as a challenging question

One of the most difficult distinctions to teach beginning students is that between truth/falsity & valid/invalid.
This page can help.

 

[Steven G]

You are given that premise (2) 3+4 is not 8
Now premise (1) : if 3+4=8,then 5+3=2 is the same as if 5+3[imath]\neq[/imath]2, 3+4 [imath]\neq [/imath]8
Continue from here.

 

[Steven G]

I'll give this a try
Since [imath]3+4 \neq 8[/imath], then you start out with a false assumption. Anything derived from a false assumption can be either true or false.
Therefore, the conclusion is that [imath]5+3[/imath] may or may not equal 2.

Where is the false starting assumption??
You are GIVEN that [imath]3+4 \neq 8[/imath]. 1st of all, it is mathematically a true statement but does that matter?

 

[Dr.Peterson]

The definition of a valid argument is the following:
the conclusion of an argument is true whenever the premises of the argument is true:
Is the following argument valid?
premise (1) : if 3+4=8,then 5+3=2
premise (2) 3+4 is not 8

Concusion: 5+3 is not 2

You can consider the question as a challenging question

You need to consider what "valid" means. You have given a correct definition.

Validity is only about the form of the argument, not its content (the truth or falsity of the statements within it). An argument can be invalid even if all its premises and its conclusion are true, as in this case.

Your argument has the form "If p, then q; not p; therefore not q." So you have to decide whether this is a valid argument, by considering whether it is possible for both premises to be true but the conclusion false. (The answer is that it is invalid; specifically, it is the fallacy of the inverse.)

I believe you have shown in the past that you know something about logic; so this should not be hard for you. Perhaps what you mean by "challenging question" is not "a question that is hard for me, for which I need help", but "a challenge question that I think you will get wrong." In either case, please tell us your thoughts about it.

 

[AvgStudent]

Where is the false starting assumption??
You are GIVEN that [imath]3+4 \neq 8[/imath]. 1st of all, it is mathematically a true statement but does that matter?

The false assumption I'm referring to is in premise (1).
"If 3+4=8, then ...", which is mathematically false, what follows may be true or false.

 

[Steven G]

The false assumption I'm referring to is in premise (1).
"If 3+4=8, then ...", which is mathematically false, what follows may be true or false.

Mathematically true or false has nothing to do with the premise being true or not.

 

[Steven G]

If I say if you study then you'll pass. That is not necessarily a true statement. You can study all you want and I bet that you'll fail a test on the understanding of Dr Wiles proof on Fermat's Last Theorem.

 

[AvgStudent]

Mathematically true or false has nothing to do with the premise being true or not.

I haven't taken a logic course so I'm not too familiar with the formal definitions.
In response to what you said, then how do you determine whether a premise is true or false?

 

[Dr.Peterson]

I haven't taken a logic course so I'm not too familiar with the formal definitions.
In response to what you said, then how do you determine whether a premise is true or false?

There are several points you need to be aware of.

First, the (first) premise of the argument is not "3+4=8", but the conditional statement "if 3+4=8,then 5+3=2".

Second, this statement is considered true because when the hypothesis of a conditional statement is false, the entire statement is considered true.

Third, as I said in post #6, the validity of the argument (which is what the question is about) doesn't depend on the truthfulness of the statements, but only on the form of the argument; a valid argument must lead to a true conclusion regardless of the truthfulness of any of its premises or conclusion.

 

[JeffM]

@AvgStudent

First, what Steve said in comment 9 confuses me as well. The modern view of mathematics has nothing to do with empirical “truth.” It seems to me that the comment in # 9 is using “truth” in two different senses. I’d start with pka’s distinction between logical validity and empirical truth.

Mickey Mouse is President of the United State as of April, 2022.
Minnie Mouse is legally married to Mickey Mouse as of April, 2022.
Therefore, Minnie Mouse is First Lady of the United States as of April, 2022.

The argument above is logically valid although neither premise is empirically true.

Mickey Mouse is not legally eligible to be president of the United States.
Minnie Mouse is not alive.
Therefore, Boris Johnson is Prime Minister of the United Kingdom as of April, 2022

The argument above is logically invalid although the premises and conclusion are empirically true.

Second, the use of the word “truth value” in logic is confusing as well given that logic is about validity rather than empirical truth. Moreover, logicians give a “truth value” of T to a statement of the form [imath]p \implies q[/imath] if p is assigned the “truth value” of F.

So, if you are using “is true“ to mean “has a truth value of T,“
then the statement “if 3 + 4 = 8, then 5 + 3 = 2” is true.

The point about “truth values” is that they are a binary categorization into T and F. This creates a psychological conundrum.

The statement ”I am going to Atlanta next week if I am then able to” is NOT a lie. Empirically, it is neither true nor false. Its empirical status is not determinable until next week. And no one would classify it as a false assertion. The binary logician says therefore that such conditional statements go into category T rather than category F. Consequently, the statement
”I am flying to Australia next week if I am then a kangaroo” is not a lie. It is nonsense, but it goes into category T. Imagine trying to get a conviction for perjury against me if I made my trip to Australia contingent on my first transmogrifying into a kangaroo.

 

[AvgStudent]

@AvgStudent

First, what Steve said in comment 9 confuses me as well. The modern view of mathematics has nothing to do with empirical “truth.” It seems to me that the comment in # 9 is using “truth” in two different senses. I’d start with pka’s distinction between logical validity and empirical truth.

Mickey Mouse is President of the United State as of April, 2022.
Minnie Mouse is legally married to Mickey Mouse as of April, 2022.
Therefore, Minnie Mouse is First Lady of the United States as of April, 2022.

The argument above is logically valid although neither premise is empirically true.

Mickey Mouse is not legally eligible to be president of the United States.
Minnie Mouse is not alive.
Therefore, Boris Johnson is Prime Minister of the United Kingdom as of April, 2022

The argument above is logically invalid although the premises and conclusion are empirically true.

Second, the use of the word “truth value” in logic is confusing as well given that logic is about validity rather than empirical truth. Moreover, logicians give a “truth value” of T to a statement of the form [imath]p \implies q[/imath] if p is assigned the “truth value” of F.

So, if you are using “is true“ to mean “has a truth value of T,“
then the statement “if 3 + 4 = 8, then 5 + 3 = 2” is true.

The point about “truth values” is that they are a binary categorization into T and F. This creates a psychological conundrum.

The statement ”I am going to Atlanta next week if I am then able to” is NOT a lie. Empirically, it is neither true nor false. Its empirical status is not determinable until next week. And no one would classify it as a false assertion. The binary logician says therefore that such conditional statements go into category T rather than category F. Consequently, the statement
”I am flying to Australia next week if I am then a kangaroo” is not a lie. It is nonsense, but it goes into category T. Imagine trying to get a conviction for perjury against me if I made my trip to Australia contingent on my first transmogrifying into a kangaroo.

Thank you, @JeffM and @Dr.Peterson, for the insights.
This was a great explanation and showed just how little I know about logic. I'm definitely intrigued to learn more about logic and arguments.

 

[JeffM]

I realize that I may have been a bit careless in my description of conditional statements.

There are four possible binary categorizations of p and q.

[math]p \in \mathbb T \text { and } q \in \mathbb T \text { MEANS } p \implies q \in \mathbb T.\\ p \in \mathbb T \text { and } q \in \mathbb F \text { MEANS } p \implies q \in \mathbb F.\\ p \in \mathbb F \text { and } q \in \mathbb T \text { MEANS } p \implies q \in \mathbb T.\\ p \in \mathbb F \text { and } q \in \mathbb T \text { MEANS } p \implies q \in \mathbb T.\\[/math]
If I give a condition and that condition is not met, I have not lied.

 

[chrislav]

Take the following argmumentcounter example)
Premise(1): lf 4+3=6,then 5+6=11
Premise(2): But 4+3 is not 6
Conclusion: 5+6 is not 11
And according to the definion of the validity of an argument the initial argument in not valid
Here we have true premises and false conclusion thus violating the definion of a valid argument
This is a counter example for the definition of the validity of the argument

 

[The Highlander]

The definition of a valid argument is the following:
the conclusion of an argument is true whenever the premises of the argument is true:

Surely it would be perfectly possible to draw an invalid conclusion from true premises?
Ergo....

 

[lex]

This is a counter example for the definition of the validity of the argument

Valid argument [imath]\Leftrightarrow[/imath] (True premises [imath]\Rightarrow[/imath] True Conclusion)
This is equivalent to:
Invalid argument [imath]\Leftrightarrow[/imath] Not(True premises [imath]\Rightarrow[/imath] True Conclusion)

You have given an example of an invalid argument where Not(True premises [imath]\Rightarrow[/imath] True Conclusion)
which is in keeping with the second form of the definition above.

 

[JeffM]

But

[math]p \implies q\\ \neg p\\ \therefore \neg q[/math]
is not a valid argument. Thus your counter example fails.

 

[chrislav]

Jeefm
If you put p=4+3=6
q=5+6=11
And not p= 4+3 is not 6
Hence not q=5+6 is not 11
You have my counter example
Thus my counter example does not fail in proving that the OP's argument is not valid

 

[Dr.Peterson]

Take the following argument: (counter example)
Premise(1): lf 4+3=6,then 5+6=11
Premise(2): But 4+3 is not 6
Conclusion: 5+6 is not 11
And according to the definition of the validity of an argument the initial argument is not valid
Here we have true premises and false conclusion thus violating the definion of a valid argument
This is a counter example for the definition of the validity of the argument

If I understand you correctly, you are demonstrating that the argument in the OP is invalid (as I said in #6, it's the fallacy of the inverse), by showing an instance of the same argument in which the premises are true but the conclusion is false.

Premise 1 is true because its condition is false; premise 2 is true; but the conclusion is false.

I agree.

I suspect people are thinking you are disagreeing with them, when you are not.