Why does the Principle of Explosion not make Mathematical Logic inconsistent? [closed]

Question

Step   Proposition  Derivation
1 ------P --------- Assumption 
2 ---- ¬P --------- Assumption 
3 ----- P ∨ Q ----- Disjunction introduction (1) 
4 ----- Q --------- Disjunctive syllogism (3,2) 

https://en.wikipedia.org/wiki/Principle_of_explosion#Symbolic_representation

(¬E) φ, ¬φ ⊢ ⊥ // contradictory premises resolve to falsum/false https://iep.utm.edu/natural-deduction/#H4

So the principle of explosion only derives its results when the above axiom or the Law of noncontradiction is ignored.

End of 5/24/2023 update that supersedes the material shown below

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In Reductio ad Absurdum (RAA), we determine that a proposition P is false when it derives a contradiction. If we use this same derived contradiction as the premises to the Principle of Explosion (POE), we now prove this same proposition P is true. How does that not make Mathematical Logic inconsistent?

Reductio Ad Absurdum
(P → (A ∧ ¬A)) ⊢ ¬P

Principle of Explosion
(1) P → (A ∧ ¬A)
(2) (A ∧ ¬A) ⊢ P
(3) ∴ P

The above seems to show that we can correctly derive both P and ¬P thus making mathematical logic inconsistent.

Logical Inconsistency
two or more propositions are asserted that cannot both possibly be true.

When the principle of explosion: (A ∧ ¬A) ⊢ B ex contradictione [sequitur] quodlibet, 'from contradiction, anything [follows]') https://en.wikipedia.org/wiki/Principle_of_explosion

This diverges from the model of the syllogism because it fails to maintain a semantic link between the terms. Because it diverges from the syllogism it is easy to see that it allows semantic nonsense to be construed as correct reasoning.

Major premise: All humans are mortal.:
Minor premise: All Greeks are humans.:
Conclusion: All Greeks are mortal.:

Each of the three distinct terms represents a category. From the example above, humans, mortal, and Greeks: mortal is the major term, and Greeks the minor term. The premises also have one term in common with each other, which is known as the middle term;

https://en.wikipedia.org/wiki/Syllogism#Basic_structure

Thus maintaining a semantic link between the premises and the conclusion.

The major premise defines a universal property M of set H.
The minor premise defines a G subset of H.

H has the property of M:
∀h ∈ H ( M(h) )

G is a subset of H:
G ⊆ H ∴ G has the property of M

Answer 1

Explosion is a property of logical consequence relations, and thus of logics, that is not trivial: Some logics have it, some don't. So there is simply no sense asking

Why can't we simply get rid of the Principle of Explosion?

If the logic you start with, say classical propostional logic, is explosive, then you cannot get rid of explosiveness and at the same time retain all properties of the intial logic. Rather what you do is change your logic. Whether that change is justified of course is determined by the field your logic is to be applied to. If your field of application is classical math, classical logic is fine and explosiveness a price worth paying, since it doesn't interfere with mathematical practice. On the other hand, if your logic is supposed to model data bases, where inconsistencies are likely to occur, explosiveness is no good. There you should choose some fragment of relevant logic like FDE.

That classical logic is explosive crucially depends on the following properties of its consequence relation:

1. Simplification: A & B ⊢ A
2. Disjunctive Weakening: A ⊢ A v B
3. Disjunctive Syllogism: A v B, ~ A ⊢ B
4. Cut rule: If M ⊢ A and N, A ⊢ B, then M, N ⊢ B

(Here M, N are sets of formulas and commas mean set union.) So, if you want to avoid explosion you must reject one of these properties. Most of the options have been played through in the literature. Parry's logic of analytic implication rejects (2), many relevant logics reject (3), Tennant's intuitionistic relevant logic rejects (4) so that logical implication is no longer transitive. But whatever option you choose, the resulting system is non-classical and that may also be a price to pay for some applications.

If you're interested in these matters of relevance you should read John Burgess's delightful piece 'No Requirement of Relevance', which appeared in the Oxford Handbook of Phil of Math and Logic.

Answer 2

Rather than give you the correct technical answer (edit: okay I ended up going into some technical details, oops), which has been provided, I think I'll try and diagnose where you're intuitions have led you astray.

It seems to me that you think that if a formal system can derive (P & ¬P), then we have to accept it and thus accept the resulting explosion. If I'm right, what you're doing is assuming that whenever we write a logical statement, we're endorsing it. But we're not, we're reasoning meta-logically. That is, to write "S ⊢ Q" is to say " suppose S ⊢ Q is a true statement in the logical system" not asserting it as true, and then trying to make it false later having already said it's true. If we were doing the latter, then yes, we'd have to accept the consequence of every logical statement we write.

But we don't have to, once we get the contradiction, accept the contradiction's truth. Instead, we can (actually, we have to) accept that whatever we used to prove the contradiction can't be true (nor derivable). What we've done is shown that some part of the set of statements IS NOT derivable because we assumed it was derivable and got contradiction. That is, what we've got is the following inference from our exercise:

Let S be a finite set of statements and Q be what we've derived from S. The question is, is S itself and instance of "⊢ S" or not? Suppose S ⊢ Q. But then suppose Q ⊢ (P & ¬P). You think that now we've got (P & ¬P), and given (P & ¬P) ⊢ A ( where A = any statement and its negation), we've now exploded the logic. But on the contrary, because assuming that "⊢ S" leads to S ⊢ (P & ¬P), we know that the assumption "⊢ S" must have been false. And because it's false, we know that the explosion "⊢ A" is also false. To accept (P & ¬P) as true would lead to explosion. But instead we've reasoned that, axiomatically, ⊢ ¬(P & ¬P), therefore anything deriving (P & ¬P) can't be derivable. Then we've shown that, were S derivable, (P & ¬P) would also be derivable. But this contradicts our axiom that ⊢ ¬(P & ¬P). From this it follows that S was never derivable in the first place. But if S is not derivable, then nor is anything of the form S ⊢ Q (Q standing for anything derived from S). But S ⊢ A is an instance of S ⊢ Q. So A is not derivable from S.

For the sake of completeness (in the non technical sense), lets also note that for every possible set of statements, if that set of statements derives A (still symbolising every possible statement, ie: explosion), the same proof above applies. Therefore, there is no possible set of statements that derive A. So not only do we know that a given set of statements S doesn't derive A, but that no set of statements derive A. Therefore "⊢ A" is ALWAYS false, and as such, this proves that explosion is impossible for our logical system. Isn't that a satisfying result we've now got ourselves to put our minds at rest?

When we write "⊢ S" in this (ludicrously informal) proof, we're not committed to it, nor any of its consequences. We just write it down as if it were the case, and as soon as we derive something impossible within the axiom schema and rules of inference our system is using from pretending its true, then we have a direct way of knowing it's false.

I hope I haven't left you in even more confusion.

Answer 3

Here is the question:

How is it that POE [the principle of explosion] does not make mathematical logic inconsistent?

If the axioms of mathematical logic were inconsistent then the principle of explosion would reduce mathematical logic to trivilism where all propositions are true since they can all be derived from that inconsistency.

However, having the principle of explosion does not make the axioms inconsistent. That would require a separate step. To show that the axioms are inconsistent one has to derive some proposition P and its negation ~P from those axioms (not from some other assumptions) prior to using that principle to derive all propositions.

Here is how Wikipedia represents that symbolically:

P, ¬P ⊢ Q

The important thing to note is that we have to first derive both P and ~P. Then we can use explosion to derive Q.

Reductio ad absurdum is a way to derive the negation of an assumption should that assumption lead to a contradiction. Here is Wikipedia's description of it:

In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity"), apagogical arguments or the appeal to extremes, is a form of argument that attempts either to disprove a statement by showing it inevitably leads to a ridiculous, absurd, or impractical conclusion, or to prove one by showing that if it were not true, the result would be absurd or impossible.

That assumption is not one of the axioms. It is only tentatively assumed. That one can derive a contradiction from that assumption and other propositions is often what one desires to do so one can derive the negation of that assumption.

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The OP adds this question:

Why is the Principle of Explosion (that starts with a contradiction) allowed when it is common knowledge that inference stops when reaching a contradiction?

One use of explosion in a natural deduction system is to handle one or both cases of a disjunction to derive a result using disjunction elimination (vE). Here is a proof showing how that works.

To eliminate the disjunction in line 1 and derive the goal P, I have to consider two cases: P and Q. The first case, P, is easy. It is already the goal I want to derive. The second case, Q, is more difficult. However, I also have an assumption on line 2, ¬Q, that I can use to derive a contradiction, symbolized by ⊥ on line 6. Now I use explosion on line 7 to derive P from that case as well. From explosion I could have derived anything, but what I need is P. So I derive that.

Having derived P in both cases I can use disjunction elimination to eliminate the disjunction on line 1, referencing both cases on lines 3-4 and lines 5-7 and derive P on line 8.

This would be one use of explosion in a natural deduction system.

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Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

Wikipedia contributors. (2019, August 15). Reductio ad absurdum. In Wikipedia, The Free Encyclopedia. Retrieved 13:25, September 11, 2019, from https://en.wikipedia.org/w/index.php?title=Reductio_ad_absurdum&oldid=910921053

Wikipedia contributors. (2019, September 6). Principle of explosion. In Wikipedia, The Free Encyclopedia. Retrieved 13:21, September 11, 2019, from https://en.wikipedia.org/w/index.php?title=Principle_of_explosion&oldid=914233106

Answer 4

• (1) P → (A ∧ ¬A)
• (2) (A ∧ ¬A) ⊢ P
• (3) ∴ P

No. The conclusion here should be P → P, via the Hypothetical Syllogism: P → Q, Q → R ∴ P → R .

Since P → P is a tautology there is no problem.

You might be more interested that (A ∧ ¬A) ⊢ P and (A ∧ ¬A) ⊢ ¬P are both valid.

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Why is the Principle of Explosion (that starts with a contradiction) allowed when it is common knowledge that inference stops when reaching a contradiction?

Because it is not common knowledge that inference stops when reaching a contradiction. The very point of PoE is that when a contradiction is derived, anything may be inferred in that context -- including contradictions.

1. A ∧ ¬A   Premise
2. P        PoE 1
3. ¬P       PoE 1

Both these inferences are valid; which means each of the derived statements are considered at least as true as the statement from which they were inferred. Since that statement is a contradiction, P and ¬P are each guaranteed to be at least false.

Answer 5

Sequitur's answer is a good one, so I'm reluctant to add another, but he has addressed only some of the problems in the question, so here goes.

*1. There is no need to 'redefine' valid inference in terms of operations that are truth-preserving. Preserving truth is exactly what classical logic does. A classically valid argument guarantees that if the premises of an argument are true, then the conclusion is true. Expressed using the language of model theory this amounts to saying that every model of the premises is a model of the conclusion. Now bear in mind that in classical predicate logic 'all' does not imply 'some', so statements of the form "every A is a B" are true in the event that there are no A's. An inconsistent set of premises has no model, and therefore in such a case it is trivially true that every model of the premises is a model of the conclusion just because there aren't any. So any argument with inconsistent premises is classically valid.

*2. As Sequitur points out, there are many logics and some feature PoE and some do not. You are at liberty to choose to use some non-classical logic, such as one of the family of relevance logics. But there is a significant price to pay. Classical logic has many desirable features. It is provably sound and complete. It is Post-complete. Substantial fragments of it are algorithmically decidable. Its propositional calculus can be expressed using boolean algebra. It forms a satisfactory basis for first order arithmetic. There is a good reason why most logicians, and the great majority of mathematicians, stick to using classical logic.

*3. One cannot simply pick and choose between rules of inference within a logic. If you wish to dispense with a rule, such as disjunction introduction, it has consequences across all of the rest of the logic. Dropping the disjunction introduction rule changes the meaning of conjunction, negation and conditionals as well. In particular, it sacrifices bivalence: you can no longer rely on the principle that if a proposition is not true then it is false and if it is not false then it is true. Relevance logics have a much more complex semantics of negation.

*4. Disjunction introduction is not as weird as it first appears. For one thing, it is truth-preserving. Constructing a truth table for A v B will show that there is no row in which A is true and A v B false. Relevance logic is not truth-preserving: its natural semantics is typically expressed as something like preservation of information in a channel between sites or possible worlds. There is a section on the semantics of relevance logic in the SEP article.

*5. Although you have edited the question so that it no longer asks whether PoE makes logic inconsistent, you continue to include the 'proof'

(1) P → (A ∧ ¬A)
(2) (A ∧ ¬A) ⊢ P
(3) ∴ P

This is incorrect. The correct conclusion is P → P, which is a tautology and unproblematic.

*6. The PoE is actually useful in classical logic. An inconsistent theory explodes and contains all formulas as theorems. The contrapositive of this is that if there is a single formula that is not a theorem then the theory is consistent. This is used to prove the consistency of theories. It was used for example by Gentzen to prove the consistency of first-order arithmetic.

Answer 6

So the question is:

WILL LOGIC STILL WORK WITHOUT DISJUNCTION INTRODUCTION?

The answer is "work for what purpose?"

What I mean by this is that what you're trying to do here (more broadly), namely "get rid" of the principle of explosion is a traveled path (well in the latter half of the 20th century) broadly known as paraconsistent logics (Note the plural.) Dropping disjunction introduction is one (but not the only) way to achieve this goal--and it yields the so-called relevant (or relevance) logics. But as with most things in life, there are tradeoffs. (And if don't mind some historical backfitting, Aristotle's syllogistic logic is also paraconsistent, by construction, although quite limited in its proof theory.)

As for what could be "wrong" with relevance logic(s), even considering the most (and first) studied fragment that with implication, in the words of Anderson and Belnap (1975; p. 350) its proof theory is "of course not as easy as truth tables".

The semantics of relevance logics are a bit "weird" and with several proposals for "natural" candidates -- some more similar to models of modal logics. Ironically perhaps, understanding possible-world semantics of relevance logics is a bit more involved than even for modal logics, because one needs a [more obscure] notion of neighbourhood semantics for the former. On this angle, Mares' 2004 book is a gentle, self-contained introduction for philosophers. To quote a relevant (pun intended) bit from it:

It turns out that no finite valued truth table can capture relevant implication.

(Alas the proof is technical and you won't find it in Mares' book, but in Anderson and Belnap.)

A more serious problem is that [introducing] quantification in relevance logic is ahem... a real pain or rather that it has some unpleasant/unexpected consequences:

Unfortunately, Kit Fine (1988a) proved that the logic RQ (the logic R of relevant implication together with some standard quantificational axioms) is incomplete over the constant domain semantics.

If this seems too abstract of a problem to you, consider something more concrete: adapting the axioms of Peano Arithmetic (PA) to RQ is no biggie (all it takes is to replace "0 is not a successor" to "0 is not its own successor"). But the theory thus obtained (let's call it "relevant arithmetic") admits as model not only natural numbers, but also complex numbers! It also follows from this that some theorems provable in "classical" PA (stemming from the Chinese remainder theorem) are non-theorems in relevant arithmetic. (See Fridman & Meyer 1992 for proofs. In view of your related question, this latter aspect might perhaps convince you of one application/use of model theory as well.)

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A further problem (see Mares, chapter 9) is that in order to have a sequent calculus with a deduction theorem, one needs to add an intensional conjunction (fusion -- &) and (not strictly necessary, but "looks nice") also an intentional disjunction (fission) on top of the "regular" (extensional) conjunction and disjunction of the relevance logic (R). In the absence of (at least one of) these additional operators in sequent calculus for R, the deduction theorem allows one to e.g. elevate the (extensional) disjunction into a deductive explosion. Fusion behaves somewhat differently than (extensional) conjunction; for example it's not idempotent, i.e. (A & A) -> A is not [a] valid [sequent] in relevance logic.(Unlike the Mares' book, the SEP treatment is a bit terse in this regard, as it only shows the Hilbert style systems e.g for R which does have one but not the other intensional operators in its axioms--one of them is technically redundant, as I said.)

Thus relevance logic plus its sequent calculus... starts to look a bit like some kind of linear logic (Although Mares doesn't make this connection in his book, the SEP treatment by the same author touches on the connection in its last paragraph though.)

Furthermore, it was shown by Read that one can take the intensional operators as a primitive and derive the (relevant, extensional) implication. (see sec. 9.6 "the Scottish plan" in Mares' book.) Also Read's own book is freely available nowadays and if/since you're interested in taking the intensional operators as primitives (given your motivation at the sequent calculus level), it may be a better read for you. See also the SEP entry on substructural logics which is also from this (sequent) perspective, where relevance logics are the sub-family that don't allow "weakening", which is what the latter article calls what you call "disjunction introduction".

Answer 7

Principle of explosion In classical logic, intuitionistic logic and similar logical systems, the principle of explosion (Latin: ex falso [sequitur] quodlibet, 'from falsehood, anything [follows]'; or ex contradictione [sequitur] quodlibet, 'from contradiction, anything [follows]'), or the principle of Pseudo-Scotus, is the law according to which any statement can be proven from a contradiction.[1] That is, once a contradiction has been asserted, any proposition (including their negations) can be inferred from it; this is known as deductive explosion.

Logical Inconsistency
two or more propositions are asserted that cannot both possibly be true.

Principle of Explosion derives P
(1) (A ∧ ¬A) ⊢ P
(2) ∴ P

Principle of Explosion derives ¬P
(1) (A ∧ ¬A) ⊢ ¬P
(2) ∴ ¬P

Since the Principle of Explosion allows P ∧ ¬P to be derived from the same premise the Principle of Explosion makes logic inconsistent.

Answer 8

The POE only came to seem legitimate once mathematicians realised it was a necessary consequence of their system of logic at the time, essentially propositional logic based on the truth table of the material implication.

As they were unable to conceive of a better system, they choose to promote the POE to the dignified position of principle, effectively making mathematical logic contradictory to Aristotle's logic.

The POE is just one of a few mathematical logic's propositions that Aristotle would have dismissed as obviously false.

Successive generations of mathematicians just learn the stuff "at school" and most of them go on repeating all their lives, as if it was the Gospel truth, what they had to learn to get their exams.

This question is one of the most frequently asked question on the Internet, and yet I never saw anyone provide the beginning of a logical justification. All people do is provide irrelevant answers or simply reassert the principle as if repetition could be convincing. There is no possible justification beyond expediency.

Still, the POE is probably the best example of the limitation of human rationality, itself essentially based on deductive reasoning: Garbage in, garbage out. In this instance, start with the premise that the truth table of the material implication is the correct model of logical implication, and get the POE, and then some.