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

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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

Answer 4

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.

Answer 5

Short Answer

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.

Answer 6

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

Answer 7

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.

Answer 8

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.

Answer 9

You ask:

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.