Mating Mechanics Would Mean Less Incest Not More

Cogito · 2021-01-14 02:57:46

DestinyCall wrote:

If you have made it this far and feel like you have a strong grasp of deductive reasoning, why not take this logic quiz?
https://global.oup.com/us/companion.web … rue_false/
And if you are still thirsty for more logic, you can follow it up with quiz on validity and invalidity! Can you spot the invalid arguments?
https://global.oup.com/us/companion.web … d_invalid/
And if you smoked the previous two tests, try this even harder (but thankfully much shorter) logic test:
https://global.oup.com/us/companion.web … d_invalid/
Enjoy

These were fun! 100% on all

DestinyCall · 2021-01-14 07:10:25

You did better than me. I got tripped up by a couple of them so I missed one question on each test. Still, that's a lot better than I would have done, if I had taken these tests a week or two ago.

I haven't thought this much about formal logic since I was in high school. Not sure if I am a better debater now, but I do know some fancy logic terminology now and I might even be able to explain my arguments more clearly.

...

Out of curiosity, Cogito, what do you think of Spoonwood's conclusion regarding the validity of arguments with a false premise?

Spoonwood wrote:

DestinyCall wrote:
"In logic, more precisely in deductive reasoning, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. It is not required for a valid argument to have premises that are actually true, but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. "
Suppose we have an argument X with a false premise. Then, for X it is impossible for the premises to be true and the conclusion nevertheless to be false. Why? Because, it's impossible for the premises to be true, since there exists a false premise. From the "if and only if" and equivalence elimination it follows that if a form makes it impossible for the premises to be true and the conclusion nevertheless to be false, then argument is valid. Consequently, X is a valid argument. X was an arbitrary argument with a false premise. Therefore, all arguments with a false premise are valid.

Spoonwood wrote:

DestinyCall wrote:
There are a couple of problems with this argument, but the most important one is that you ignore that when testing for validity, you do not consider the truth of the premises. So if we assume there is an argument X with a false premise, we would then IGNORE that the premise is false when checking if the form of the argument was valid or not.
If X has a false premise, then the conjunction of the premises implies the conclusion (by the deduction meta-theorem). Consequently (by the inverse of the deduction theorem), it follows also that the argument is valid.

After doing a bit of digging, I was able to find a similar argument regarding contradictory premises.

If the argument's structure makes it impossible for all premises to be true at the same time, then the argument would be "valid but unsound". This can happen if one or more of your premises form a contradiction. Since the contradiction make it impossible for all premises to be true, it is also impossible for all premises to be true while the conclusion is false. This meets the requirements for logical validity, while simultaneously failing the requirements for logical soundness.

However, Spoon's argument isn't quite the same. Contradictory premises are a structural problem. False premises are a content problem. The description of validity makes it clear that premises need not actually be true, because you are assessing the form of the argument, not its meaning. So even if some or all premises are actually false, you would consider the argument as though all of its premises were true. Like pretending that there is a parallel world were giraffes are purple or the moon is made of real cheese or whatever is necessary to satisfy the conditions of your argument. In such a world, almost anything could be true. If it is still possible to arrive at a false conclusion, even though all premises are completely "true", then the form is invalid.

I don't feel like Spoon's argument actually supports his conclusion, as presented, but I'm really struggling to clearly articulate where and why it is faulty. However, I do know that it is definitely possible for an argument to have a false premise and also be invalid, so his conclusion appears to be false at face value. And a false conclusion would mean that the argument cannot possibly be sound.

Any thoughts?

Cogito · 2021-01-14 12:35:11

DestinyCall wrote:

Out of curiosity, Cogito, what do you think of Spoonwood's conclusion regarding the validity of arguments with a false premise?

Like you say, Spoon is starting with the idea that contradicting premises can prove anything. That is, if an argument contains two contradictory premises than it is 'valid', as the conclusion (any conclusion!) follows from the contradiction. Such an argument is certainly not sound.

The reach is in the implication that a false premise necessarily contradicts the others, so that "the conjunction of the premises implies the conclusion" - or at least that is what I think Spoon is saying.

Importantly, declaring all arguments with false premises as 'valid' is a counterproductive thing to do. If I don't know if all premises are true, but I can prove the argument is invalid, then I can move on from that argument. The alternative is that we have to know the truth of all premises before evaluating the validity of an argument, which is just silly.

To steal an example from one of those quizzes, consider this:

> Jim is between fifty and sixty years old. Jan is older than Jim. So, Jan is older than sixty.

This is an invalid argument, because Jim could be 51 and Jan could be 52 and both premises would be true but the conclusion would be false.

Spoon is essentially saying "Well Jim is actually 40, therefore this is a valid argument".

By bringing this extra premise in (Jim is 40) you do introduce a contradiction, and so can prove anything, and so the (new!) argument is valid.

Spoonwood · 2021-01-14 15:49:51

Cogito wrote:

If I don't know if all premises are true, but I can prove the argument is invalid, then I can move on from that argument. The alternative is that we have to know the truth of all premises before evaluating the validity of an argument, which is just silly.

One has to know all of the premises true (or be able to prove all of them true) in order to prove an argument invalid. Knowing the truth of the premises before evaluating the validity of an argument consists in what one does by appealing to truth tables. That does happen.

Spoonwood · 2021-01-14 16:02:24

Cogito wrote:

> Jim is between fifty and sixty years old. Jan is older than Jim. So, Jan is older than sixty.
This is an invalid argument, because Jim could be 51 and Jan could be 52 and both premises would be true but the conclusion would be false.

It could be that Jim is 59 and Jan is 61. In which case, the argument would be valid, as it would be impossible for all of the premises to be true, and the conclusion to be false.

So, no, the argument is not invalid. It's not valid either, since it does come as possible for the premises to hold and the conclusion to fail. The argument has an indeterminate status.

DestinyCall · 2021-01-14 20:39:30

You can't change the actual premises of an argument when checking for validity, only their truth value.

The premises of the argument are still "Jim is between fifty and sixty years old." and "Jan is older than Jim." And the conclusion is still "Jan is older than sixty."

If Jim is 59 and Jan is 61, both premises are true and the conclusion is true. But the argument is invalid (and unsound). Because it is still possible for the premises to be true and the conclusion to be false. Perhaps Jan lied about her age and she is actually 59 and a half, despite claiming to be 61. The original premises are both true, but the conclusion is false. The structure of the argument is invalid, because it was POSSIBLE for Jan's age to be less than 60 while both of the argument's premises were true. A valid logical structure would not have allowed this outcome.

Now, you could say "Jim is 59 and Jan is 61. So Jan is older than sixty." This is a NEW argument with a new structure and new premises. And this new argument is valid. If both premises are true, it is not possible for the conclusion to be false. In fact, you don't even need to mention Jim at all. The conclusion follows just from Jan's age alone.

This argument is valid, not because Jan is actually 61, but because IF she was 61, the conclusion can't be false. It is impossible for all premises to be true while the conclusion is false, so it is a valid structure and if the premise(s) are also true, the conclusion must be true, which would also make this argument sound.

Keep in mind, invalid arguments can also have true conclusions. In fact, valid arguments with false premises can have true conclusions! This does not make the argument invalid (or valid).

For example, in our new argument, if Jan's age is actually 63, one of the premises would be false (i.e. "Jan's age is 61") but the conclusion is still true. This change does not affect the structure of the argument so it does not impact validity. But this does have an impact on soundness.

It is important to remember that a valid argument with false premises would, of course, be unsound, even if the conclusion happened to be true. It doesn't do much good to arrive at the right answer by mistake. Your argument is still badly reasoned and fails to provide solid support for your conclusion. In a legal setting, that would be like having a "hunch" that someone was the murderer, but you lack solid evidence. Even if you are completely right, you still need to back up your conclusion properly to convince other people.

Validity promises that, if the argument is valid, true premises cannot lead to a false conclusion. An argument is valid, if and only if, it is impossible for the conclusion to be false if all the premises are true. Validity does not guarantee that only a valid argument will lead to a true conclusion or that a valid argument can only have a true conclusion.

This means that you still must establish the actual truth value of your premises to confirm that the argument is not just valid, but also, sound. Soundness requires both validity AND true premises. A sound argument cannot have a false conclusion. So if the conclusion is false, the argument is not sound.

And that is bad.

Spoonwood · 2021-01-14 22:02:17

DestinyCall wrote:

If Jim is 59 and Jan is 61, both premises are true and the conclusion is true. But the argument is invalid (and unsound).

An argument with all true premises and a true conclusion is both valid and sound. So, in such a case, the argument is both valid and sound.

DestinyCall wrote:

Because it is still possible for the premises to be true and the conclusion to be false.

If Jim is 59, and Jan is 61, it is impossible for the premises to be true and the conclusion is false, because the conclusion is true. If you have a true conclusion, it's impossible for it to be false. No statement can be different than what it is. Since it is true, it can't be false, which follows from the law of identity.

DestinyCall wrote:

Keep in mind, invalid arguments can also have true conclusions.

Do NOT keep that in mind, as it's simply not true. An argument is *only* invalid if all of it's premises hold true, and it's conclusion is false. Otherwise, the argument is valid.

"The moon is made of green cheese. Therefore, one of the United States Presidents name was Reagan." is a valid argument.

DestinyCall wrote:

In fact, valid arguments with false premises can have true conclusions! This does not make the argument invalid (or valid).

An argument with a false premise is valid, by virtue of having a false premise, since it's not possible for it to have all true premises and a false conclusion, it can't be invalid. Consequently, it's valid.

DestinyCall wrote:

For example, in our new argument, if Jan's age is actually 63, one of the premises would be false (i.e. "Jan's age is 61") but the conclusion is still true. This change does not affect the structure of the argument so it does not impact validity.

If the argument has a true conclusion instead of a possibly false one, then it's conclusion is not possibly false. Thus, it has a different structure (it can only have some possible rows of its truth table, instead of more possibilities if the truth values are unknown precisely, but known to be either true or false). Also, an argument is only invalid if it has all true premises and a false conclusion. Thus, the argument is valid.

DestinyCall wrote:

It doesn't do much good to arrive at the right answer by mistake. Your argument is still badly reasoned and fails to provide solid support for your conclusion.

Arguing from false premises is fine with respect to the reasoning, because reasoning only concerns moving from the set of premises to the conclusion.

Evidence would be lacking, but that only concerns the premises, not the reasoning, since the premises, by definition, didn't involve reasoning. Only the inference from premises to conclusion involves reasoning.

DestinyCall wrote:

An argument is valid, if and only if, it is impossible for the conclusion to be false if all the premises are true.

So, if it is impossible for the conclusion to be false if all premises are true, then an argument is valid (by equivalence elimination). In other words, if, (if all premises of an argument are true, then it is impossible for the conclusion to be false), then an argument is valid. Now, suppose that some premise of a given argument X is false. So, not all premises of X are true. Then, the conditional, C "if all premises of X are true, then it is impossible for the conclusion to be false" is true. Why does that conditional hold? Because "if not p, then p, then q" is a tautology. 'p' is 'all premises of X are true'. Since some premise is false, 'p' is false, and thus 'not p' is true. C is the antecedent of this conditional: "if, if all premises of an argument are true, then it is impossible for the conclusion to be false, then an argument is valid." That conditional holds by definition, and its antecedent holds also by the above. By detachment it follow that X is valid. X was an arbitrarily selected argument with some false premise.

Therefore, all arguments with a false premise are valid.

DestinyCall · 2021-01-15 00:04:00

Spoonwood wrote:

An argument with all true premises and a true conclusion is both valid and sound.

That statement is false. Both valid and invalid arguments can have all true premises and a true conclusion. The actual truth or falsehood of the statements is not what determines if an argument is valid.

And no ... "The moon is made of green cheese. Therefore, one of the United States Presidents name was Reagan." is a not valid argument. The form of this argument allows a true premise and a false conclusion, so it is definitely invalid.

You are allowing the actual truth value of the statements in the argument to distract yourself from the possible truth values. This is FORMAL LOGIC. The focus is on form, not meaning.

Logical structure is vitally important when assessing the quality of your reasoning. It is like the skeleton of your argument. Look past the meat and focus on the bones. If your argument has strong bones, you can build a solid body of evidence off it. If the bones of your argument are full of cracks, your whole line of reasoning will fall apart at the lightest touch.

Let's go back to basics ...

All poodles are dogs.
Fido is a poodle.
Therefore, Fido is a dog.

ALL X ARE Y
F IS X
THEREFORE F IS Y

This is a valid argument. It is impossible for the conclusion to be false, if all the premises are true.

Even if you replace the individual statements with false statements, the underlying argument remains valid and unchanged.

All birds are dogs.
Spoonwood is a bird.
Therefore Spoonwood is a dog.

ALL X ARE Y
F IS X
THEREFORE F IS Y

This is a valid argument. The premises are actually false, but that doesn't matter for validity.

....

All poodles are dogs.
Fido is a bird.
Therefore, Fido is a dog.

ALL X ARE Y
F IS H
THEREFORE F IS Y

This argument is invalid. It is possible for the conclusion to be false, even if all the premises are true.

Even if we replace the individual statements with true statements (or false statements), the underlying argument remains invalid and unchanged.

All birds are dogs.
Spoonwood is a poodle.
Therefore Spoonwood is a dog.

ALL X ARE Y
F IS H
THEREFORE F IS Y

This is an invalid argument. The premises might be true or false, but that doesn't matter for validity. What matters is it is possible for all premises to be true, yet the conclusion is still false.

The logical structure of the sentence fails to preserve the truth of the premises, so it fails the validity test.

DiscardedSlinky · 2021-01-15 00:06:39

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fug · 2021-01-15 01:44:17

DiscardedSlinky wrote:

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Please do not ree at the people trying to educate Doug. By teaching him how to make a proper argument we might one day see threads that actually make sense and are relevant to the game and forums instead of his cringe posting.

Bless all those out there trying to educate this poor poor spoon.

Cogito · 2021-01-15 02:46:29

I think the key takeaway for the 'truthiness of premises affects the validity of an argument' is that as soon as you assert that a premise is false you have introduced a new premise and hence have a new argument. Assessing the validity of this new argument (it will always be valid because we introduced a contradiction) says nothing about the validity of the original argument. This is why validity is assessed under the assumption that all premises are true, and it's only structural contradictions that are important from a contradictions point of view.

Let's look at an argument with a structural contradiction:

A is TRUE
A is FALSE
therefore X is TRUE

This is a valid, unsound argument, as it contains contradicting premises.

Now we have an invalid argument where the conclusion does not follow from the premises:

A is TRUE
if A is TRUE then B is TRUE
therefore X is TRUE

If we assert that the second premise is false we get a new argument:

A is TRUE
if A is TRUE then B is TRUE
if B is TRUE then A is FALSE <-- this is our new assertion
therefore X is TRUE

We now have a valid argument because of the contradiction, but of course it is unsound.

Last edited by Cogito (2021-01-15 03:47:36)

Spoonwood · 2021-01-15 05:36:34

DestinyCall wrote:

That statement is false. Both valid and invalid arguments can have all true premises and a true conclusion. The actual truth or falsehood of the statements is not what determines if an argument is valid.

Check the truth tables. Truth and falsity of statements determines whether or not an argument is valid or sound.

DestinyCall wrote:

And no ... "The moon is made of green cheese. Therefore, one of the United States Presidents name was Reagan." is a not valid argument. The form of this argument allows a true premise and a false conclusion, so it is definitely invalid.

The form here only allows a false premise and true conclusion. So, it's definitely valid.

DestinyCall wrote:

You are allowing the actual truth value of the statements in the argument to distract yourself from the possible truth values. This is FORMAL LOGIC. The focus is on form, not meaning.

Formal logic does have a focus on meaning in the form of semantics and model theory.

DestinyCall wrote:

Let's go back to basics ...
All poodles are dogs.
Fido is a poodle.
Therefore, Fido is a dog.
ALL X ARE Y
F IS X
THEREFORE F IS Y
This is a valid argument.

Those aren't basics. It involves a term logic or quantifier logic, which makes for more than propositional logic.

DestinyCall wrote:

ALL X ARE Y
F IS H
THEREFORE F IS Y
This argument is invalid. It is possible for the conclusion to be false, even if all the premises are true.
Even if we replace the individual statements with true statements (or false statements), the underlying argument remains invalid and unchanged.

The original argument above involves variables. If you replace the individual *forms* of statements of the premises with false statements, then the truth values of those statements no longer can be more than one possible value, and instead constants. A constant false proposition implies any proposition at all. Consequently, the argument is valid. Again, it's impossible for all of the premises to be true and the conclusion false, because by definition at least one proposition is false and thus it's impossible for all premises to be possibly true.

DestinyCall wrote:

The logical structure of the sentence fails to preserve the truth of the premises, so it fails the validity test.

If the premises include a false statement, then one of the proposition has the structure of a constant. That's a *different* structure than if the argument has premises which are not constants and the conclusion is not a constant also.

Also, it's rather telling that you completely ignore this argument Destiny:

"So, if it is impossible for the conclusion to be false if all premises are true, then an argument is valid (by equivalence elimination). In other words, if, (if all premises of an argument are true, then it is impossible for the conclusion to be false), then an argument is valid. Now, suppose that some premise of a given argument X is false. So, not all premises of X are true. Then, the conditional, C "if all premises of X are true, then it is impossible for the conclusion to be false" is true. Why does that conditional hold? Because "if not p, then p, then q" is a tautology. 'p' is 'all premises of X are true'. Since some premise is false, 'p' is false, and thus 'not p' is true. C is the antecedent of this conditional: "if, if all premises of an argument are true, then it is impossible for the conclusion to be false, then an argument is valid." That conditional holds by definition, and its antecedent holds also by the above. By detachment it follow that X is valid. X was an arbitrarily selected argument with some false premise."

Spoonwood · 2021-01-15 05:41:33

Cogito wrote:

I think the key takeaway for the 'truthiness of premises affects the validity of an argument' is that as soon as you assert that a premise is false you have introduced a new premise and hence have a new argument.

Well, if 'truthiness' gets replaced by 'trutfulness' I would agree.

But also, as soon as you have a premise of an argument as false, the argument is valid, since it's impossible for all of the premises to be true (since one of them is false). Thus, it's impossible for all of the premises of the argument to be true and the conclusion false. Thus, the argument is not invalid. All arguments are valid or invalid. And the argument is not invalid. Therefore, the argument is valid.

Last edited by Spoonwood (2021-01-15 05:47:10)

DestinyCall · 2021-01-15 08:26:23

Spoonwood wrote:

DestinyCall wrote:
That statement is false. Both valid and invalid arguments can have all true premises and a true conclusion. The actual truth or falsehood of the statements is not what determines if an argument is valid.
Check the truth tables. Truth and falsity of statements determines whether or not an argument is valid or sound.

I don't think you are using truth tables correctly.

Spoonwood wrote:

DestinyCall wrote:
You are allowing the actual truth value of the statements in the argument to distract yourself from the possible truth values. This is FORMAL LOGIC. The focus is on form, not meaning.
Formal logic does have a focus on meaning in the form of semantics and model theory.

You have missed the point.

Spoonwood wrote:

DestinyCall wrote:
Let's go back to basics ...
All poodles are dogs.
Fido is a poodle.
Therefore, Fido is a dog.
ALL X ARE Y
F IS X
THEREFORE F IS Y
This is a valid argument.
Those aren't basics. It involves a term logic or quantifier logic, which makes for more than propositional logic.

So you are saying I need to go even more basic. I was afraid of that.

....

I think we need to establish when and why we ask if an argument is valid or sound or strong.

When assessing the quality of an argument, we ask how well its premises support its conclusion. More specifically, we ask whether the argument is either deductively valid or inductively strong.
A deductive argument is an argument that is intended by the arguer to be deductively valid, that is, to provide a guarantee of the truth of the conclusion provided that the argument’s premises are true. This point can be expressed also by saying that, in a deductive argument, the premises are intended to provide such strong support for the conclusion that, if the premises are true, then it would be impossible for the conclusion to be false. An argument in which the premises do succeed in guaranteeing the conclusion is called a (deductively) valid argument. If a valid argument has true premises, then the argument is said also to be sound. All arguments are either valid or invalid, and either sound or unsound; there is no middle ground, such as being somewhat valid.
Here is a valid deductive argument:
It’s sunny in Singapore. If it’s sunny in Singapore, then he won’t be carrying an umbrella. So, he won’t be carrying an umbrella.
The conclusion follows the word “So”. The two premises of this argument would, if true, guarantee the truth of the conclusion. However, we have been given no information that would enable us to decide whether the two premises are both true, so we cannot assess whether the argument is deductively sound. It is one or the other, but we do not know which. If it turns out that the argument has a false premise and so is unsound, this won’t change the fact that it is valid.

Source: https://iep.utm.edu/ded-ind/

Notice that in this example argument the ACTUAL truth of the premises is never established, but we are able to IMMEDIATELY determine that the argument is valid, based on form alone. That is because it is not necessary to obtain information about the truthfulness of the argument's content before determining the validity of argument's form. More importantly, the validity or invalidity of the argument would not change based on the results of our findings, if we do learn the premises are true or false. Learning that the premises were all true would not make the argument valid if the structure was invalid. Learning that one of the premises was false would also not change the validity of the argument either. The logical form of the argument does not change based on the actual truth value of its premises and conclusion. And validity is determined by logical form, not content.

This is a major flaw in your previous arguments. Whenever you base a conclusion on the idea that you can change a deductive argument's validity in this way, you are making the whole argument unsound. To change the validity of an argument, you MUST alter the structure of the argument. This could be done by adding or removing premises and altering the form of the premises. But it cannot be done by declaring one or more of the premises actually true or false, unless you intend to add that information as a new premise to your original argument ... which would, of course, change the argument's structure.

...

I think the root of the problem goes back to one of the definitions for validity provided earlier in the post, because it wasn't sufficiently clear that you are not analyzing the ACTUAL truth of the premises in the real world, but all possible truth values for that argument, based on the logical form. Basically, you are expected to ask yourself, if it would be possible, in a world or situation were all the premises are true, for the conclusion to be false. This hypothetical situation is not affected by the actual truth of the premises, as we know them. Our understanding of the truth might be flawed or change over time. It can be incomplete or imperfect. We can't necessarily trust that a statement actually holds the truth value that we assign to it. But if it is impossible, based on the form of the argument itself, for true premises to produce a false result, then we are able to show that, IF the premises are actually true, then the conclusion must follow by logical necessity.

Here are a couple additional definitions of deductive validity that help to round out the full meaning and correct usage of this concept.

The concept of deductive validity can be given alternative definitions to help you grasp the concept. Below are five different definitions of the same concept. It is common to drop the word deductive from the term deductively valid:
An argument is valid if the premises can’t all be true without the conclusion also being true.
An argument is valid if the truth of all its premises forces the conclusion to be true.
An argument is valid if it would be inconsistent for all its premises to be true and its conclusion to be false.
An argument is valid if its conclusion follows with certainty from its premises.
An argument is valid if it has no counterexample, that is, a possible situation that makes all the premises true and the conclusion false.

All of these statements are true for valid arguments. Ultimately, we use validity to test the logical structure of the argument to confirm that true premises would lead to a true conclusion. After the structure has been confirmed as valid, we check the content of the argument to assess if the premises are true. If logical structure ensures that a true conclusion will follow with certainty from true premises, we can make the logical inference that the conclusion MUST be true, so long as we are reasonably certain that all premises are true. That is why when you use deductive reasoning, the goal is to achieve a SOUND argument that is valid and has all true premises.

I suspect that part of the problem is that you are trying to apply logical techniques to the definition of validity in a way that they were never intended to be used. And also, that you keep ignoring that when you judge validity, you are looking at all possible versions of the argument, based on it's structure, not just the single set of statements in front of you. That's why truth tables are useful, because they allow you to consider all the different ways that the premises and conclusions could interact, hypothetically, if they were true or false.

Spoonwood wrote:

DestinyCall wrote:
ALL X ARE Y
F IS H
THEREFORE F IS Y
This argument is invalid. It is possible for the conclusion to be false, even if all the premises are true.
Even if we replace the individual statements with true statements (or false statements), the underlying argument remains invalid and unchanged.
The original argument above involves variables. If you replace the individual *forms* of statements of the premises with false statements, then the truth values of those statements no longer can be more than one possible value, and instead constants. A constant false proposition implies any proposition at all. Consequently, the argument is valid. Again, it's impossible for all of the premises to be true and the conclusion false, because by definition at least one proposition is false and thus it's impossible for all premises to be possibly true.

See above regarding how validity works. You are not judging the actual truth of the statements. You are considering if it is hypothetically possible for a situation to exist were all the premises are true but the conclusion is false. People tend to get confused by their perceptions of what is or is not true in the real world, so they have a hard time imaging a world in which "All birds are dogs" or "the moon is made of green cheese" or whatever, which is one of the reasons why formal logic is frequently expressed using formulas. Not only does this break the argument down into functional units, but it also eliminates unconscious biases due to perceived truth or falsity.

ALL X ARE Y
F IS H
THEREFORE F IS Y

This is the part of the argument that we are interested in when we are discussing validity. Not the actual statements used in the premises and conclusion. Just the "bones" of the argument.

Looking at this argument, if "All X ARE Y" is true and "F IS H" is true, is it IMPOSSIBLE for "F IS Y" to be false? Does the structure of the argument prevent the conclusion from being false when all the premises are true? If the answer is no, then the argument's logical structure is invalid.

Spoonwood wrote:

DestinyCall wrote:
The logical structure of the sentence fails to preserve the truth of the premises, so it fails the validity test.
If the premises include a false statement, then one of the proposition has the structure of a constant. That's a *different* structure than if the argument has premises which are not constants and the conclusion is not a constant also.

You are overthinking it. When you are judging validity, you are not interested in ambiguous fickle words and weird fleshy concepts, just the elegant forms hidden beneath the surface. So you need to strip away the meat and just look at the bare bones of the argument. Ignore whether or not you BELIEVE a statement is true or false. You don't really know and if you are a serious logician, you probably don't care. Just look at the interaction between the different parts of the argument. That is how truth tables were meant to be used. As a way to compare all the possible outcomes from all the different ways that a logical argument could be formed. The actual content of your argument is important too ... but it is worthless floppy meat without a good skeleton to move it around. You need to establish validity before you start worrying about true premises.

Once you know that a particular form is valid, you can use the same structure over and over in future arguments and, as long as you don't change any of the structural elements, it will stay valid. Then it is just a question of determining if the premises of a given argument are true to decide if you have constructed a sound argument.

Spoonwood wrote:

Also, it's rather telling that you completely ignore this argument Destiny:
"So, if it is impossible for the conclusion to be false if all premises are true, then an argument is valid (by equivalence elimination). In other words, if, (if all premises of an argument are true, then it is impossible for the conclusion to be false), then an argument is valid. Now, suppose that some premise of a given argument X is false. So, not all premises of X are true. Then, the conditional, C "if all premises of X are true, then it is impossible for the conclusion to be false" is true. Why does that conditional hold? Because "if not p, then p, then q" is a tautology. 'p' is 'all premises of X are true'. Since some premise is false, 'p' is false, and thus 'not p' is true. C is the antecedent of this conditional: "if, if all premises of an argument are true, then it is impossible for the conclusion to be false, then an argument is valid." That conditional holds by definition, and its antecedent holds also by the above. By detachment it follow that X is valid. X was an arbitrarily selected argument with some false premise."

I'm not ignoring this argument. I am waiting for you to reach a point where you will be able to understand why it is unsound. This argument is based on a false assumption regarding validity and how it is tested. As long as you hold onto that assumption, I will not be able to explain the problem to you in a way that you will understand it. Once you figure it out, I probably won't even have to explain why your conclusion is false. It will be self-evident.

The first step to understanding is passing a simple logic quiz:
https://global.oup.com/us/companion.web … rue_false/

It is from chapter 4 in the textbook "Introductory to Philosophy" so I promise it is not that difficult. It might even be fun.

DestinyCall · 2021-01-15 08:27:33

P.S. - What is trutfulness?

Spoonwood · 2021-01-16 09:29:10

DestinyCall wrote:

This is a major flaw in your previous arguments. Whenever you base a conclusion on the idea that you can change a deductive argument's validity in this way, you are making the whole argument unsound. To change the validity of an argument, you MUST alter the structure of the argument.

No, it's not a flaw. An argument with a known false premise has a different structure than the one you put up as an example where the premises are not known as true or false. An argument with a known false premise has a constant as a premise. There is no way to change it's validity. It ends up automatically valid. As one tautology says " if falsum, then 'p' ". In other words, if you have a constant false proposition, then you can deduce any proposition whatsoever. Thus, all arguments with a false premise are valid.

DestinyCall wrote:

Basically, you are expected to ask yourself, if it would be possible, in a world or situation were all the premises are true, for the conclusion to be false. This hypothetical situation is not affected by the actual truth of the premises, as we know them.

If one of the premises is false, it's impossible to have all of the premises hold true.

DestinyCall wrote:

But if it is impossible, based on the form of the argument itself, for true premises to produce a false result, then we are able to show that, IF the premises are actually true, then the conclusion must follow by logical necessity.

If one of the premises of the argument is false, then based on the form of the argument itself, its impossible for only true premises to produce a false result, since the argument doesn't have all true premises. Consequently, the argument is valid by form alone.

If you have a false premise for an argument, the conditional "if the premises are actually true, then the conclusion must follow by logical necessity" holds, because its antecedent is false, and all conditionals with a false premise are true.

DestinyCall wrote:

I suspect that part of the problem is that you are trying to apply logical techniques to the definition of validity in a way that they were never intended to be used.

Those logical techniques are intended to get applied as I've used them. Again, the philosophy pages I linked to said explicitly that if you don't find a row of a truth table where there exists a falsity for the last column (and the truth table is complete), then the argument is valid.

DestinyCall wrote:

And also, that you keep ignoring that when you judge validity, you are looking at all possible versions of the argument, based on it's structure, not just the single set of statements in front of you.

No, I have explicitly said before that if an argument has a known false premise, then it has a constant as a premise, having a different structure than one with a variable as a premise.

DestinyCall wrote:

An argument is valid if the premises can’t all be true without the conclusion also being true.

If one of the premises of an an argument X is false, then all of the premises can't all be true. Consequently, the conditional if the premises can't all be true without the conclusion also being true follows as true also. By detachment, the argument X is valid. X was an arbitrary argument with a false premises. Therefore, all arguments with a false premise are valid.

DestinyCall wrote:

An argument is valid if the truth of all its premises forces the conclusion to be true.

If an argument has a false premise, then the truth of all of the premises of the argument is false. So, the conditional holds, and the argument is valid.

DestinyCall wrote:

An argument is valid if it would be inconsistent for all its premises to be true and its conclusion to be false.

If have an argument has a false premise, then it's not consistent for all its premises to be true, since we would have a contradiction by having such. Thus, the conditional here also holds.

DestinyCall wrote:

An argument is valid if its conclusion follows with certainty from its premises.

Falsum implies all propositions. Thus, from a false premise, the conclusion of the argument follows with certainty. Therefore, the argument is valid.

DestinyCall wrote:

An argument is valid if it has no counterexample, that is, a possible situation that makes all the premises true and the conclusion false.

An argument with a false premise has no possible situation that makes all the premises true, since one of the premises is false by definition. Thus, there is no counterexample. Therefore, the argument is valid.

DestinyCall wrote:

All of these statements are true for valid arguments.

All of those statements imply that if an argument has a false premise, then it is a valid argument.

Also, and I haven't made this argument before, note that if an argument is invalid, then it only has true premises. There's a logical law/tautology that says "if, (if 'not p', then 'q'), then (if 'not q', then 'p')". By instantiation, "if, if an argument X is not valid, then X only has true premises (as premises), then if X does not only have true premises, then X is valid." Thus, it follows that if an argument X does not only have true premises, then X is valid. When does an argument X not only have true premises? When it has at least one false premise. Thus, an argument X with a false premise is valid. X was an arbitrary argument. Therefore, all arguments with a false premise are valid.

DestinyCall wrote:

You are overthinking it.

This phrase signals a logical cop out.

DestinyCall wrote:

When you are judging validity, you are not interested in ambiguous fickle words and weird fleshy concepts, just the elegant forms hidden beneath the surface. So you need to strip away the meat and just look at the bare bones of the argument. Ignore whether or not you BELIEVE a statement is true or false. You don't really know and if you are a serious logician, you probably don't care.

I don't believe for a second you have a clue what you're talking about here. You can write words on a page. You have dogmatically asserted that an argument can have a false premise and not be invalid, in spite of several proofs now provided above that an argument with a false premise is valid.

Also, logicians don't ignore whether or not they believe a statement true or false. Investigations in axiomatic logic (almost) always start from tautologous forms, which are always true once understood. Otherwise the investigation would be unsound (soundness in the sense that no deduction of a false premise can occur). Lack of metalogical soundness of a system makes for a serious issue.

DestinyCall wrote:

You need to establish validity before you start worrying about true premises.

Validity is easily established at the formal level by having a small number of truth-preserving rules. If a logical system has detachment as it's only rule of inference, then validity is much easier than checking that all premises hold true.

DestinyCall wrote:

I'm not ignoring this argument. I am waiting for you to reach a point where you will be able to understand why it is unsound. This argument is based on a false assumption regarding validity and how it is tested. As long as you hold onto that assumption, I will not be able to explain the problem to you in a way that you will understand it. Once you figure it out, I probably won't even have to explain why your conclusion is false. It will be self-evident.

You haven't provided a false assumption. You don't have one. You engage in a logical cop out here, instead of engaging in reasoning.

Also, if you were honest here, you would also have to contend with the argument that "if an argument is invalid, then it only has true premises as premises. Therefore, an argument with at least one false premise, is valid." And check the truth table for CCNpqCNqp, or probably more how you would see it notated, ((~p->q)->(~q->p)), if you doubt that I used a tautology there.

One could even encapsulate all forms of validity as follows: an argument is valid if it has a false premise, or it has a true conclusion.

Last edited by Spoonwood (2021-01-16 09:40:29)

Lava · 2021-01-16 17:37:17

This is such a pog thread.

DestinyCall · 2021-01-16 17:54:19

I remember pogs. Never understood the appeal of collecting them, tho.

DestinyCall · 2021-01-17 09:22:53

Okay ... let's take this one step at a time, because I want to make sure that everything is covered.

In deductive logic, arguments are composed of statements. Statements are sentences that can be either true or false. You can't base an argument on a question, like "who are you?" or an exclamation like "Wow!" or a string of unconnected words like "Mop flip cat jumpy." or whatever. You have to make a clear statement that can be realistically assigned a truth value. However, SOME statements are structured in such a way that they can ONLY be true or false. This is still allowed and they can be used to form an argument, but they can only ever result in a single truth value, because it isn't possible for the other value to result from that statement's logical structure. Such a statement could be described as logically true or logically false. I would also call them "structurally" true or "structurally" false, because they are true or false on a structural level.

An example of a statement that is always TRUE would be "Black is black". This statement has the form "A = A", so it is always true for any value assigned of A. This kind of a statement can be called a tautology. For example, the statement "For all x, the conjunction of Fx and not Fx is false" would be considered a tautology, because you basically just defined a conjunction. This statement will always be true based on the rules of logic. Tautologies can also be much longer or embedded within a larger statement or complex argument. In natural language, a tautology is an expression or phrase that says the same thing twice, just in a different way.. For example, "In my opinion, I think ..." or "You gotta do what you gotta do." or "I am either here or I am not here." It is basically repetitive redundancy that extends the length of your sentences without meaningful purpose and makes your sentences unnecessarily long. Since tautologies don't really add any new information, they can usually be removed without any change in meaning, however they are occasionally useful to clarify ambiguous terms or emphasize a point.

The opposite of a tautology is a contradiction. Contradictory statements are always logically false. The logical structure of the statement prevents it from being true. One of the simplest contradictions is "Black is not black" or, as a logical formula, "P is not P". Keep in mind, there is more than one use of the term "contradiction". It can also be used to describe two or more premises that are in conflict with each other. When used to describe a single premise it means that the statement is unconditionally false - in other words, it is a self-contradictory premise. This can be generalized to a collection of statements, which is then said to "contain" a contradiction.

The awesomely-named "Principle of Explosion" describes what happens when you try to formulate a logical argument by using one or more self-contradictory statements.

That's right. It explodes.

I warned you to never divide by zero, but did you listen? Of course not. Now we need to buy a new calculator.

Actually, the Principle of Explosion means that if it is possible to use the rules of deduction to prove a contradiction within a system of reasoning, it then becomes possible to "prove" any statement at all within that system (whether it’s actually true or not). In particular, if you start by assuming a self-contradictory statement, you can prove anything. That's right ... anything is possible if you assume that up is NOT up or black is NOT black.

https://xkcd.com/704/

In formal logic, this means that an argument with contradictory premises or at least one self-contradictory premise is logically valid, but structurally unsound. It kind of goes without saying that this is a very bad thing. It ruins the logical foundation of your argument.

As a demonstration of the principle, consider two contradictory statements—"All lemons are yellow" and "Not all lemons are yellow"—and suppose that both are true. If that is the case, anything can be proven, e.g., the assertion that "unicorns exist," by using the following argument:
We know that "Not all lemons are yellow," as it has been assumed to be true.
We know that "All lemons are yellow," as it has been assumed to be true.
Therefore, the two-part statement "All lemons are yellow OR unicorns exist” must also be true, since the first part is true.
However, since we know that "Not all lemons are yellow" (as this has been assumed), the first part is false, and hence the second part must be true, i.e., unicorns exist.
In a different solution to these problems, a few mathematicians have devised alternate theories of logic called paraconsistent logics, which eliminate the principle of explosion. These allow some contradictory statements to be proven without affecting other proofs.

https://en.m.wikipedia.org/wiki/Principle_of_explosion

As you can probably imagine, this is really dangerous territory for philosophers and logicians because if your proofs contain even one contradiction, any reasoning based off that proof is thrown into doubt. Historically, the discovery of contradictions such as Russell's paradox at the foundations of mathematics threatened the structure of mathematics as we knew it. Resolving these contradictions was paramount to reestablishing sound logical principles for many mathematical proofs.

https://en.m.wikipedia.org/wiki/Russell's_paradox

....

Okay ... so some statements are logically true and some are logically false. What about the rest? Most statements are either true or false. But it is not necessarily clear which one it is. And more over, whether or not they are true can change over time and depend on a variety of other factors. The statement "It is raining" might be true today, but false tomorrow. Or it might be true where I live, but not where you live. It is probably raining somewhere and it has probably rained at some time, but is the statement "it is raining" true or false right here and right now? Likewise, if I say something like "There are three fruits in this basket", you won't necessarily know if that is true or false. If you look inside the basket and see three tomatoes and three pineapples ... are there three fruits or six? Or maybe more? Tomatoes can be categorized as either fruit or vegetable. Pineapples are a multiple fruit, formed by several smaller fruits fusing together. If we can decide on the definition of "one fruit", what happens if I take out one of the fruits and replace it with a banana? Or what if the basket falls over and all the fruit falls out ... now there is no fruit in the basket, regardless of what was true earlier. Truth can be hard to pin down, even for something as simple as a basket of fruit.

Our world and our understanding of the world is constantly changing. Truth is a moving target. It is not always easy to determine definitively what is or is not true. And even if you think you have it all figured out ... you might be wrong.

Thus, in formal logic, most statements are reduced down to variables and logical connections in order to describe the fundamental relationship between the different moving parts in a statement or series of connected statements. In this way, we can simplify complex concepts down to just the most basic structural elements. The "skeleton" of the argument, as I have previously described it.

Now, a funny thing happens when you strip away the meat and just leave the bones - a lot of arguments look fundamentally similar at a structural level, even though the content of the arguments might be dramatically different.

All bats are mammals.
Mammals are warm-blooded.
Therefore, bats are warm-blooded.

All helicopters are walnuts.
Walnuts are brown.
Therefore, all helicopters are brown.

These two arguments are very different, but they share the same structure. It is easy to see how you could form a variety of different arguments by swapping out the individual pieces without significantly changing that underlying structure.

All T are Q.
Q are X.
Therefore, All T are X.

One of the fundamental principles of logic is that whether an argument is valid or invalid is determined entirely by its form. In other words, validity is a function of form, not content.

This means that if an argument is valid, then every argument with the same form is also valid. And if an argument is invalid, then every argument with the same form is also invalid.

So once you determine that a particular form is definitely invalid, you can be sure that any concrete argument based on that form will also be invalid.

Let's do another example ...

If it is a sunny day, then I always wear a hat.
It is a sunny day.
Therefore, I am wearing a hat.

This argument involves a conditional statement. In common language, these kinds of statements usually take the form "If X then Y". In logic, they are sometimes called inferences. If X is true, you can infer that Y must be true. This kind of relationship is described using an arrow pointing from one variable to the other, like so P->Q. The first variable is called the antecedent. The second one is the consequent. It is important to recognize that IF/THEN statements in formal logic are very rigid and uni-directional. This means that IF the antecedent is true, THEN the consequent must also be true. But the relationship is only one way, so if the consequent is true, the antecedent is not guaranteed. It might be true or false, we don't know.

We can use a truth table to show all possible outcomes of a simple conditional statement (P->Q). A truth table shows what would happen if the values P and Q are true or false. From there we can easily see all the ways that the whole statement could end up true or false.

P Q | P->Q
T T T
T F F
F T T
F F T

As you can see, when the antecedent (P) is true and the consequent is true (Q), the conditional statement is true. But if the antecedent is true and the consequent is false, the conditional statement is false. That is because the conditions were met, but the expected result did not occur, so the inference was wrong. Also, notice that if the antecedent is false, the conditional statement is still considered true.

Think of conditional statements like a promise. I promised that, if it was a sunny day, I would definitely wear a hat. If it is not a sunny day, I don't have to wear a hat (P is false and Q is false). But if I felt like wearing a hat on a non-sunny day, that would be okay too (P is false and Q is true). I didn't promise to not wear a hat unless it was sunny, so I kept my promise. But if I didn't wear a hat on a sunny day, the promise would be broken - and the conditional statement would be false.

Now for the real question. Is this argument valid?

Let's break it down into its logical form:

IF P, THEN Q.
P.
Therefore, Q.

Now let's consider all the possibilities using a truth table.

If it is possible for both premises to be true when the conclusion is false, then the argument is invalid. In this case, there is only one scenario where both premises would be true (if both P and Q are true) and in that situation, the conclusion would also be true (since Q is the conclusion). It is not possible for both premises to be true and the conclusion to be false, so this form is valid.

Now let's look at a similar argument with a slightly different form.

If it is a sunny day, then I always wear a hat.
I am wearing a hat
Therefore, it is a sunny day.

Is this argument valid?

IF P, THEN Q.
Q.
Therefore, P.

Time for another a truth table.

As you can see, row three of the table shows that our two premises (P->Q) and Q are both true, but our conclusion P is false. This is not good. It means that this structure is invalid. The conclusion does not logically follow from the premises.

In fact, this is an example of a well-known formal fallacy sometimes called "Affirming the consequent". Any argument that takes this basic form has this same flaw and is invalid.

If A is True, then B is True.
B is True.
Therefore, A is True.

It is fallacious because the truth of the premises do not necessarily lead to the truth of the conclusion. As I mentioned earlier, this kind of conditional relationship is unidirectional. I didn't promise that I would never wear a hat on a non-sunny day. I only promised that I would definitely wear one if it was a sunny day. the truth of the consequent is independent from the truth of the antecedent.

Now let's try this one ...

If it is a sunny day, then I always wear a hat.
It is not a sunny day.
Therefore, I am not wearing a hat.

I bet you can guess this one based on what you already know about similar structures.

If A is true, then B is true.
A is false
Therefore, B is false.

Is this argument valid?

Let's look at truth tables again.

Once again, row three shows all premises are true and the conclusion is false. Therefore, this argument is not valid. Why? Because the conditional statement only promises that IF the antecedent is true, the consequent must also be true. If is not a sunny day, so I can wear a hat or not wear a hat. Doesn't matter. I am not obligated to do anything.

This is another classic formal fallacy, called "Denying the antecedent". Any argument that uses this form will be invalid, because you can't assume that the consequent is false simply because the antecedent is false. That's not how this kind of conditional relationship works.

....

So what does all this have to do with the original topic of this discussion? Absolutely nothing, because I didn't mention incest even once.

What does it have to do with Spoonwood's conclusion that "All arguments that have a false premise are valid"? Well, it means that an argument with a false premise and an argument with a true premise can be either valid or invalid, depending on their logical form. Validity is a function of form.

The only kind of false premise that really matters to validity is a contradiction - a premise that is logically false and cannot be true on a structural level. Otherwise, if a premise in the concrete argument is known to be false but it is NOT a contradiction of itself or another premise, we would still consider all possibilities when evaluating the argument's logical structure. Since we are not testing the validity of the concrete argument, but rather the underlying structure of that argument, in order to verify that true premises will logically lead to a true conclusion, we assume all variables can be either true or false. Then we can construct a truth table that covers all possibilities for that structure. This is true even if we know which premises are actually true or false for the concrete argument currently under-evaluation.

Here is another example:

If dinosaurs are alive today, then I have a pet dinosaur.
I have a pet dinosaur.
Therefore, dinosaurs are alive today.

Is this argument valid or invalid?

If it helps you to decide, her name is Rocky.

I'll even add that to the argument:

If dinosaurs are alive today, then I have a pet dinosaur.
I have a pet dinosaur.
My pet dinosaur's name is Rocky.
Therefore, dinosaurs are alive today and one of them is named Rocky.

What do you think? Valid or Invalid?

Now, what if I admit that I lied about having a pet dinosaur. And her name is actually Samantha.

Does this change the validity of my argument? Is it more or less valid now that you know that Rocky/Samantha is a complete fabrication and dinosaurs do not roam the earth?

It shouldn't. Because validity is a function of form. And the form of this argument is invalid. Therefore, invalid arguments can have false premises and the cake is a lie.

My pet dinosaur told me so, and she always speaks the truth.

Last edited by DestinyCall (2021-01-17 10:06:37)

Spoonwood · 2021-01-17 12:53:29

DestinyCall wrote:

An example of a statement that is always TRUE would be "Black is black". This statement has the form "A = A", so it is always true for any value assigned of A. This kind of a statement can be called a tautology.

No, that's not how a tautology gets defined. A tautology, in propositional logic, consists of a well-formed formula which is true for all assignments of truth values to the variables. "A=A" is not a tautology in first-order logic: https://math.stackexchange.com/question … gy-why-not Specifically, "A=A" in first-order logic with equality, would just be an instance of a propositional variable, and there is no way to get it otherwise from anything but a propositional variable. But, no propositional variable is a tautology, so "A=A" is not a tautology in first-order logic, since it can't get obtained from a propositional tautology using the additional rules, and symbols, of a sound first-order logical system.

DestinyCall wrote:

For example, the statement "For all x, the conjunction of Fx and not Fx is false" would be considered a tautology, because you basically just defined a conjunction.

No, that wouldn't be a tautology either. Negation doesn't mean "is false". NKpNp or ~(p^~p) are tautologies, which involve negations, and so are NK(phi)xN(phi)x or ~(Fx^~Fx), where I'm using (phi) instead of the Greek letter 'phi'.

DestinyCall wrote:

Tautologies can also be much longer or embedded within a larger statement or complex argument. In natural language, a tautology is an expression or phrase that says the same thing twice, just in a different way.. For example, "In my opinion, I think ..." or "You gotta do what you gotta do." or "I am either here or I am not here." It is basically repetitive redundancy that extends the length of your sentences without meaningful purpose and makes your sentences unnecessarily long. Since tautologies don't really add any new information, they can usually be removed without any change in meaning, however they are occasionally useful to clarify ambiguous terms or emphasize a point.

If you meant to compare them as if they were similar in this respect, no. Tautologies in classical propositional logic can't get removed without changing meaning. Really, if you removed a single tautology from classical propositional logic, you get an entirely different system of logic. There exist logics which don't have all the tautologies of classical logic also. But, if you added particular tautologies to them, you would add new information to those logical systems. There also exist tautologies in classical propositional logic which don't repeat information (though some can get said to do so).

DestinyCall wrote:

When used to describe a single premise it means that the statement is unconditionally false - in other words, it is a self-contradictory premise.

I think that if a well-formed formula is a contradiction, it just means that for all valuations of the variables, the well-formed formulas is false. I don't know what "self-contradictory" would mean here, as the formulas don't speak.

DestinyCall wrote:

The awesomely-named "Principle of Explosion" describes what happens when you try to formulate a logical argument by using one or more self-contradictory statements.
That's right. It explodes.
I warned you to never divide by zero, but did you listen? Of course not. Now we need to buy a new calculator.

This analogy doesn't make much sense. Division by zero in mathematics doesn't get used, because it's illegal (assuming that the structure has two elements... division by zero in a single element structure poses no problems). The principle of explosion does get used in studies of formal logic.

DestinyCall wrote:

Actually, the Principle of Explosion means that if it is possible to use the rules of deduction to prove a contradiction within a system of reasoning, it then becomes possible to "prove" any statement at all within that system (whether it’s actually true or not). In particular, if you start by assuming a self-contradictory statement, you can prove anything.

More like it means that if you can produce a contradiction, then you can deduce any well-formed formula within the scope of that contradiction. OR it means that if you can produce a falsity, then you can deduce any well-formed formula within the scope of that falsity (ex falsum quodlibet), or both.

DestinyCall wrote:

In formal logic, this means that an argument with contradictory premises or at least one self-contradictory premise is logically valid, but structurally unsound.

I wouldn't say that. There exist formal proofs where contradictory premises used, temporarily. They don't end up structurally unsound.

DestinyCall wrote:

As you can probably imagine, this is really dangerous territory for philosophers and logicians because if your proofs contain even one contradiction, any reasoning based off that proof is thrown into doubt.

No. Natural deduction proofs commonly use proofs by contradiction. Hypotheses get eliminated because of sub-proofs having contradictions. Also one axiom in one axiom set for classical propositional logic could get notated as ((~p->~q)->((~p->q)->p)) (I'd prefer to just write CCNpNqCCNpqp)... and if you understand what that means, you can tell that if we have the negation of a proposition implying contradictory consequents, then we can accept the proposition. The key lies in keeping track of scope. Russell's paradox doesn't keep track of scope whatsoever, as I've understood it.

DestinyCall wrote:

Most statements are either true or false.

Propositional logic does not suggest this. If you look at truth tables of formulas having less than 20 symbols, you'll likely find that most formulas are neither tautologies, nor contradictions, and instead contingent in that they may hold true or may hold false. Thus, most statements appear to be neither true nor false, if we can stop assuming a priori that statements are either true or fasle.

DestinyCall wrote:

And more over, whether or not they are true can change over time and depend on a variety of other factors.

There is no sense of time in classical logic. Things are not true today and false tomorrow in classical logic. All statements basically have to get completely contextualized and put in a time framework, or have to hold true outside of time and outside of a context to make sense in classical logic.

DestinyCall wrote:

Truth is a moving target.

Truth in classical logic is no such thing. It is constant. Were it a moving target, it wouldn't be so simple to find and generate tautologies as to have already gotten automated several times, and automated rather successfully.

DestinyCall wrote:

What does it have to do with Spoonwood's conclusion that "All arguments that have a false premise are valid"? Well, it means that an argument with a false premise and an argument with a true premise can be either valid or invalid, depending on their logical form. Validity is a function of form.

In each of your examples you assumed all of the premises true. If an argument has a false premise, you can't use any variables for the false premises. You can only use the logical constant of falsity/falsum.

DestinyCall wrote:

Since we are not testing the validity of the concrete argument, but rather the underlying structure of that argument, in order to verify that true premises will logically lead to a true conclusion, we assume all variables can be either true or false.

If you're not testing the validity of the concrete argument, then claiming it valid or invalid doesn't have a solid basis. In other words, if you claim a concrete argument valid or invalid, then you need to test the concrete argument for validity, not something else.

Also, if a premise is known to be false, then it doesn't have the structure of variables. By classifying its structure as involving variables, you've engaged in a type error since constants are not variables. Again, if a premise of an argument is known to be false, then you need to analyze the validity of the argument by appealing to a constant.

DestinyCall wrote:

If dinosaurs are alive today, then I have a pet dinosaur.
I have a pet dinosaur.
Therefore, dinosaurs are alive today.
Is this argument valid or invalid?

You don't have a pet dinosaur, at least according to conventional meaning of the word 'dinosaur'. Therefore, you have a false premise of "I have a pet dinosaur". That premise has the structure of a constant, and thus can't get accurately analyzed by appealing to variables. Since it has a false premise, it is impossible for all of the premises to hold true and its conclusion to be false. So, it's not possibly invalid. Every argument is either invalid or valid. Therefore, it's a valid argument.

It doesn't matter whatsoever whether it's conclusion holds true or not.

You want to claim that it involves "affirming the consequent"? But "affirming the consequent" is *only* relevant when we can meaningfully assume we had true premises, which we didn't have here.

The form can get described as follows, where "0" means falsity:

(P->0)
0
-----
P

The form of the argument thus is valid.

Does it look like it has the form of "affirming the consequent"? But, it also has the form ex falsum quodlibet along with an additional premise. Ex falsum quodlibet is a valid form of argument, even in the present of another premise.

DestinyCall wrote:

Now, what if I admit that I lied about having a pet dinosaur. And her name is actually Samantha.
Does this change the validity of my argument?

No, because what matters involves whether the premises were true or false in reality, not what you claim about them.

It still holds that if an argument has a false premise (in reality), its impossible for all of the premises to hold true and the conclusion to end up false. Thus, such an argument will be valid.

DestinyCall wrote:

It shouldn't. Because validity is a function of form.

Yes, but it is not a function of forms involving variables necessarily.

Validity can involve constants also.

You know that principle of explosion you talked about above Destiny?

Well, there's one form with variables:

CNpCpq (~p->(p->q))

The other involve, which more precisely encapsulates the "ex falsum quodlibet" idea, is a logical constant:

C0p (0->p).

'0' there is a constant.

And you simply can't evaluate whether or not 'C0p' is a tautology or contradiction by appealing to variables. You have to appeal to the constant '0'.

Same thing happens when you have an argument with a known false premise. You can't appeal to variables everywhere to evaluate it honestly. You have to appeal to a constant only when evaluating that argument.

Last edited by Spoonwood (2021-01-17 13:09:07)

DestinyCall · 2021-01-17 17:01:12

Thank god. It took us forever to get here, but with the help of Samantha, I think we are finally getting close to common ground. I don't think we are quite there yet, but it might actually be possible to settle our differences. I wasn't sure if it was possible before. I'm trying not to get too excited.

I think we have been talking at cross-purposes for most of this argument. Imagine we are discussing a piece of fruit and I say "This piece of fruit is red." And you respond, "No, you are wrong. The fruit is orange." And then you add "It is also a type of citrus fruit." And I respond "That is an idiotic thing to say. This is definitely not a citrus fruit. Citrus having thick rind and juicy pulp. And they usually grows in tropical regions. This fruit has a thin skin and it is most often grown in cold temperate regions." And you respond with "No, the skin is 35 millimeters wide. It is not thin. And it contains bitter oils, so it is often removed prior to eating." And then I say "Wait, wait, wait. What are you even talking about? You don't always peel this fruit for fresh eating. And the skin is certainly not that thick. Also, it does not contain bitter oils. Maybe you are thinking of the seeds - they are removed and discarded before eating because they contain trace amounts of arsenic." And then you respond with "Arsenic is not found in the seeds of this fruit, as shown by this reputable source on botony."

And then we both suddenly realize we were talking about apples and oranges the whole time and live happily ever after.

....

I don't disagree with your conclusion when you make all of your assumptions clear. The reason we were disagreeing this whole time is because I am holding a different set of assumptions.

I assume that in the real world, outside of the clean and rigid rules of deductive reasoning, "truth" is rarely known with a high enough degree of certainty to be considered a constant. You are correct that formal logic is black and white, true or not true, valid or invalid. But the real world is muddy. In order to KNOW that something is 100% true, you must make assumptions and your assumptions can be wrong.

So when considering the majority of "arguments that have a false premise", I would assume that the "known false premise" still might be true or false. I keep an open-mind to the possibility that my assumed knowledge might be at fault, not my opponent's reasoning skills.

I mentioned the principle of charity earlier in the discussion for a reason. Applying the principle of charity means you take the time to look at your opponent's deductive argument under the generous assumption that they believe it is a sound and well-reasoned argument. If it is not, you do your best to construct a sound argument from what they appear to be trying to argue and check back to confirm that you understood their position correctly and described it accurately. That way, you are not shadow-boxing with a weaker version of their true position. And you are not arguing about completely unrelated points and trivial flaws in their argument that do not refute the main premises.

When arguing with other people, you frequently fail to view their arguments in a charitable light and also fail to clearly explain your own assumptions, which leads to inexplicable deductive leaps, a lot of unnecessary disagreement, and hilarious misunderstandings. Just like the ones that started this argument off in the first place. I don't expect you to change your ways, so I won't bother giving you any real advice on how you could avoid this issue in the future. Maybe you like to win every argument by virtue of being the last one still arguing, rather than because you have presented the best argument and convinced other people to agree with your position. I can understand that. Winning feels great.

Long story short, this argument has been a complete waste of time, except that I had a lot of fun and learned more about formal logic than I will every need to know in the real world.

So all and all, I'd say it was great!

DiscardedSlinky · 2021-01-18 06:47:02

Anyone here want some gamer girl toe nail clippings?

Cogito · 2021-01-18 13:36:59

Spoonwood wrote:

DestinyCall wrote:
The awesomely-named "Principle of Explosion" describes what happens when you try to formulate a logical argument by using one or more self-contradictory statements.
That's right. It explodes.
I warned you to never divide by zero, but did you listen? Of course not. Now we need to buy a new calculator.
This analogy doesn't make much sense. Division by zero in mathematics doesn't get used, because it's illegal (assuming that the structure has two elements... division by zero in a single element structure poses no problems). The principle of explosion does get used in studies of formal logic.

Watch out, the mathematics police are coming for you!

As a trained mathematician, the idea of anything being 'illegal' is silly, though the term has it's place as a shorthand that get's the point across in some contexts.

The analogy made a lot of sense to me, as when you evaluate f = 1/x as x approaches 0, f goes to infinity (positive or negative depending on which side you come at it from) - literally exploding from the number line!

Thus I have nitpicked your nitpick, and by the principle of nitpick ad absurdum have rendered all your nitpicks in this thread void.

All substantive arguments remain unvoided.

Last edited by Cogito (2021-01-18 13:37:20)

DestinyCall · 2021-01-18 17:06:54

Don't divide by zero

DestinyCall · 2021-01-18 17:09:13

Cogito wrote:

All substantive arguments remain unvoided.

But if you remove all the nit-picks, red herrings, and non sequiturs ... would anything be left?

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