4 very easy arguments. Are they valid?

[Speakpigeon]

I have a question.

If a valid deductive argument entails a contradiction, does that guarentee that a premise is false?

Of course not.

But the only way an argument is set up rationally is with true premises.

What's your preferred paradox, Unter?
EB

 

[Speakpigeon]

We can’t do that with just any ole use of “valid.”

Any "ole"?

I'm not sure how to read that...
EB

 

[fast]

I have a question.

If a valid deductive argument entails a contradiction, does that guarentee that a premise is false?

Of course not.

But the only way an argument is set up rationally is with true premises.

I don’t see it as irrational that an argument (especially with good form) has false premises.

P1: whoever burnt up in that fire suffered a horrible death
P2: Joey burnt up in that fire
Therefore, C: Joey suffered a horrible death.

That’s not irrational just because P1 is false; it so happens that he had already died a peaceful death before the fire.

 

[Treedbear]

That is not why they are valid. In fact, whether they contain a contradiction is not related to validity. The reason that they are valid is that the premises entail the conclusion.

...

The fact that the conclusion is a contradiction is not relevant as to whether it is valid. But perhaps, you have other concept of validity in mind. If so, what do you understand as "valid"?

Ok now I see that valid wasn't the right word. Perhaps "unsound" is more appropriate. If you don't mind I found this on another forum (my bolding):

Think about the definition of validity: if the premises are true, then the conclusion is true as well. To show that an argument is invalid, you need to find an interpretation that makes the premises true but the conclusion false. If you start with a valid argument, can you add further premises that make all of the premises true but the conclusion false? – possibleWorld Jun 3 '17 at 13:02

In classical logic no, but you can make a sound argument unsound by adding false premises. The difference between soundness and validity is usually ignored colloquially, but logical validity is neutral on the truth of the premises, it only cares whether inference would preserve that truth. Contradictory premises, however, would make any argument from them unsound. Moreover, in classical logic due to the law of explosion anything can be validly inferred from contradictory premises. – Conifold Jun 4 '17 at 22:03

I would say that A4 is not valid because it appears to me that two or more premises contradict each other. In particular P1 and P2".


The fact that these premises are valid when they appear in A1, A2, and A3 has no bearing on whether A4 is valid.

Premises are not the sort of thing that can be valid or invalid. Premises can be true or false. But arguments can be valid or invalid. Assuming that A1, A2, and A3 are valid, it follows at once that A4 is also valid, since the premises of A4 entail the conclusion (I already explained why in my previous reply to you, and in the hidden part of the OP).
...

So I also found this:

valid ...
(logic) Of an argument: whose conclusion is always true whenever its premises are true.
An argument is valid if and only if the set consisting of both (1) all of its premises and (2) the contradictory of its conclusion is inconsistent.

For A4 it doesn't seem that a contradictory of the conclusion can even exist. It brings me back to my initial misgivings about the syllogistic form, and to the concept of the Principle of explosion as mentioned by the quote from the other forum:

The principle of explosion ... or the principle of Pseudo-Scotus, is the law of classical logic, intuitionistic logic and similar logical systems, according to which any statement can be proven from a contradiction. That is, once a contradiction has been asserted, any proposition (including their negations) can be inferred from it. This is known as deductive explosion. The proof of this principle was first given by 12th century French philosopher William of Soissons.

 

[fast]

I see that seemingly relevant premises are distracting. It’s like writing a paper and finding supporting information for a conclusion. Throwing in irrelevant information doesn’t distract from the flow, but relevant information in an argument forces us to focus on soundness instead of form.

P1: when something moves, it makes a sound
P2: ants move
So far, we have every reason to conclude ants make sound

P3: oranges are red
That does not distract us from “ants make sound”
P4: oranges are not red
Still, we have no reason to reject “ants make sound”
P5: ants don’t move
SO WHAT! Its still the case that if P1 and P2 are true, it’s still the case that ants make sound.

The key is to look for premises given that if true guarentees the conclusion. Finding opposing premises is distractive, but we’re inadvertently changing focus from form to a trusted guarentee (or validity to soundness)

 

[steve_bank]

Deriving support from a conclusion is what call bootstrapping an argument.

 

[untermensche]

I have a question.

If a valid deductive argument entails a contradiction, does that guarentee that a premise is false?

Of course not.

But the only way an argument is set up rationally is with true premises.

What's your preferred paradox, Unter?
EB

There is a paradoxical reaction in some people to stimulants.

We use thinks like Ritalin, a stimulant, for hyperactivity.

 

[fast]

Deriving support from a conclusion is what call bootstrapping an argument.

For, not from

 

[Speakpigeon]

Deriving support from a conclusion is what call bootstrapping an argument.

It depends how you mean but "deriving support from a conclusion" is definitely at the heart of the method of proof used in logic since probably Aristotle but at least since the Scholastics.

I guess what you mean may be that affirming the consequent is bad logic.

But we will never know what you mean.
EB

 

[Angra Mainyu]

For A4 it doesn't seem that a contradictory of the conclusion can even exist. It brings me back to my initial misgivings about the syllogistic form, and to the concept of the Principle of explosion as mentioned by the quote from the other forum:

The principle of explosion ... or the principle of Pseudo-Scotus, is the law of classical logic, intuitionistic logic and similar logical systems, according to which any statement can be proven from a contradiction. That is, once a contradiction has been asserted, any proposition (including their negations) can be inferred from it. This is known as deductive explosion. The proof of this principle was first given by 12th century French philosopher William of Soissons.

I do not understand what you mean by "For A4 it doesn't seem that a contradictory of the conclusion can even exist.", but the proof of the validity of A4 does not depend on the Principle of Explosion.

 

[Treedbear]

For A4 it doesn't seem that a contradictory of the conclusion can even exist. It brings me back to my initial misgivings about the syllogistic form, and to the concept of the Principle of explosion as mentioned by the quote from the other forum:

The principle of explosion ... or the principle of Pseudo-Scotus, is the law of classical logic, intuitionistic logic and similar logical systems, according to which any statement can be proven from a contradiction. That is, once a contradiction has been asserted, any proposition (including their negations) can be inferred from it. This is known as deductive explosion. The proof of this principle was first given by 12th century French philosopher William of Soissons.

I do not understand what you mean by "For A4 it doesn't seem that a contradictory of the conclusion can even exist.", but the proof of the validity of A4 does not depend on the Principle of Explosion.

I was referencing the definition I quoted:

...

valid ...
(logic) Of an argument: whose conclusion is always true whenever its premises are true.
An argument is valid if and only if the set consisting of both (1) all of its premises and (2) the contradictory of its conclusion is inconsistent.

...

I assumed you'd agree with it. When I try to apply it to the A4 conclusion:

C’’’: Joe is not an elephant, and Joe is an elephant.

all I'm able to come up with is "Joe is an elephant, and Joe is not an elephant." Assuming that the order isn't important it appears to be identical. Therefore the test for validity cannot be satisfied. Therefore to the question in the OP, although the premises are all valid, A4 is not. Maybe that's a technicality. What do you think?

My comment about the Principle of Explosion was an aside referencing the comments from the other forum that I quoted, and actually led me to a number of very interesting theories of logic as they apply to contradictions. Unfortunately it turned out to be a tangent due to my own difficulty in accepting that two premises cannot be contradictory.

 

[Angra Mainyu]

Assuming that the order isn't important it appears to be identical. Therefore the test for validity cannot be satisfied. Therefore to the question in the OP, although the premises are all valid, A4 is not. Maybe that's a technicality. What do you think?

The order is not important. However, the test for validity in the definition you provided can be and is satisfied. The test says that " set consisting of both (1) all of its premises and (2) the contradictory of its conclusion is inconsistent." That set is as follows:

P1: Joe is either a squid or a giraffe.
P1’: A giraffe is not an elephant.
P1’’: An elephant is not a squid.
P2’’: Joe is an elephant.
¬C’’’: It is not the case that (Joe is not an elephant, and Joe is an elephant).

Now, the set in question is in fact inconsistent. There are different ways to see this, and one of them is precisely to derive C''' from the premises. Since C''' is a contradiction, that implies that the set of the premises alone is inconsistent, and hence, if we add the negation of the conclusion to that set (or anything else, for that matter), then the set remains inconsistent (if you add something to an inconsistent set - whatever you add -, you still have an inconsistent set).

Another way of looking either the matter is to use truth tables, considering in this case the quantifiers too (i.e., "A giraffe is not an elephant" should be understood as "For all X, if X is a giraffe, then X is not an elephant.").

Unfortunately it turned out to be a tangent due to my own difficulty in accepting that two premises cannot be contradictory.

But two premises can be contradictory. For example, you can choose as the first premise "Joe is a squid" and the second premise "It is not the case that Joe is a squid".

 

[Speakpigeon]

Are the following arguments valid?

First argument (A1).

P1: Joe is either a squid or a giraffe.
P2: Joe is not a squid.
C: Joe is a giraffe.

A2:

P1’: A giraffe is not an elephant.
P2’: Joe is a giraffe.
C’: Joe is not an elephant.

A3:

P1’’: An elephant is not a squid.
P2’’: Joe is an elephant.
C’’: Joe is not a squid.

A4:

P1: Joe is either a squid or a giraffe.
P1’: A giraffe is not an elephant.
P1’’: An elephant is not a squid.
P2’’: Joe is an elephant.
C’’’: Joe is not an elephant, and Joe is an elephant.

That's very interesting.
EB

 

[Treedbear]

Assuming that the order isn't important it appears to be identical. Therefore the test for validity cannot be satisfied. Therefore to the question in the OP, although the premises are all valid, A4 is not. Maybe that's a technicality. What do you think?

The order is not important. However, the test for validity in the definition you provided can be and is satisfied. The test says that " set consisting of both (1) all of its premises and (2) the contradictory of its conclusion is inconsistent." That set is as follows:

P1: Joe is either a squid or a giraffe.
P1’: A giraffe is not an elephant.
P1’’: An elephant is not a squid.
P2’’: Joe is an elephant.
¬C’’’: It is not the case that (Joe is not an elephant, and Joe is an elephant).

Ok, I get it now. But why offer the other arguments A1, A2, and A3 as if they were necessary when A4 can stand on its own?

Now, the set in question is in fact inconsistent. There are different ways to see this, and one of them is precisely to derive C''' from the premises. Since C''' is a contradiction, that implies that the set of the premises alone is inconsistent,

Ok, I now know the difference between contradiction and inconsistent.

and hence, if we add the negation of the conclusion to that set (or anything else, for that matter), then the set remains inconsistent (if you add something to an inconsistent set - whatever you add -, you still have an inconsistent set).

I see that, but is there ever any reason for adding the negation? Is it just a method used for eliminating a contradiction from the conclusion?

Another way of looking either the matter is to use truth tables, considering in this case the quantifiers too (i.e., "A giraffe is not an elephant" should be understood as "For all X, if X is a giraffe, then X is not an elephant.").

Unfortunately it turned out to be a tangent due to my own difficulty in accepting that two premises cannot be contradictory.

But two premises can be contradictory. For example, you can choose as the first premise "Joe is a squid" and the second premise "It is not the case that Joe is a squid".

Thanks, I see that now. I've always avoided the use of syllogisms. Is there a good primer on the web? The syntax is my main barrier at the moment.

 

[Angra Mainyu]

Ok, I get it now. But why offer the other arguments A1, A2, and A3 as if they were necessary when A4 can stand on its own?

Mainly because A1, A2 and A3 are short and very easy arguments, and I thought the chances that readers would realize that they are valid was much higher than if I had posted only A4. Now, if a reader understands that A1, A2 and A3 are valid, I think that improves my chances of successfully explaining to them why A4 is also valid.

I see that, but is there ever any reason for adding the negation?

I wasn't going to add it, but since you provided a test for validity and asked me about it, I addressed the matter. The test for validity that you provided requires that one adds the negation of the conclusion: The test says "An argument is valid if and only if the set consisting of both (1) all of its premises and (2) the contradictory of its conclusion is inconsistent." In other words, an argument is valid if and only if the set of statements consisting on all of the premises of the argument and the negation of the conclusion, is an inconsistent set of statements. So, in order to check the validity of a deductive argument by the method you provided, one needs to add to the premises, the negation of the conclusion.

Is it just a method used for eliminating a contradiction from the conclusion?

No. The contradiction is not eliminated. Adding the negation of the conclusion is the way of applying to an argument the test you provided, regardless of whether the argument happens to have a contradictory conclusion.

Thanks, I see that now. I've always avoided the use of syllogisms. Is there a good primer on the web?

Here you can find a short introduction to validity and soundness, with links to some of the relevant definitions and also to more sophisticated analysis if you later want to study this matter and related one in more detail (for example, this one).
Here is an even shorter intro (just the basics, but at least you can test whether you're getting it); also, here you can get some basic info on the technical vs. non-technical distinction.

 

[Treedbear]

...
I wasn't going to add it, but since you provided a test for validity and asked me about it, I addressed the matter. The test for validity that you provided requires that one adds the negation of the conclusion: The test says "An argument is valid if and only if the set consisting of both (1) all of its premises and (2) the contradictory of its conclusion is inconsistent." In other words, an argument is valid if and only if the set of statements consisting on all of the premises of the argument and the negation of the conclusion, is an inconsistent set of statements. So, in order to check the validity of a deductive argument by the method you provided, one needs to add to the premises, the negation of the conclusion.

So if I have all the terms straight, A4 is valid but its conclusion contains a contradiction demonstrating that there is an inconsistency in the set of premises. Is there anyway to determine where the inconsistency lies? I mean what I had earlier called a contradiction between P1: Joe is either a squid or a giraffe, and P2’’: Joe is an elephant, now seems like the obvious source. But I imagine there are cases when it is not so obvious.

...

Thanks, I see that now. I've always avoided the use of syllogisms. Is there a good primer on the web?

Here you can find a short introduction to validity and soundness, with links to some of the relevant definitions and also to more sophisticated analysis if you later want to study this matter and related one in more detail (for example, this one).
Here is an even shorter intro (just the basics, but at least you can test whether you're getting it); also, here you can get some basic info on the technical vs. non-technical distinction.

Thanks. I'll be reading through them.

 

[Angra Mainyu]

So if I have all the terms straight, A4 is valid but its conclusion contains a contradiction demonstrating that there is an inconsistency in the set of premises. Is there anyway to determine where the inconsistency lies? I mean what I had earlier called a contradiction between P1: Joe is either a squid or a giraffe, and P2’’: Joe is an elephant, now seems like the obvious source.

The contradiction does not need to be between two premises. It turns out that in this particular case, all of the premises are needed to have a contradiction: you remove one, and the set of premises is consistent, so you cannot derive a contradiction from them. To say that Joe is an elephant and a giraffe is not a contradiction without a premise saying (say) that elephants are not giraffes (you could argue that it is a contradiction by the meaning of the words; I would argue it is not, but in any case, the contradiction here was obtained by the form of the argument).
Consider the following example:

R1: X is either 1 or 3.
R2: X is not 1.
R3: X is not 3.

The set or premises {R1, R2, R3} is inconsistent, but there no subset of two elements that is inconsistent. In other words, you can derive a contradiction from R1, R2 and R3, but not from R1 and R2, or from R1 and R3, or from R2 and R3. You need all three to get a contradiction. And similarly, in A4, you need all premises to get a contradiction.

Thanks. I'll be reading through them.

You're welcome.

 

[ruby sparks]

C contradicts itself.

No, it does not. But C''' does.

Yes, whoops. I meant C'''.

And does not seem to follow from P1 or P2’’.

No?

It does follow from P1, P1', P1'', and P2'', as explained in detail above - well, in detail if you realize that A1, A2 and A3 are valid. If you do not, then if you let me know, I will explain why those are valid as well.

I get that A1, A2 & A3 are valid.

So, it is the case that an argument (in this case A4) can still be valid even if the conclusion is a contradiction of itself, is that it?

If yes, I'm bound to ask, what is the point? I don't mean that rudely. I mean, so what if it's valid?

Or to put it another way, and using layman's language, if you fed this into a 'logic machine', and the machine was set up to detect logical flaws, and setting aside the issue of truth, a red button would flash on the machine to indicate that something was wrong with the argument, even if it was valid. Could I even use a layman's term and say the argument is nonsensical (even if valid)?

I guess you are merely using this argument as an example of what validity, of itself, is.

Also, quick query. Does P2’’ not contradict P1/P1'/P1" (as a set of premises)? I mean, can you even have (or use, or introduce) P2" along with (after having used) P1, P1' & P1" without invoking a contradiction in your set of premises? If that makes technical sense, which it may not, and if there is such a thing as contradictory premises (which it seems to me there should and can be).

In other words, if (emphasis if, because logic is not my area of any expertise) there is a contradiction there also, again, would a red button on the hypothetical logic machine not start to flash even before you finished inputting, ie before you even got to the (separately, of itself) contradictory conclusion C'"?

Unless you switched off all the machine detectors (including 'detect truth' and 'detect contradictions') except for 'detect validity at the end'. Which I'm guessing is sort of your point. And if it is, I'm not suggesting it's pointless. Because saying that might even itself be a contradiction. Lol. What I mean is, I'm not suggesting it's useless. It may in fact be very useful, or hypothetically or theoretically useful, in ways and for things or processes (perhaps even involving computing) that I am not familiar with or that I appreciate or understand. It is certainly interesting (to me) in any case.

ETA: I have not read all the intervening posts, so it may be that you have already dealt with some or all of the above.

 

[fast]

If the goal is to walk like a normal human being, it’s best to have both a left leg AND a right leg.

If the goal is to construct a sound argument, it’s best to have both valid structural form AND factually true premises.

The point is that there are TWO individually necessary conditions for the beloved sought out deductively SOUND argument.

P1: giraffes are taller than turtles.
P2: at least some vehicles have motors.
P3: cats eat.

That’s 3 premises with no conclusion. No conclusion, no inference. No inference, no argument. What’s the problem? No problem.

P1: giraffes are taller than turtles.
P2: at least some vehicles have motors.
Therefore, C1: cats eat.

In long form, it looks like this:

Giraffes are taller than turtles, and at least some vehicles have motors; therefore, cats eat. That says a whole heck of a lot more than: giraffes are taller than turtles, at least some vehicles have motors, and cats eat.

The word “therefore” signals a conclusion, and a conclusion implies an inference to an argument. How can those premises tie together such that we can derive the conclusion? We can’t because the structural flow of the argument is such that the truth of the conclusion cannot be properly gleaned from the premises, and that is despite the truth of both the premises and the conclusion.

“Valid” speaks to form and form alone.

If two experts were speaking about form and decided to use an example argument that had true premises, you might hear them regard the argument as valid if the form is good, but remember, sound implies valid, but valid doesn’t imply sound. If we know the premises are true, we immediately learn the argument is sound upon hearing that it’s valid.

A person listening in on their conversation would not be incorrect to understand the argument is valid, but they would fail to grasp the ramifications had the premises been false if they confused soundness for validity.

 

[ruby sparks]

A5

P1: Joe is not an elephant
P2: Joe is an elephant
C: Joe is an elephant and Joe is not an elephant

(My P1 seems to be a summary of P1, P1' & P1" in A4?)

Valid? Yeah, I think.



And?