Principle of explosion

Someone has randomly interjected a statement into an argument that the principle of explosion doesn't work with anything possessing material form, if god exists.

I've no idea why this is being directed at me as I had never even heard of this before, much less said anything about it.

Looking into it, it just appears to be a non sequitur which is then claimed to be valid logic because "reasons".

Convince me I'm wrong.

I've never heard of a principle of explosion, but I suppose that someone could write a paper or blog post about it. Sounds like the kind of thing we debated in my college dorm after midnight when we were too tired to think straight.

But thanks for writing, sorry i can't help.

I’m assuming this is in reference to the Big Bang with a rather ignorant interpretation that it was both big and a bang. Of course this is now widely refuted and challenged by some astrophysicists like Victor Stenger who postulates that ”inflation” was neither big nor much of a bang or any kind of explosion whatsoever.

Here, The claimant seems to be equating explosion with god’s supernatural powers which sounds very close to an ad hoc rescue .

An Ad Hoc fallacy is when a person gives an explanation for an event and the explanation is written or said as an argument for the event. When a person poses an explanation that is disputed by evidence the person has to resort to untestable answers to salvage their claim.

What you are describing seems to be Appeal to Faith<>. It may also be Appeal to Self-Evident Truth<> or Appeal to Stupidity<>.

Hi, Bryan!



Explosion refers to the process whereby any statement can be proven from a contradiction. I have occasionally cited this principle while completing proofs in symbolic logic. One of the best ways I can attempt to explain the principle of explosion is by discussing the Material Implication and Disjunction.

The material implication (IMP) is a formal inference rule, just as modus ponens is a formal inference rule. IMP is an inference to a disjunction from a conditional statement or vice versa. A disjunction is an "either...or" kind of statement, such as the cat will either come inside or go hungry. A conditional statement is an "if--then" kind of statement, such as if the cat does not come inside, then the cat will go hungry. In order to correctly restructure a conditional statement into its disjunctive form, negate the statement in the antecedent and make the statement in the consequent a disjunct. For example, the conditional statement "if P then Q" implies the disjunctive statement “Either Not P or Q". The antecedent of the conditional statement is negated ("P" is negated to "Not P") and the consequent (Q) becomes a disjunct. That is IMP applied to the conditional statement if P then Q to derive Not P or Q.

Conversely, in order to correctly restructure a disjunction into the form of a conditional statement, take one of the disjuncts, negate it, and put it in the antecedent of the conditional statement. Take the other disjunct, without negating it, and put it in the consequent. To use the previous example, “Either Not P or Q” implies “if P then Q”. One of the disjuncts is negated (the first one, in this case; "Not P" is negated to "Not Not P", and the negations cancel out to simply "P") and then put in the antecedent. The other disjunct, “Q” in this case, is not negated and is put in the consequent of that conditional statement. This is the material implication applied to the disjunction Not P or Q to derive if P then Q.

Here is one more examples:

“Not P or Not Q” implies “if Not Not P then Not Q”. And vice versa.

Essentially, you may validly infer a disjunction from a conditional statement or a conditional statement from a disjunction.



Why is the material implication a valid inference form?

It is valid to infer a disjunction from a conditional statement because of the law of the excluded middle. The law of the excluded middle states that, for any statement "P", "either P or not P" is true. Now, the conditional statement "If P then Q" states that the truthfulness or occurrence of "P" is sufficient for the truthfulness or occurrence of "Q". So, given the conditional statement, the first disjunct in the expression of the law of the excluded middle (any statement “P”) can be extrapolated to any statement “Q”, so that the new disjunction reads “either Q or Not P”. With the law of the excluded middle in mind, one can see why the following argument is valid. In the layout below, I put the law in brackets to make it easier to visualize its role in the argument.

1. If P then Q
[P or Not P]
Therefore
2. Q or Not P

Therefore,
3. Not P or Q.

(Statement 3 is just to switch the variables around for alphabetical order). Basically, given statement 1, the "P" in the expression of the law of the excluded middle extrapolates to "Q".

On the converse, it is valid to infer a conditional statement from a disjunction because of what it means for a disjunction to be true. For a disjunction to be true, at least one of the disjuncts must be true. When proving the validity of an argument, the premises are assumed to be true, so let us assume that Not P or Q is a true disjunction for the sake of argument. Because Not P or Q is true, either Not P must be true or Q must be true, so that the entire disjunct is true. If we assume that it is not the case that Not P, then logically we must assume that Q is the case, because one of the two disjuncts is true and Not P isn't the true one. If the true disjunct isn't Not P, then the true disjunct would have to be Q. In other words, given Not P or Q, it follows that if not Not P, then Q. The negations (the two "nots") cancel each other out so that what follows from the disjunction may be understood as "if P then Q". The layout is below.

4. Not P or Q.
Therefore,
5. If P then Q.

An important feature about Disjunction is that it also has an inference rule. The rule is called "Addition", by some logicians. Addition occurs when you add a disjunct to a statement. Assume that the sun is bright. To perform Addition on this premise, you may connect this statement to another statement using an "either...or" connective. For example: either the sun is bright or Asia is the smallest continent in the world. The statement that we assumed about the sun has been connected to the statement about Asia, using an "either...or" connective. Of course, it is false that Asia is the smallest continent in the world, but the disjunction is true because at least one of its disjuncts is true, which is enough for the whole disjunction to be true---the sun is bright. Indeed, I may Add any statement at all to a true premise of an argument and that new, larger premise will still be true. I can say that either the sun is bright or Asia is the smallest continent in the world or Thomas Jefferson was the first president of the United States or "Tuesday" is spelled with a "g", and this entire lengthy statement is still a true disjunction because as least one disjunct is true (the sun is bright), regardless of the absurdity or truth-value of the other disjuncts.


With an explantion of the material implication and disjunction behind us, I can better attempt to basically explain the principle of explosion.

A contradiction proves anything. For instance, I can construct a VALID argument (in classical logic) proving that if am not wearing shoes, then I will live to be a thousand years old. This is my argument:

6. I am wearing shoes.
Therefore,
7. If I am not wearing shoes, then I will live to be a thousand years old.

The argument assumes that I am wearing shoes. To simultaneously assume the falsehood of statement 6, then, is a contradiction. And from a contradiction, anything follows. That is what the conclusion is saying. It is saying that given statement 6, if we also assume the negation of the statement 6 (thereby bringing in a contradiction), then I will live to be a thousand years old; given P, it follows that if not P, then Q.

Let me prove the validity.

8. I am wearing shoes.
9. I am wearing shoes or I will live to be a thousand years old. (take statement 8, perform Addition by adding "I will live to be a thousand years old")
7. If I am not wearing shoes, then I will live to be a thousand years old. (take the new statement 9, perform material implication).

As you can see, there are only two steps for deriving the conclusion, and they draw upon the concepts of disjunction and material implication, which is why I reviewed these two concepts to start with. By performing Addition on statement 8, I gain an “either…or” statement. Then using material implication, this “either…or” statement validly implies the conditional statement in the conclusion. This is also a SOUND argument for statement 7. The premise is true: in fact, I am wearing shoes. Because the argument is not only valid but contains all true premises, we are logically compelled to accept that if I am not wearing shoes, then I will live to be a thousand years old. You may even let the first premise—and the antecedent of the concluding statement—say something such as “the sun is bright”, so that you are certain the first premise is true (because you can’t see that I am wearing shoes). Statement 7, must be accepted as true, at least in classical logic. I have explored alternative logical systems that certain philosophers of logic, such as pragmatist philosopher C. I. Lewis, have proposed in place of systems that use the material implication. Many logicians, however, continue to teach the material implication in formal logic, along with its oddities, such as explosion, as a valid inference rule in conducting proofs. With an understanding of IMP and disjunction, it becomes clear as to how a contradiction formally proves any statement.

Similarly, you may be interested to know that every argument with a logically false statement as a premise is valid. For example,

10. Two plus Two equals Seven.
Therefore,
11. Daffy Duck is the president of Hawaii.

An argument is valid if and only if it is impossible for all of the premises to be true and the conclusion false. In the above argument, it is impossible for all of the premises to be true; Two plus Two equals Seven cannot be true. Therefore, it is impossible for all of the premises to be true and the conclusion false. Therefore, the argument is valid.

Similarly, every argument with a logical truth as its conclusion is valid. For example,

12. Red dragons are battling green dragons under the streets of Paris.
Therefore,
13. Six is Three times Two.

It is impossible for the conclusion to be false. Therefore, it is impossible for the premises to be true and the conclusion false. Therefore, the argument is valid.

In sum, explosion is a bizarre concept and only one of the several more counterintuitive aspects of classical logic. Grasping the material implication, disjunction, and the meaning of validity are important for wading through some of these perplexities.

Thank you, Bryan.

From, Kaiden