# Can it be proven from pure logic that at least one thing exists?

Can pure logic alone prove that at least one thing exists? And if so, how about at least two, three, ..., infinitely many objects? Personally, I believe that logic can't even prove that at least one thing exists. But what have philosophers of logic written about this matter?

• Most logics just assume at least one thing exists. The logics that don't make that assumption are called "free logics." "Exists" here is in the mathematical sense, though, not the sense of existing in physical reality. Commented Aug 5 at 23:59
• After writing this question, this question is a thing that exists, isn't it? So the question in itself is a proof, or the question needs to be made more clear. Commented Aug 6 at 13:17
• This question makes no sense right now - it does not say whether with "one thing exists" you mean "one object within the realm of logic (in the mathematical sense)" or "one object in the physical universe". Leaving the people writing answers to guess is not effective, or even cruel, looking at some of the existing answers.
– AnoE
Commented Aug 6 at 14:50
• if talking about mathematical objects, in Zermelo-Frankel set theory, the axiom of infinity asserts the existence of a set with specific properties. Once you have a set, the axiom (schema) of separation gives you the empty set using any contradiction like `x != x` as the predicate. Ofc, whether ZF is consistent is a rather sticky question, so you might not believe that the things that "exist" in ZF, "exist" in whatever sense you mean in this question. But to take a very easy case, if by "exist" you mean "exist in set theory" then everything in the constructible universe is proven to exist :-) Commented Aug 6 at 17:54
• @JD, I mean it makes no sense for the StackExchange format to have a question that is so vague that every answer just picks their interpretation and goes ahead. Didn't mean it doesn't make philosophical sense to think about this kind of question.
– AnoE
Commented Aug 7 at 9:57

To expand a little on causative's comment. In standard logic, it is a convention to assume that the universe of quantification is non-empty, which is to say that at least one thing exists. It we don't make this assumption then the basic rules of inference for first order logic become more complicated. For example, we can prove that something exists using natural deduction as follows:

``````1. (∀x)(x = x)        axiom of identity
2. a = a              1, universal instantiation
3. (∃x)(x = a)        2, existential generalisation
``````

This proves that some thing exists, that we have chosen to call 'a'. We could choose to formulate a logic that does not permit this inference, but we would have to change our rules and make them unnecessarily fiddly for most purposes.

We might justify the assumption that the universe is non-empty by asking why you would want to reason about a universe where nothing exists. If you wish to allow that there are things that have possible or potential existence without commitment to whether anything actually exists, then there are logics for doing this. They are called free logics, and they have different rules for handling existence and names. Also, there is a thing called inclusive logic or universally free logic that allows for an empty universe.

• Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Philosophy Meta, or in Philosophy Chat. Comments continuing discussion may be removed. Commented Aug 7 at 18:49

Just the fact that you are pondering this question confirms that you exist – according to Descartes, see Cogito ergo sum.

Also Anselm considered his ontological argument to be a proof of the existence of God just by logic, see Proslogion.

• @Rushi It is my answer to the OP's question "But what have philosophers of logic written about this matter?" - One can also add Anselm's attempt of an ontological proof of God's existence. - I for myself, I'm neither convinced by Anselm nor by Descartes. Commented Aug 6 at 5:36
• @Rushi: This doesn’t assume Descartes — it’s presenting an argument, and attributing it to Descartes. It does assume the reader accepts the correctness of the argument — but no more than any conceivable argument relies on the reader accepting its basic validity. (Cf. Lewis Carroll’s What Achilles said to the Tortoise.) Commented Aug 6 at 14:55
• @PeterLeFanuLumsdaine The thing with Descartes' radical scepticism is it only works for him. Because when he starts from saying I assume nothing, for him it's true. But when Jo or you repeat it as an argument or whatever it contradicts itself and loses its authenticity. Remember: Descartes called it Meditations and rightly so. It's not generalizable philosophy Commented Aug 6 at 17:04
• @Rushi: Descartes isn’t asking us to just accept the results of his argument; he’s inviting us to repeat his thought-experiment ourselves, and draw our own parallel conclusions. Descartes concluded that his own thinking self exists, in at least some sense. Parallel to that, I can conclude that I must, in some sense, exist; you can conclude the same for yourself, and so on. Commented Aug 6 at 17:10
• @Rushi "Everything that has a beginning has an ending. Make your peace with that and all will be well." Commented Aug 7 at 23:34

Easy one: In order to prove anything from "pure logic", one has to assume the existence of something called "pure logic". Hence from pure logic one can derive the existence of at least one thing: logic itself. QED

• That is, you have to assume that logic is consistent. Otherwise, the things you conclude from this logic may not be true. Commented Aug 6 at 14:46
• As @returntrue wrote, you have to assume consistency. But you also have to assume that self-reflective statements (statements about pure logic) can be made (without leading to contradictions) in "pure logic". This is not at all self-evident. Commented Aug 7 at 12:24
• The assumption of abstract object realism is -- strongly disputed in philosophy. While I agree with the assumption, it is very far from universally accepted. Relying upon a disputed premise makes a proof -- not at all compelling. Commented Aug 8 at 15:57
• @Dcleve If logic does not exist, how could one infer anything from it? Commented Aug 8 at 20:04
• That one can use logic -- was not considered an effective refutation of materialism, despite your "logical" reasoning basically calling for a more complex ontology than either material or mental monism. It took a stronger argument-- that physics shows that matter is not essential, to accomplish the rebuttal of materialism. Similarly, most physicalists today do not consider physicalism to be matter/abstraction dualism (despite my confidence that physics makes no sense unless one recognizes this). Again, convincing me is not sufficient. Commented Aug 8 at 20:13

logic by definition, I'll refer to it as a tool even though it may not be as accurate, is a tool that infers the truth of one statement based on previous statements which are given to be true. there cannot be a truly logical statement or expression that does not make assumptions or use previously proven statements which are known to be true. as in:

• the forecasters are always right;
• the forecasters predicted that it will rain tomorrow

therefore it will rain tomorrow.

in conclusion, in order to prove something from pure logic, you have to make an assumption first, and an assumption is by definition not a definitively true thing, therefore in any logical system there has to be an axiom at the bottom, and unproven and unprovable truth.

• Welcome! Make sure you take some time to read the FAQ by clicking on the question mark in the menu bar. :D
– J D
Commented Aug 6 at 16:02

No.

We cannot fully justify any assumptions, per the Munchausen Trilemma. https://philosophy.stackexchange.com/a/64646/29339 And one must start with assumptions to then do logic.

Additionally, there are an infinity of DIFFERENT logics, not, as is assumed in the question, One True Logic. https://www.cambridge.org/core/journals/think/article/guide-to-logical-pluralism-for-nonlogicians/EDFDFA1C9EB65DB71848DABD6B12D877

Given an infinity of different logics, one cannot actually "prove" anything using "logic". One can only provide proofs for "logic X under assumptions Y", where as noted above, the assumptions are not "logically" or otherwise justified, as likewise the selection of logic is not justified.

There is no need for a long answer.

• I always wished we could make it logically impossible to ask inane questions, but, alas... We'll have to settle for pointless. Commented Aug 7 at 23:26
• +1 I was wondering if you were going to throw your hat in the ring on this one. So "Given an infinity of different logics, one cannot actually "prove" anything using "logic"..." is a fascinating answer. I'm confused by it though. One cannot prove anything using logic, but one can provide logic proofs with extra-logical assumptions. But if cannot prove anything in logic, how is it that it you call it a logic proof? I see scare quotes, but I don't know how to read them.
– J D
Commented Aug 8 at 2:53
• @JD Logic proofs are far more limited than normally presumed in general usage. As in "IF we assume conditions A, B, C hold, AND logic system Z applies, THEN we can construct a proof within Z that is valid and shows D is the case" is the best we can ever come up with as far as "logical proof". There are no general logical proofs. We can also usually show that A, B and C are suspect, and Z is not applicable across our whole universe. Commented Aug 9 at 18:41
• @Dcleve In other words, given all of the potential flaws of a proof, it is as the logical mechanism of proof is too fallible to be considered proof in a more general epistemological sense?
– J D
Commented Aug 9 at 18:46
• @JD -- Yes. I am a radical empiricist with respect to rationalism. I consider rational claims to basically reduce to empirical claims, and rationalism is a mistaken approximation, and is at its core analog rather than absolutist. Commented Aug 9 at 19:20

Gorgias the Sophist says ...

1. Nothing exists
2. Even if something exists nothing can be known about it
3. Even if something can be known about it, it can't be communicated
4. Even if it can be communicated, it can't be understood
• This reminds me of what I heard a lawyer illustrate as a stereotypical defense where the defendant is unwittingly conceding a little ground with each subsequent denial: "That's not my dog; and if it is my dog, he didn't bite you; and if he bit you, he was provoked." Commented Aug 9 at 20:52
• @PeterRankin 🤭 hahaha Commented Aug 9 at 21:58

No, it cannot. We must axiomatically declare that something exists, or we might obtain free logics instead.

Of course, this depends on one's choice of logic, since choosing a logic includes choosing axioms. If one chooses axioms like classical logic, then the Axiom of Existence is independent of the other axioms, meaning that the other axioms are valid without Existence: pure classical logic cannot otherwise prove that at least one thing exists. For a formal statement, see the note of Levein 2005 that Metamath axiom `ax-6` is independent of its neighboring axioms.

Note that the situation is not known to be so simple for intuitionistic logic in general. In intuitionistic Metamath, axiom `ax-i9` is not known to be independent of axiom `ax-4`, and the root cause is likely Metamath-specific choices of quantifier introduction combined with the intuitionistic need for weaker axioms in general.

• And free things are worth every cent. Commented Aug 7 at 23:28
• Especially free things that don't cost a penny (since there isn't even one penny in the whole wide void world). (My wife once watched over my shoulder what I was writing, and commented: "That seems really indecent: public void... People shouldn't do that.") Commented Aug 8 at 2:50
• @mudskipper programmers are a naughty bunch :-) Commented Aug 8 at 11:59
• @mudskipper Sometimes it's unavoidable but usually it should be private or at least protected. Commented Aug 8 at 20:06
• @JimmyJames ERROR: StackExchange overflow: class control pun counter limit exceeded. BDFL has hardcoded limit to unity. Please RTFM.
– J D
Commented Aug 10 at 4:15

Can pure logic alone prove that at least one thing exists?

Logic doesn't prove things. We do, using hopefully logically arguments.

(a) You prove a conclusion true by deriving it, hopefully logically, from premises that we believe are true. (b) So we cannot prove anything logically without assuming something first. (c) To prove that something exist, we have to be able to assume first that something exists. (d) So, we cannot prove that one thing exists without assuming first that one thing exists. The proof will be valid, but it is probably not what you are asking for. (e) Still, we do know that something exists, so we can assume that something exists, which proves logically that at least something exist. If you don't like this proof, there is no other available.

And if so, how about at least two, three, ..., infinitely many objects?

If time is infinite, and if there is an infinite number of things, then it is at least just conceivable that the number of things that we are, in theory, able to prove is infinite.

Personally, I believe that logic can't even prove that at least one thing exists.

See above.

• If "one thing exists" is meant in the sense of "there is at least one thing in our empirical universe", then this is probably the simplest and best answer to the OPs question. If "one thing exists" is meant in the sense that mathematicians sometimes use (for instance as axiom "There is an empty set" or "There is an infinite set"), this answer is also valid (but see @Bumble). Commented Aug 7 at 16:51

Can it be proven from pure logic that at least one thing exists?

Yes, and we can use the meta and object language distinction to prove a thing exists, where this answer is the meta language, and we consider for an object language a simple linguistic proof. For the purposes of this answer, the metalanguage is the answer to your question, and we define the object language to be a fragment of text where existence is demonstrated by modus ponens which is symbolically `(P->Q,P)->Q`.

P1: If a gavagai meets the requirements of a existence proof, the thing exists.
P2: A gavagai meets the requirements of the an existence proof.
Therefore:
C: A gavagai exists.

Now, if we accept modus ponens, and we accept a gavagai is a thing, then we have proven using "pure" logic (which I take it as a mental and not physical method), that at least one thing exists, in this case a gavagai. Of course, this is a sentential logic proof where our proof of existence is constrained to the use of propositions. It would be an entirely different matter to establish proof with a different evidential theory.

Of course, our proof was only a proof existence in the sense it would satisfy a logician, and there are other proofs for existence depending on how one interprets the term 'existence' in your original question. This wouldn't satisfy the existence of a gavagai for a scientist, because in that case, the existence would require a physical standard of proof, at a minimum, an operational definition and some empirical evidence in addition to a logic proof.

• Logicians are easily satisfied, I guess. Commented Aug 7 at 23:30
• @ScottRowe Sure. They can simply appeal to Plato's Forms to solve all of life's problems. Scientific realists on the other hand are accountable to the physical universe. That why I find mathematical physicists such a weird bunch. There's dark matter over there! There are strings over here! Look, it from bit in the participatory universe if you gaze at your navel long enough! ; )
– J D
Commented Aug 8 at 0:47