Arguments with Inconsisent Premises

Essentially a valid argument is an argument where it would be impossible for the conclusion to be false if all of the premises are true.

So ponder this syllogism:

P1) Premise 2 is false.

P2) Premise 1 is false.

C) Pigs can fly.

This syllogism apparently satisfies the conditions for a valid argument because since both of the premises contradict each other there is no possible condition where both of the premises are true. It apparently doesn't matter what the conclusion is. Since the premises contradict each other there is no condition in which both of the premises are true and the conclusion is false; therefore the argument is valid.

I discussed this the other day with someone and I felt like we didn't really come to any conclusions. Personally I don't really understand how these arguments are valid because the conclusion does not deductively follow from the premises. I would like to hear other peoples thoughts on this apparent paradox because I'm not sure I have fully wrapped my head around it.

Quote:

---

Originally Posted by stormcrow

...these arguments are [not] valid because the conclusion does not deductively follow from the premises

---

http://img.photobucket.com/albums/v6...ndofthread.gif

Exactly. If your conclusion doesn't follow the premise you are not making an argument your just playing games.

You've stumbled upon the principle of explosion, which states that anything follows from a contradiction. The name comes from the observation that, if we allow a contradiction into a logical system, the system "explodes into triviality."

The specific argument you gave is actually kind of interesting because it's not how the principle of explosion is usually derived. The usual method is pretty straightforward and is illustrated nicely in the wiki link from above (see the example about lemons and Santa Claus.) It looks like you've found a different way of proving the same thing, which is neat.

We start by defining what is a valid argument. Formally, let P = "all of the premises are true" and C = "the conclusion is true." An argument is valid iff P ⊃ C. Now, the statement P ⊃ C is equivalent to the statement ¬(P & ¬C), which we can see by breaking the two statements down to truth tables:

Spoiler for truth tables:

In these tables, 1 = True and 0 = False.

Code:

---

|==========================|
|P | C | P & ¬C | ¬(P & ¬C)|
|==========================|
|1 | 1 |  0    |    1    |
|--------------------------|
|1 | 0 |  1    |    0    |
|--------------------------|
|0 | 1 |  0    |    1    |
|--------------------------|
|0 | 0 |  0    |    1    |
|--------------------------|


|===============|
|P | C | P ⊃ C  |
|===============|
|1 | 1 |  1    |
|---------------|
|1 | 0 |  0    |
|---------------|
|0 | 1 |  1    |
|---------------|
|0 | 0 |  1    |
|---------------|

---

Since ¬(P & ¬C) and P ⊃ C share the same truth conditions, they are equivalent statements.

So if we can show that, for a given argument, it cannot be the case that the premises are true and the conclusion is false, this is equivalent to showing that the conclusion is entailed by the premises, which is our definition of validity.

As you observed, any argument with contradictory premises can never satisfy P & ¬C, because it can never satisfy P. So for any argument with contradictory premises, it is necessarily the case that ¬(P & ¬C). So any such argument is necessarily "valid." QED.

One difficulty of course is that we can, by this same logic, easily construct a counterargument with contradictory premises that "proves" the opposite of what was just proved (in this case, that pigs can fly). We have indeed exploded into triviality.

@Alric
Technically the syllogism I used was actually a valid argument although it is not a sound argument. The validity of a deductive argument rests on its logical form not its correspondence to the external world. For example:
All A’s are B.
C is an A.
Therefore C is a B.

This argument is valid. We could replace the symbols and say that:
All cats are dogs.
Doug is a cat.
Therefore Doug is a dog.

Technically this is a valid argument, the conclusion follows from the premises and the conclusion is necessarily true but the argument is not sound because cats are not dogs.

@DuB

Thanks for dropping some knowledge on me DuB, that cleared up a lot of questions I had. The guy I discussed this with called it the paradox of entailment which seems to me to be basically the same as the principle of explosion.

So “either lemons are yellow or Santa Claus exists”? But since lemons are not yellow (P2); Santa Claus necessarily exists? This all seems really foreign to me…

Just a minor digression but I’m not familiar with using this ⊃. I was under the impression that this was used to denote supersets; I have never seen it used before like that. I’m guessing it can be used to denote a material implication as well? I’m also not familiar with using binary in truth tables; I guess I still have a lot to learn.

I realize that this contradiction arises from the ambiguity of the definition of a valid argument but I think this issue has some epistemological baggage as well. I think this problem stems from the law of noncontradiction preventing a proposition from being A and ¬A. Do you think that the law of noncontradiction applies elsewhere other than formal logic? I was thinking about what instance might contradict the law so I was thinking along the lines of a photon being a wave and a particle although I’m not sure if that is quite the best example. Maybe this one. If someone is in a boat (in a lake or ocean) they are in the water and not in the water…again this could just be a trivial trick of natural language.

Also how is the principle of explosion dealt with in modern logic? I believe that traditionally paradoxes in logic and math are dealt with by introducing hierarchies (like Russell’s paradox with Type theory).

Also it seems to me that the principle of explosion only arises in deductive arguments where (deductive) arguments can only be valid or invalid (and sound or unsound). The notion of validity is not really an issue for inductive arguments however because they deal with uncertainty. Inductive arguments employ the notion of probability so they are only certain to a degree. Even strong inductive arguments are invalid because there is always a chance that the premises are true but the conclusion is false. Take the famous example:

P1) 90% of humans are right handed.
P2)Joe is human
C) Joe is right handed.

The p-value is 90% so this would be a pretty strong inductive argument but there is a chance that he is in fact left handed despite the fact that premises 1&2 are true so it is still invalid. I don’t think the principle of explosion would present a problem to logic if we used probabilistic and inductive logic.

Quote:

---

Originally Posted by stormcrow

The guy I discussed this with called it the paradox of entailment which seems to me to be basically the same as the principle of explosion.

---

Right, as far as I know, they are the same thing. Wiki: paradox of entailment.

Quote:

---

Originally Posted by stormcrow

I’m guessing it can be used to denote a material implication as well?

---

Right. It's a convention you may run into. I use it because it's the one adopted in the text that I learned formal logic from.

Quote:

---

Originally Posted by stormcrow

If someone is in a boat (in a lake or ocean) they are in the water and not in the water…again this could just be a trivial trick of natural language.

---

That is my suspicion.

Quote:

---

Originally Posted by stormcrow

Also how is the principle of explosion dealt with in modern logic? I believe that traditionally paradoxes in logic and math are dealt with by introducing hierarchies (like Russell’s paradox with Type theory).

---

Well, most commonly, by just not allowing contradictions. Apparently so-called paraconsistent logics deal with this issue in a more explicit way, but I don't know anything about those.

Quote:

---

Originally Posted by stormcrow

Also it seems to me that the principle of explosion only arises in deductive arguments where (deductive) arguments can only be valid or invalid (and sound or unsound). The notion of validity is not really an issue for inductive arguments however because they deal with uncertainty.

---

I'm not sure. I think the key part of the paradox arises because a proposition is allowed to have multiple simultaneous truth values. One way of interpreting probability is that instead of assuming everything is either True or False, we allow an infinity of truth values between 0 and 1. But it seems that it would still be contradictory in inductive logic to say that a proposition is true with, say, both 0.4 probability and 0.7 probability, just as it was contradictory in deductive logical to say that something had truth values of both 0 and 1. It's not immediately obvious to me how one would use this to derive absurd conclusions in the inductive context, but maybe it also works there. I will have to think about it later.

If you think about it ⊃ as a superset symbol is actually the same thing as an implication symbol. That's why it's used; consider sets of events.

Another great thread stormcrow. I tried to reply to it earlier but I became a bit tangled up in Dub's relating it to the principle of explosion, because I too thought it was more about the paradox of entailment. I need to think about it some more but don't have time right now, possibly tomorrow.

I wish that more threads like this were created. Thanks DuB for the explanation and the link (although the superset notation confused me too). I'd never heard the paradox before.

Technically, all arguments with false premises are valid in formal logic, as can be seen from DuB's second truth table. But so are circular arguments, among others. We're really just saying "If it were true that both p and ¬p were true, then q would be true." But that statement is irrelevant because the premises aren't (and can't be) true. I suspect it only seems so wrong because we're still used to hearing valid defined as something like 'correct' in everyday speech.

Sorry for the confusion. In the future I'll use the → operator.

Valid argument? It's 'possible' for the conclusion to be false. Therefore, Stormcrow, your exemplified argument is invalid. What hype is there?

Quote:

---

Originally Posted by Malac Reborn

Valid argument? It's 'possible' for the conclusion to be false. Therefore, Stormcrow, your exemplified argument is invalid. What hype is there?

---

As explained earlier, the criterion for validity is not simply that it is impossible for the conclusion to be false, but rather that it is impossible for the conclusion to be false AND the premises true. By this standard definition, the argument is valid; it is impossible for the conclusion in the OP to be false and its premises true, by virtue of the fact that it is impossible for both of its premises to be true.

I'm not really a logician so my terminology may be off...

Quote:

---

Originally Posted by stormcrow

P1) Premise 2 is false.

P2) Premise 1 is false.

C) Pigs can fly.

---

This is tricky because the premises of the syllogism are referencing other premises.

I've never seen this before and my initial reaction is to not allow it. The premises in syllogisms will involve statements about sets and elements of those sets, not other premises. There are a fixed amount of possible syllogisms. Have you coordinated this with one?

Also, it's impossible to plug anything into it.

Consider what P1 or P2 could even be? If P1 is some actual statement with variable content then it is no longer the statement that P2 is false. Same for P2.

Quote:

---

Originally Posted by PhilosopherStoned

This is tricky because the premises of the syllogism are referencing other premises.

I've never seen this before and my initial reaction is to not allow it.

---

The self-referential nature of the premises is indeed tricky, and possibly should not be permitted. I think the more important point for the purposes of this discussion is just that they are mutually contradictory. We can replace them with something like P1 = the sky is blue, P2 = the sky is not blue, and be in the same position as we are now. Maybe something like that is in fact a cleaner way to formulate this.

Sorry to derail this for a second but...

Quote:

---

Originally Posted by Malac Reborn

Valid argument? It's 'possible' for the conclusion to be false. Therefore, Stormcrow, your exemplified argument is invalid. What hype is there?

---

Why are you using light pink on a white background? Are you intentionally trying to irritate everyone and encourage people to ignore your posts?

No.
In fact, it's OK for you to ignore it. I'm only interested in one person here.

I await DuB's reply. I might learn something from this.

EDIT: Nvm. I see what the hype is about now.

Doesn't a contradiction of premises automatically prove that not both premises are true? It seems like the rule is purely hypothetical. The claimed rule would make more sense if it were that if two premises were true and in contradiction, anything could be the case. Such an absurd situation would prove metaphysical chaos and that anything can be the case because reality is not constrained by metaphysical order?

However, it seems to strongly relate to the paradox "This statement is false." Is that statement false? I think the paradox majorly calls into question the validity of the law of the excluded middle, which essentially says that either A is the case or A is not the case.

Quote:

---

Originally Posted by Universal Mind

The claimed rule would make more sense if it were that if two premises were true and in contradiction, anything could be the case.

---

That is essentially what it does say. Another way to phrase this would be: if you accept both of the contradictory premises to be true, it can be shown that you must also accept any other arbitrary proposition as true (if you are to be logically consistent). Obviously if you reject one of the premises, then you are not obligated to accept any conclusion that follows from the argument. This phrasing frames the issue in terms of argumentation rather than in terms of what "really is" or is not the case, but it is the same issue.

I simply looked at it in the sense of ~(A * B) //valid argument's conditions

If an argument's premises can't all be true, then the true value of B is false. Regardless of the true value of whether the conclusion is false(A), the negation encompassing the false conjunction, thus makes the statement/conditions true. Which, in turn, means the argument is valid.

Any feedback on this, DuB?

I think we're on the same page but I can't tell for sure. What are you using A and B for exactly?

A is a constant for "the argument's conclusion is false," and B for "the argument's premises are all true." If I would allow a biconditional, then attached to them would remain C for "the argument is valid." So, its extended logical form is ~(p&q)<->r, substituted by the sentence ~(A&B)<->C.

It feels weird falsifying B, since it feels I have to assume it true to test the measures of validity. Like was mentioned, if it's the case that I must assume truth for a contradiction, then to understand how the argument's valid, I feel as I'm missing, not the intuitivity that if contradictions can exist, then should everything else may exist, but something conceptual. Something about the relationship between the conclusion and presumptious contradiction, yielding validity, is unsettling in me.

It is impossible for both premises to be true. One must be true while the other one is false. There is no way to know which is which. Suppose premise 1 is true and premise 2 is false. What is the problem? Suppose premise 2 is true and premise 1 is false. What is the problem? The situation turns into a philosophical clusterfuck only under the assumption that both are true or both are false.

http://imgs.xkcd.com/comics/formal_logic.png

Quote:

---

Originally Posted by PhilosopherStoned

http://imgs.xkcd.com/comics/formal_logic.png

---

Did you honk? I couldn't hear it. Could you turn up the sound to where I can hear it from Mississippi? It will take a few hours for the sound waves to get here.

Quote:

---

Originally Posted by Universal Mind

It is impossible for both premises to be true. One must be true while the other one is false. There is no way to know which is which. Suppose premise 1 is true and premise 2 is false. What is the problem? Suppose premise 2 is true and premise 1 is false. What is the problem? The situation turns into a philosophical clusterfuck only under the assumption that both are true or both are false.

---

Yeah, this ^

I'm not particularly familiar with formal logic, but does it not follow, linearly, from Premise 1 to Premise 2 and so on? Such that accepting Premise 1: 'Premise 2 is false' invalidates the truth of Premise 2, i.e. it is irrelevant what Premise 2 now states? It is seemingly impossible to accept both premises unless you can somehow accept them in parallel, simultaneously.

Though if obvious contradiction is allowed in formal logic.....

This is a fascinating Wikipedia article on the "Liar Paradox," which is essentially, "This statement is false." The article does not address the paradox this thread is about specifically, but it involves analysis of many of the related issues.

Liar paradox - Wikipedia, the free encyclopedia