# Irresistible Force Meets Immovable Object

Updated on August 31, 2014

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## An ancient quandary.

Remarkably, an ancient phrase that means "contradictory" or "nonsensical" - "zìxīang máodùn" (自相矛盾), originates literally in the phrase (in Chinese) "spear and shield." The ancient fable goes like this: a merchant was selling both the spear that no shield can withstand and the shield that no spear can smash. What happens if you use that spear against that shield?

That the phrase "spear-and-shield" is used in Chinese to indicate "contradiction" is suggestive. This is what we want to analyze here. In a sense, we have started at the end but let's also take some lessons out of this already. It seems that the logic of this Chinese language, does not catch us by surprise - so, no nonsense about cultural dependence of meaning can fly in this instance (which does not prove, of course, that it cannot happen in general.) Our own logical puzzle about an irresistible force meeting an immovable object seems exactly parallel to the logical challenge presented in the Chinese story. Notice, also, that, if the above information is accurate, the Chinese phrase that has originated from the story does not exactly see a puzzle or challenge: rather, the point is that we do have a contradiction - not a problem or puzzle to solve. Nor do we have a paradox. In discussions about the force-object combination for us, respondents often will see it as a genuine problem or they might declare that there is something paradoxical about it. This could be happening because the case is not understood; or because some words - like "paradox" - are misused. Let us take a closer look into all this. For now, mark that the problem is not in physics or in how nature works! Or is it? It is not - but let's see how we can understand this.

What if the laws of nature make it possible to have such a force - an irresistible force - and such an object - an immovable object? Still, IF it is the case that we end up in a contradiction, that doesn't change even if, hypothetically, it is naturally possible to have such items. This shows that the issue is logical. It can happen - it turns out - that we have a physics that might be forcing on us a revision of logic itself! This is a radical and controversial point. It might sound wrong if you think hard about it. Einstein was uncomfortable with it. Yet, one branch of modern physics, Quantum Physics, runs us into that kind of situation. Nevertheless, you see that, even in such a cutting-edge instance, we are discussing logic.

The bottom line is this: if we don't understand it, how can we also "pretend" we understand when we do the physics? Contradictions are incomprehensible. To state both that "it is Monday right here now" and "it is not Monday right here and now (for the same indexicals of place and time)" leaves us completely in the dark. The comprehension we are talking about is not a matter of psychology: subjectively, one might go through the motions of "getting it" but they are not really getting it. It cannot be understood.

Let us be more careful about important details. Strictly speaking, we are talking about logical possibility. We take contradictions to be logically-impossibly-true: a contradiction cannot be logically-possibly true; it cannot be true in any situation that is logically possible. This is not a matter of imagination but of logic. You can surely write a creative short story about the spear that smashes every shield there can ever be and the shield that withstands every spear there can ever be: then, have the two meet and the spear hit the shield with force. What will happen? How would you continue the story? But, look: if we have a logical problem here, there is nothing creative imagination can do to resolve it! You need to understand this.

Surely, one could continue the story somehow. There might be cartoons in which the "irresistible force" moves the "immovable object." Something then happens. The author of the story might continue. The question is this: is what happens next logically possible or not? If it is not logically possible (can never be true), yet it is presented as true, people can "buy" it but it is still not true; it is absurd and remains (logically) absurd and believing it does not make it not being absurd. This is key. We cannot say "this may be logically impossible to you but it is not logically impossible to me." Try an example to see this. It is like "a triangle has three angles." This sentence cannot logically-possibly be true. Why? It seems simple now. It is the meanings of the words - the meanings of "angle", "three", and "triangle." The issue here, least you are confused, is that we cannot ever have a private language. The meanings of words are to be taken as settled and fixed - not open to arbitrary subjective revision. Otherwise, we could not use language to communicate. Sure, meanings of words change over time - well, if "triangle" ever meant "having four angles" then it is the sentence "a triangle has three angles" that would be logically impossibly true or necessarily false. Same game!

Let us learn this: let's say that the name of the meaning expressed by a sentence that can be used rightly in language is "p." p can be true or false - not both and not neither. A contradiction occurs when any such meaning, p, is put down as both true and false. This cannot logically be true in any logically conceivable setup. "p and not-p" is like a "triangle does not have three angles." In "p and not-p" the key words (whose meanings makes the whole impossible to be true) are not words like "triangle" or "angle": it is the words "not" and "and." These words are called logical constants. In the triangle-sentence, it was non-logical words whose meanings fixed it - made the whole meaning impossible to be true. Here, in "p and not-p" it is the logical words "not" and "and" that do the same.

In language, contradictions are asserted. But this is not the point. Such sentences are not correctly assertable. Those who assert them do so wrongly. There is a school of thought that treats, at least certain, contradictions differently but that is an advanced and more sophisticated subject in logic and, surely, we have to start with more fundamental matters. The Chinese we saw in the opening of this entry takes the spear-and-shield situation to be a contradiction and, as such, logically impossible. Do you know what this suggests, by the way? That, to this extent, the logic of the Chinese langauge has only "true" and "false" - so anything that is put down as both true and false (a contradiction) is automatically ruled out. This is what we mean by logical nonsense or logical absurdity. There seems to be a moral to the story too. The merchant who promises both an irresistible spear and an unbreakable shield is the ultimate con-man charlatan -- like our own advertisers, for the most part, because many many commercials, when analyzed, can be shown to imply contradictions! A rational person should not accept as true something that is false. Most propositions we make with sentences in everyday language are contingently true-contingently false. It is possible for such a proposition to be true; and it is possible for it to be false (not in the same situation in which it is true - that would be a contradiction! - but in some other situation.) It is not irrational to take a contingency for true, or for false (unless I have seen the evidence for it or against it.) But it is automatically irrational to take a contradiction for true! Because a contradiction, as we have seen, can never possibly be true - it is not a matter of what situation we are talking about. In the spear-shield case, it is sort of dramatically obvious that we have a contradiction but more often than not it is not so obvious. The ideal rational person does not "buy" it but the everyday reasonable person may still fall for it; but, then, the catch is that we need logic to assist us so that we act ideally.

Now, there may be solutions to the spear-shield (or the force-object) challenge. Let's be careful here. If it is provable that we have a contradiction, there can be no solutions. Some presumed solutions (we will check them out) work by redefining some of the words in the story. But, then it is not IT that is solved; a different story is created.

For example:

1. When irresistible force meets immovable object, then the force bounces off but still keeps on going while the object is not moved; this, however, redefines the terms of the story: "irresistible" implies continuation in the same direction, so bouncing off is not permitted. Sure, there is no contradiction if "irresistible" means "not ever been made to come to a stop." This is a weaker meaning, though. The punch of the story requires the stronger meaning - that "irresistible" also includes not being made to bounce off or to move in some other direction instead of going through.
2. What if they both disappear when they meet? This is not logically absurd. Notice, logically absurd (like a contradiction, which cannot possibly be true) is not the same as "exotic," "weird," "very unlikely," etc. So, sudden disappearance seems physically impossible - but is it logically impossible? I'd say, no. Is this a solution - which would show that we have a solvable challenge and not a logically impossible situation? But disappearance suggests failure to achieve what the words "irresistible" and "immovable" promise. I'd think that our story has between the lines: "if irresistible, then successful in keeping on moving." This sentence is not like " a triangle has three angles" (which is logically necessarily true given the meanings of the words in it.) Or is it? It hinges on the meaning of the word "irresistible" and also in what the context suggests about the use - meaning of the word in the story. If this sentence ("if irresistible, then successful in keeping on moving"), then we will end up with a contradiction again. Because, if it disappears, the force is not successful, and so not irresistible; but it is supposed to be irresistible; so, both it is and it is not irresistible.
3. Any other solution you might think of, see if it falls into the same trap - if it forces a redefinition of the terms or some falsification of a principle that seems to be inserted in our story.

Sometimes, people say that we have a paradox here. Let's be clear first on what "paradox" means. Here is the definition we are using: a paradox is a proof that is impervious to criticism but proves a conclusion that cannot be true. For instance, one of the famous ancient proofs by Zeno of Elea that conclude that nothing can possibly move. Yet, it cannot be true that nothing can move. Yet, supppose that there is no flaw in Zeno's proof. That is a genuine paradox. If there is a flaw, then we don't have a genuine paradox. The proof doesn't work, and that's it. We had a pseudo-paradox or a puzzle that we have solved, It is possible, of course, that we might think we have a genuine paradox only because no one has detected the flaw yet. It is a deep question whether there are any genuine paradoxes - or just puzzles we haven't been able to solve yet by finding the flaw in those proofs.

Why would the force-object depiction be a paradox. The point must be that such a situation is logically possible but, also, we can prove that it is not logically possible. But why are we saying that this situation is logically possible? Maybe it seems possible that we could have an irresistible force and an immovable object - but are we right in assuming this? Why not say, instead, that it is impossible for both those things to exist and this is what our story shows?

Let's do some analysis of the story. It seems that the two cases (and there are more such stories, by the way, some of them ancient) - the shield-spear and the force-object - are of the same kind. So, it shouldn't matter which one we analyze. A pedantic point to make is that "force" here is itself an object - an unstoppable object.

## Analysis

1. There is an object x, such that, for every object y it meets, given that x is not the same with y, x goes through y.
2. There is an object z, such that for every object y it meets, given that z is not the same as y, y cannot possibly go through z.
3. Let's call the object in (1) "a". So, from (1): a is such that, for any y, given that y is not the same as a, if they meet, a goes through y.
4. Now, let's call the object in (2) "b." From (2) we have: b is such that, for any y, given that y is not the same as b, if they meet, y cannot possibly go through b.
5. Note that our "y" can be anything! So, we choose a from (3) and we plug it into the y of (4). We have (see (4)): b is such that, given that a is not the same as b, if a and b meet, a cannot possibly go through b.
6. Our "y" can be anything, as we know. Now we take b from (4) and we plug it into (3). So, (3) now gives us: a is such that, given that a is not the same as b, if a and b meet a goes through b.
7. The story tells us that a and b are not same.
8. Given (7) and (5): when a and b meet, a cannot possibly go through b.
9. Given (7) and (6): when a and b meet, a goes through b.
10. Let us ASSUME that a and b meet. This is the scenario we are contemplating.
11. From (10) and (8): a cannot possibly go through b.
12. From (10) and (9): a does go through b.
13. From (12): since a goes through b actually in the scenario, it is possible for a to go through b.
14. From (13) and (11) - we join them together: a cannot possibly go through b and a goes through b.
15. (14): that's a contradiction - logically absurd proposition; we can symbolize this with "!"
16. The ASSUMPTION (10) has taken us to a contradiction. Every other line in this proof is accounted for - warranted or justified. (10) was the only tentative assumption. So, we have (10) => !. This means that the proposition made by (10) is logically impossible - cannot be true.
17. So, from (10)-(16): (10) cannot be saying something true. So, the scenario given by (10) - that "a and b meet" - is logically impossible.

What if the proof has a flaw? We need to know what was used in the proof and, so, what we can give up to get rid of the proof. It seems, however, that only valid rules of inference have been used. The method that took us from (10) - the tentative assumption - to the end - falsification of the tentative assumption - is a famous proof method we find again and again in Math and in many famous philosophic texts: names for this proof method are: Reductio ad absurdum, Proof by Contradiction, Indirect Proof Method. Without accepting this proof method, we cannot have most of our math.

Other rules that were used - they are all valid:

1. We used the rule of universal substitution: for "any y" we plugged in the name of any thing. This seems right. A wrinkle here - an advanced topic - is: what if the predicates you have (for example, flying-horse in "all flying-horses") do not have any existent things having those attributes? In our case, this would not matter. We could actually write out a proof in which our reductio (see above) is applied to the existential proposition "there is an x such that is an irresistible force, and there is such a y such that y is an immovable object." We could get a contradiction, as we did above. Then we have to negate "there is an x such that x is an irresistible force and there is such a y such that y is an immovable object." This is interesting. Here we need a rule of inference known as DeMorgan's Law: negation of both gives us either not-the-first or not-the-second. So, we would prove that, logically, it is impossible for both these objects to exist - at least one of them (possibly both) do not exist!
2. We used a rule known as existential substitution. Notice that we plugged in for "some x" or "some z" labels or names that mark objects. The catch is that for "some x" we should be putting in an object that is specific and we don't know anything else about it. For instance, "someone is a professional football player in this room": you are not at liberty to plug in any name you like from the names of all the people in the room. Assume also that you don't know which one is the football player. You mark the spot, so to speak, by using a letter (a label, or name) that is just plugged in: "d is a professional football player in this room." Here is the catch: any other names you may have used already in the proof - we CANNOT use any of them to plug into this existential sentence. The "d" I used should be NEW. This makes sense: we don't have justification for using any one name we like here. This is unlike the case of "all x" where we can indeed use any name we like. That was called, above, universal substitution (or instantiation.) What we are discussing here is the rule for existential substitution.
3. We used a valid rule of inference known as Modus Ponens: given, p=>q and p, we can correctly infer q.
4. Except for the tentative assumption we made - which we then negated on the basis of the absurdity we got - all the other assumptions were justified. For instance, the assumption that the things - named "a" and "b" in our proof - are not the same thing was an assumption given in the story. Incidentally, you would also get into absurdity by taking it to be the same thing that has both attributes!

© 2014 Odysseus Makridis

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