Suppose I want to prove that negative numbers exist. Well, I could easily do that using a mathematical proof. However, all I would be doing is adding another logical object to a list of known logical objects, none of which appear outside in the real world. It's a bit like proving that you lost in a game of chess by invoking the rules.

Using logic or mathematics to prove things does not relate to the real world directly. You cannot prove objects exist in the real world by using logic because no matter how cunning you are, it still might be the case that the objects do not exist. It is possible that no physical objects exist, but that would not affect your logic.

This explains why all attempts to prove God exists fail. You can't do it just by using logic. You could demonstrate that God exists, but nobody has managed to do that under laboratory conditions. All we have are stories that contradict each other.

So my question is: Can we logically prove that anything exists?

  • Welcome to the Philosophy SE. To get the responses you’re looking for, you’ll probably need to rephrase this a little, and to turn it from a statement of opinion into a clearer, answerable question. – MarkOxford Aug 4 at 13:06
  • I made an edit. You may roll this back or continue editing. You can see the versions by clicking on the "edited" link above. – Frank Hubeny Aug 4 at 13:26
  • No, we can't. But it is not necessary to do so; we exist; planets exist; electrons exist, and so on. – Mauro ALLEGRANZA Aug 4 at 15:22
  • What are your axioms? How do you define "exists"? Philosophers have been trying to answer those questions for millennia, and so far there's no clear winner. That makes your question meaningless. – Sam Aug 7 at 21:21
  • It would help if you were to define 'exists'. Without a definition the question is ambiguous. if you are speaking of metaphysics where 'exists' means 'exists fundamentally' or 'is irreducible', then it is not possible to prove anything exists by any method. Which is just as well since otherwise the Perennial philosophy would be falsifiable. – PeterJ Aug 8 at 11:54

IMO, there are two related but different issues here.

We prove statements : in math and logic we prove a theorem from axioms. There is no way of proving a statement "from scratch", i.e. without assumptions.

The basics of "deductive sciences" are well-known since Aristotle and the same A discussed the issue of infinite regress in the foundations of knowledge.

Every theory (including math and logical ones) can prove the existence of something only in the context of the axioms presupposed by the theory itself.

The same for the deductive arguments about the existence of God; see e.g. Spinoza's Ethics : it deduces the existence of God from axioms and definitions.

In conclusion, there is no absolute (i.e. unconditional, not relying on some assumptions) proof that something exists.


A different topic regards facts : we do not prove e.g. the fact that Napoleon was the Emperor of France from some axiomatic theory of Emperors.

We can assert it because it is a well-known historical fact.

In a similar way, we may appeal to the Bible and say that it provides historical grounds for the existence of God.

But, in both cases, they are not "logical proofs" at all.

  • But shouldn't be this theory be something in any possible context of the axioms? – Francesco D'Isa Oct 1 at 17:35

"If a man will begin with certainties, he shall end in doubts; but if he will be content to begin with doubts, he shall end in certainties." — Bacon, Francis, Graham Rees, and Lisa Jardine. The Oxford Francis Bacon. Clarendon Press, 1996.

-

"If you tried to doubt everything you would not get as far as doubting anything. The game of doubting itself presupposes certainty." — Wittgenstein, Ludwig, et al. On certainty. Vol. 174. Oxford: Blackwell, 1969.


One argument is as such: To absolutely doubt anything and everything, thus namely radical skepticism, results in destroying your basis of inference of radical skepticism by the very nature of doubting itself.

The point I am trying to make is that the very act of doubting can be argued to have tautological implications, such as the Wittgenstein quote above.

(https://plato.stanford.edu/entries/logic-epistemic/)

I use the term, tautological, in the non-pejorative sense as used in epistemology.


"To doubt everything or to believe everything are two equally convenient solutions; both dispense with the necessity of reflection." — Poincaré, Henri. Science and hypothesis. Science Press, 1905.

  • And more generally: the Private Language argument. Once you begin to think, you presuppose the community of concept builders and their context. "To make an apple pie from scratch, first create the universe" - Carl Sagan – CriglCragl Aug 8 at 15:20
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    Thanks for sharing that. :) My answer can certainly be expanded further. – Tautological Revelations Aug 9 at 4:40
  • I added another quote and citation for that quote. It elaborates on this issue. – Tautological Revelations Oct 10 at 22:46
  • @CriglCragl May I suggest writing on this as your own answer either/or suggesting an edit to my answer on the issue of the Private Language argument? – Tautological Revelations Oct 11 at 18:35

The Italian philosopher Umberto Eco offered a funny and insightful answer to this question:

Perché c’è dell’essere piuttosto che nulla? Perché sì.

Translated, it means: "Why is there something instead of nothing? Because."

One can be skeptical about everything, even about skepticism itself, as suggested by Vladimir Nabokov in Transparent things:

Men have learned to live with a black burden, a huge aching hump: the supposition that "reality" may be only a "dream." How much more dreadful it would be if the very awareness of your being aware of reality's dreamlike nature were also a dream, a built-in hallucination!

Everything, except that something exists – indeed, even a mistake or an illusion are something.

So, let's say that,

(1) Nothing exists.

it naturally follows that

(2) (1) is something

therefore

(3) Something exists.

Even if we doubt the working principles of logic, our doubts and mistakes would still be something.

  • 1
    Skepticism is fun, but if you are skeptical about the petrol gauge in your car, be prepared for a very long walk home. – user34467 Aug 9 at 21:58

I would agree with your suggestion: with logic alone, we cannot prove the existence of anything ... we need to make at least some observation.

Indeed, Descartes needed to note that he had thoughts (through introspection, a kind of internal observation) in order to prove the existence of those very thoughts (and, consequently, himself ... as the thought-haver)

Also, a quick note on the existence of mathematical objects. When mathematicians talk about existence (there exist perfect numbers; there does not exist a largest prime number, etc.), they are indeed not talking about 'real world' kind of existence, but rather are talking about logical consistency with some set of axioms. That is, mathematicians simply postulate the (logical) consistency of objects (e,.g. numbers), and then see what logically follows ... and what follows from, e.g. the Peano Axioms, is that there are perfect numbers, but no largest prime numbers. But yeah, in the end, this kind of 'mathematical existence' is not the same as the kind of 'real word existence' as we normally think of it, you're right.

  • Of course if a thing has contradictory attributes, then we can use logic to show that it doesn't exist. – user34467 Aug 12 at 8:56
  • For example, married bachelors do not exist. God commanded Moses to kill the Midianites because they were the enemies of the Jews. He also said love your enemies, do good to those that hurt you, turn the other cheek. – user34467 Aug 12 at 8:58
  • It is not possible to love and hate your enemies. – user34467 Aug 12 at 9:00

We can't prove that negative numbers exist using mathematical proof; they are simply implications of our axioms.

Similarly, in life, nothing can be proven. But this begs the question: what would we gain from proving existence? Sure it would provide closure and confirmation of universal truths, but how would it benefit society in any way?

I agree, however, that it is very interesting to think about how many things we believe will never be and, more importantly, can never be proven.

  • Would you have any philosophers who take a similar view to your view? Quotes and references would help strengthen the answer since it would make it more than an opinion and give the reader a place to go for more information. – Frank Hubeny Aug 4 at 18:44
  • Actually, simple implications of axioms are exactly what mathematical proofs are. And in Real Analysis classes you cover the proof that negative numbers exist. – Alexander Aug 6 at 3:25
  • It is very easy to prove negatibe Numbers exist. Consider the equation x+y=0. If y is greater than zero, then x must be less than zero, which by definition makes x a negative number. Alternating current can be represented mathematically and you can plot this representation on a graph. If you plot a complete cycle, you have to allow negative numbers into your representation. It cannot be done without negative numbers. Mathematically, that is proof. If you claim negative numbers do not exist, then much of mathematics would have to be abandoned. – user34467 Aug 6 at 13:25
  • You can set up an axiomatic arithmetic system without negative numbers. It would be pretty much useless as an axiom system, but you could do it. – David Thornley Aug 7 at 15:19
  • I was using the axiomatic arithmetic system that we already have. Of course you can make up some other system, but once we have agreed on the system's rules, we can then proceed to prove propositions within the system. It's a game. Don't change the rules to make a point. In chess, there are rules. You can't win by changing the rules. It's not allowed. – user34467 Aug 11 at 10:27

This isn't quite a proof, but it seems that in the act of attempting to formulate any proof at all, we must presuppose that there exists something that is capable of doing logical reasoning; otherwise, the whole endeavor would be futile. We sometimes catch mistakes in what we thought was a solid proof, and so it is not clear that we can decisively rule out the possibility that any logical thinking we have ever done is fundamentally mistaken. This is not to advocate extreme skepticism, but it does seem that to get started on any logical reasoning (or at least to have any faith in such reasoning), we have to implicitly assume that something capable of such reasoning exists.

(This seems somewhat different from the Cartesian argument that I exist, which still allows for the possibility that I am -- and perhaps everything in existence is -- in general a highly flawed logical reasoner. On the other hand, the Cartesian argument is at least an argument for directly establishing something, whereas the above is just saying that there is little point in not making a certain assumption.)

  • It is very easy to prove negatibe Numbers exist. Consider the equation x+y=0. If y is greater than zero, then x must be less than zero, which by definition makes x a negative number. Alternating current can be represented mathematically and you can plot this representation on a graph. If you plot a complete cycle, you have to allow negative numbers into your representation. It cannot be done without negative numbers. Mathematically, that is proof. If you claim negative numbers do not exist, then much of mathematics would have to be abandoned. – user34467 Aug 6 at 13:24
  • It is interesting to note that invented concepts have to be defined, whereas natural things are described. For example, we define traffic regulations but describe accidents. When we define God, we choose those attributes that we think God aught to have. For example, God is good, powerful and knows everything. The fact that we make these decisions proves God is a fabrication of our brains and not a real thing. You cannot describe God because God is not available for examination. – user34467 Aug 21 at 9:40
  • I could easily define what I mean by the term "King of France" but to say the the King of France is bald, suggests that the King of France exists, but he doesn't. – user34467 Aug 21 at 9:47
  • Sup pose God showed up and said "I don't know everything. I don't know the future", would you accept this or argue with God ? – user34467 Aug 21 at 9:54
  • Or perhaps you would say, "You are not God, because I've decided that God knows everything". – user34467 Aug 21 at 9:56

Can we logically prove that anything exists?

Before answering the question, we should agree about what we're talking. Discussion upon subjective understanding is useless. My perspective comes from the systems theory.

  1. Existence: this is the concept I prefer: an object system exists for a subject system if the subject can interact with the object. So, if the moon exists for me, that's because I can interact in some way with it (i.e. I receive light, I react creating a symbol in my mind for it). Ergo, existence is subjective. Right, example: God exists for some people (they say they interact with him), and doesn't exists for others. Another example: the quantum of energy started to exist as soon as we have been able to interact with it... on paper, with math. Another: cogito ergo sum means "thinking is interacting with myself, ergo, everytime I think, I exist. Problem: can we be objective about existence? Objectivity is only possible as a shared subjectivity. Even if I try to express my feelings or logical conclusions about the existence of something, they are valid for you only if you are able to be coherent with my understanding. There is no physical bounds between your mind and mine, that would be objectivity. We both follow similar (or different) thinking paths, and agree about the existence of the moon, therefore it objectively exists due our capability of having coherent subjectivities.

  2. Anything, what is it? Here, you are talking about things. Again, things are mass, delimited by mental boundaries. I'm sure that you and me don't agree about what a chocolate box or a tree are. Does the tree include its water? If so, at what distance? Isn't that subjective? Where do clouds end? If that's difficult to know, then, where do rocks end? (think of it: rocks are also fuzzy ideas, but limits are defined at a smaller scale, they seem more coherent than vapor, but both are just atoms). Things can be studied by the systems theory, due to they fit perfectly the definition of system. Things can be abstract, when they have no correspondence, or a loose correspondence to matter, or concrete, if they correspond more or less to an amount of matter. But things exist only in our minds. Matter is just physical interaction (Feynman said something equivalent).

  3. Logically, prove, sounds the same to me: use reason, following logic rules (the rules of nature, that we have learned and systematized, as a tool and a language: math). A logic system is just a calculator: requires of an input value to generate an output result. E.g. "if I think, I exist" is my logic; "I think" is the value I input to the system. The system tells me that I exists. So, What's the input for my logical system to assess if I can prove existence? Again, a subjective valoration. Is that valid?

So, the only possible existence raises in our mind, as a subjective idea, and can be proven using a formal construct named logic, but it requires subjective valorations as inputs. Under such constraints, it's only up to you to decide, propose a logic system and test it using observations raising from your subjective experience.

This largely depends on what you mean by ‘logically prove that anything exists’ (and a bit on what you mean by ‘logical object’). Let’s consider two alternatives.

If you mean: ‘showing that it is a logical truth that something exists’, then the answer is no, or almost no. If a is an individual constant of given logical language, a = a is a logical truth, and so is the existential sentence ∃x x = a – which effectively say that a exists. So, it is a logical truth that something (or, some thing) exists; but that’s as far as it goes: we can’t prove the existence of a second thing, nor can we prove that the existing thing is a physical object.
Some logicians feel that even this is too much. Logic should be subject neutral, they argue, to the extent that the existence of no thing should be a logical truth. This type of approach is known as Free Logic.
On the other hand, Logicists and Neo-Logicists hold that numbers exist of logical necessity, and have tried to show that their existence follows from what they take to be laws of logic. If this can be done successfully, numbers would be 'logical objects' - but that certainly doesn't mean that their existence is trivial! This is one of the best-developed, but also one of most complex and complicated views in formal logic and philosophy of mathematics.

Alternatively, by ‘logically prove that anything exists’, you might mean: ‘prove deductively as opposed to inductively that something exists’. The next question is then what you want to prove this from, i.e.: What are the premises in your proof? If they are once again only logical truths, we are back to the above situation. However, logical truths aren’t the only truths. Take the claim ‘Proxima Centauri is a red dwarf’ as an example. This is certainly not a logical truth, but a truth nevertheless. Now, both the existence of Proxima Centauri itself, and the existence of red dwarfs, follow from this statement. Importantly, they follow by means of a logical rule of inference, or deductive rule, known as existential generalisation.
Just because we’ve used a logical rule in the proof doesn’t mean that the conclusion are truths of logic: that Proxima Centauri and red dwarfs exist, are contingent truths and not logical ones. So, we can prove deductively that something exists, even though the existence of that something isn’t itself a truth of logic. (But that's all we wanted.)
Of course, we haven’t proved the existence of something that we previously thought didn’t exist. However, Quine in his famous paper On What There Is points out that the same method will sometimes uncover ontological commitments that we previously didn’t think we had to make. In particular, he argues (roughly) that because the laws of physics involve numerical statements (like e.g. ‘The speed of light is 299.792.458 m / s’), physics commits us to the existence of numbers. This is called the Indispensability Argument.

You mention God, and proofs of the existence of God; but as it stands, I’m not sure you’re your question is specific to God’s existence. Also, people have certainly done more (whether successfully or unsuccessfully) than tell stories about God: proofs of the existence of God abound, and some even claim to be ‘logical’ or at least ‘analytic’ or ‘a priori’. – Whether these proofs are sound, is a different question, of course.

  • The proof of the logical truth that something (or, some thing) exists, deriving from a=a the existential sentence ∃x (x = a) obviously relies on the assumption that the constant a of the language has a reference, and this in turn is simply the assumption that something exists. Thus, there is no "absolute" proof that something exists. – Mauro ALLEGRANZA Aug 4 at 15:59
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    @Mauro ALLEGRANZA I’m not denying that. In fact, that’s the point I was trying to make when I mentioned Free Logic. Of course, some hold that classical logic is true an absolute sense, in which case there is an absolute proof that something exists. I’m not advocating that view here, but not everybody is a pluralist about logic. – MarkOxford Aug 4 at 19:22
  • Agreed; I'm not a pluralist about logic. What I'm trying to stress is that also in the realm of logic is difficult to assert that we can have an "absolute" (i.e. unconditional) proof of existence of some sort. :-) – Mauro ALLEGRANZA Aug 5 at 9:36
  • @Mauro ALLEGRANZA That’s fair enough. Although, that point isn’t limited to existence proofs. There are statements that aren’t existence claims, which hold in classical logic but not in e.g. 3-valued logic. So, if by no absolute proof we mean no proof that’s valid in every (sensible) logic, then we are left with very few absolute proofs / absolute theorems – in which case someone might say that our notion of absoluteness isn’t very interesting / useful. Still, I think we’re in agreement. – MarkOxford Aug 5 at 10:59
  • Some people say you cannot get something from nothing, yet they are very happy with the idea that God made the universe from nothing ! If it is true that you can't get something from nothing, then it immediately follows that something has always exist. We know that matter/energy exists and if it didn't come from nothing then it has always existed. Theists say that there must have been a beginning, but they are quite happy to believe their magical friend had no beginning. – user34467 Aug 6 at 13:32

I believe you are adding to the confusion by adding "logically" to the question. Can we prove that anything exists?
First of all, there is no absolute proof of any kind. However, there is proof by the preponderance of the evidence!
What this means, is that if you want to prove anything (to a given degree), you need evidence (observations), rather than logic! Of course, you could use logic to gather and present the evidence, but logic by itself, would not be enough.

  • People often claim something about the "nature" of God when trying to prove God exists. This is backwards. You first have to show that God exists before you can examine God's nature. Besides, you should be using the term "supernature". Of course, if you've simply defined God to be what you want, then all you are doing is expanding on the imaginary attributes of an imaginary being. You can say that an angel has wings with feathers without having to admit that such a creature must be a bird and without having to worry about how angels put on their clothes. – user34467 Aug 24 at 13:19
  • Can pigs fly ? If you say it is the nature of pigs that enables them to fly, then you are making an extraordinary claim, unless you mean cartoon pigs, in which case it's a trivial claim. – user34467 Aug 24 at 13:39

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