Questions tagged [proof]
For questions about the correctness of a proof or the nature of proofs in general.
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Does Gödel’s incompleteness theorems imply the necessity of an infinite recursive hierarchy of “proofs”, and that any “proof” is relative?
My understanding is that Gödel’s incompleteness theorems state that the consistency of any axiomatic system cannot be proven within that axiomatic system, and requires a stronger axiomatic system in ...
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The question about the existence of an infinite non-trivial controversy
I wonder if there are some examples in mathematics or non-mathematics of two people who cannot settle an argument, or in other words, what I am specifically looking for is that every argument one of ...
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What is the difference between negation-eliminiation ¬E and contradiction-introduction ⊥I?
I don't understand the difference between the rules negation elimination and contradiction introduction. I am using the Open Logic Project's natural deduction proof checker proof checker. The rules on ...
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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 ...
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Has anyone ever studied which proof types are feasible for which theorems in mathematics? If not, why not?
For instance, when asked to prove that sqrt(2) is irrational, we go straight for the proof by contradiction where we assume it’s equal to a/b in lowest terms and end up with a and b not being in ...
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How do you prove mathematical induction without the notion of a set?
EDIT - Peano's axioms for N can't be used to answer this question, because they assume induction. So what axioms can be used? I am thinking the following:
P1. x ∈ N iff x=1 ∨ ∃y (x=y' ∧ y ∈ N)
P2. 0'...
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How can objects be nonexisting?
A square circle. Obviously, this is contradictory, but i feel odd saying it doesnt exist as well. thats not the bestw ay to say it.
but, then again, whatg do we even mean in mathematics or logic by ...
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Why take Panpsychism seriously?
This question is about Panpsychism, but it is also a question I have more generally about the requirement for proof in philosophy.
Panpsychism is the view that mentality is fundamental and ubiquitous ...
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Approaching the knowledge of knowing using premises
The only thing(knowledge) that I know is that I don't know anything/I know nothing. (Universal Quant.). Does this mean that I don't even know that I know to begin with?
The only thing(knowledge) that ...
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What is the Normal Form of Proofs in Intuitionist Logic?
I came across the concept of a normal form of proofs in Neil Tennant's A New Unified Account of Truth and Paradox (2015). I did a quick scan on SEP, and it seems to be a concept specific to his ...
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How to prove the world is real, and one isn’t just hallucinating everything? [duplicate]
I'm not asking whether other minds are real or not, I'm asking whether the whole shebang is "real".
Google Gemini says it's impossible to know/prove it, and basically says to "move on&...
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How is this logic valid?
An excerpt from Logic 2010:
In particular, what is confusing is that it permits assuming the conditional but then reaching a contradiction to prove the conditional. In my experience, that is not a ...
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Converting a Euclidian proposition to a syllogism format
I am attempting to analyze Euclid's proof demonstrating that the interior angles of a triangle are equal to two right angles (book 1, prop 32). In particular, I'm looking for a way to convert the ...
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Can someone help me with this? Logic using natural deduction
Does anyone know what I am doing wrong? I think I am missing how to pass line 12 of the subproof to the general proof. Can someone show me whats wrong? Thank you.
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Trying to prove ¬A ∨ B from A → B, but a bit stuck on how to end the proof
This is my first time trying out proofs, and I am not entirely sure what I am doing wrong, so criticism is happily received!
My proof so far is as follows:
A → B
Subproof 1:
2. A ∨ ¬ A [Assumption ...
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Do the incompleteness theorems need the provability predicate to be expressed, or can they be expressed via just ⊢?
In his "Epistemic Set Theory," William Reinhardt says:
It is the purpose of this paper to formulate axioms for Gödel's modal operator B for provability (see [3], [8]) in the context of set ...
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Can science prove statements to be true?
I know that mathematics can prove certain statements to be true, which we call theorems. But what about science? Can science prove statements to be true? Note, just because a statement can't be proven ...
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Do people who "withhold judgement" also have a burden of proof?
I will illustrate my question with several examples involving 3 individuals: A, B, C.
Example 1: The shape of the Earth
A defends the claim that the Earth is round.
B defends the claim that the Earth ...
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What is the burden of proof? Has this principle ever been challenged?
I have been surprised to find that some people doubt this principle. Onus probandi incumbit ei qui dicit, non ei qui negat - the burden of proof lies with the speaker, not with the one who negates. I ...
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Do atheists bear the burden of proof in showing why/how the reasons presented by theists are unconvincing?
In conversations and debates between atheists and theists, is it enough for the atheist to assert that they are skeptical of theism without providing justification, or does the atheist bear the burden ...
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Can fatalism be disproved by fate-changing magic?
In a world where magic exists, fate-spinners are people with supernatural powers that influence the chances of events happening.
Their power has been proven by countless experiments where they compare ...
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Why not move from proof numbers to theories instead of theories to proof numbers?
In mathematics, they do this thing where they figure out what are called "proof-theoretic ordinals" (see this section of the SEP article on proof theory for background details) of theories, ...
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Is non-deterministic automated reasoning a viable strategy for solving problems in mathematical logic?
EDIT 2023/10/06
There are objections that this is too technical to be philosophy, and while I've seen questions on this forum go far beyond what I'm asking here in set theory, computability theory, ...
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Proof of the existence of God?
Here it is, the long-awaited proof for the existence of God (for your consideration).
I have taken the liberty of defining discretely what God is, without which there is no question to be answered (...
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Does theism have the burden of proof?
I have heard that agnosticism seems to be the only position with respect to god that doesn’t have a burden of proof. What I find troubling about this is most people do not as a practical matter think ...
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Is there a way to confirm something material is infinite?
Mathematics is full of immaterial examples of infinities. However, is it possible to confirm or prove something material is infinite? Or, can we only conjecture they are?
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Proof for predicate logic
Prove the following formula in Fitch format:
∃x(∀y(P(y)→y=x)∧P(a)) |= ∀x∀y(¬(x=y)→(¬P(x)∨¬P(y)))
I tried to use universal introduction as my main rule but didn't know how to proceed
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Peirce cuts (mirrored) + demi-negation = demisets?
[Note: I found one essay, about Aristotle, that used the word "demiset," although at a glance it seemed like they might've been substituting this terminology for a counterpart to the subset/...
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Would the imaginary unit be the truth-value of sentences formed using √𝐧𝐨𝐭?
Section 4.3 of "Sentence Connectives in Formal Logic" discusses a concept of demi-negation or what is (for the sake of the text) resolved to a concept of "the square root of negation&...
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Is the idea of weak and strong evidence incoherent?
Hypotheses are either true or not. Justin is either a murderer or not. There is either a God or not.
If statements are either true or false, how can it be the case that certain forms of evidence ...
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Why do some proof method feel more satisfying than others?
Let's say we are asked to show that 1+2+3.. =n(n+1)/2, then a very simple way to prove this is to use induction. The proof is simple, consider P(1) and show P(n+1) from P(n). However, it feels quite ...
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Is solipsism truly unprovable?
I know I’ve asked a lot of these but this one I’m hoping to be definitive.
I have heard pretty much everyone say that it’s unprovable, that you can’t know for sure if it’s true or not because of the ...
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Jurisprudence and logic: Is it a necessary criterion for a claim to be declared sound that there be no evidence to the contrary as to its soundness?
So, I've been generating various arguments (such as related to the synthesis of legal arguments), and I have been doing my best to figure out how to declare that a particular claim is not sound. For ...
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Can, "This problem is unsolvable," be used to formulate the first incompleteness theorem in erotetic logic specifically?
Assumptions/definitions: the Gödel sentence is informally equivalent to, "This sentence can't be proved in system X," where X is appropriately specified. Since that sentence can itself be ...
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Does the preface paradox undermine long mathematical proofs?
Descartes, IIRC, somewhere says something about the vagaries of memory influencing our justification for believing in our memory, and thence for believing in proofs involving many steps that we have ...
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Does this proof of God I made work? [closed]
I worked up this proof of God, and I want to know if the proof works. Here is the proof:
Something which is unlimited is limited by not having limits. If limits were to be imposed on something which ...
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Why is Diogenes the Cynic's solution to Zeno's Dichotomy Paradox insufficient?
According to Wikipedia's discussion of Zeno's Dichotomy paradox (emphasis mine),
According to Simplicius, Diogenes the Cynic said nothing upon hearing Zeno's arguments, but stood up and walked, in ...
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Proving God and believing only provable things
This question has two, related, parts.
Part 1.
I recently have been on some discussions regarding proving God exists and proving that Jesus is really the son of God; intertwined with that, there was a ...
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Can someone formulate John Pollack's argument in understandable terms for laymen?
http://www.strongatheism.net/library/atheology/ontological_argument_for_nonexistence/
By the way, I am not asking whether this argument is sound or not. I just want to have it formulated in ...
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Need help with this Symbolic Logic Proof please
I am having trouble solving this proof. Line 5 is wrong, I know it's Demorgan's Law, but the proof machine doesn't accept that as an answer. I think it only accepts ~Elim, vElim, vIntro, ~Intro, &...
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Fitch Proof Help, Conclude ~B from ~(A > B) [closed]
I'm in the process of learning fitch proofs and I've come across one I'm having trouble setting up.
Premise: ~(A > B)
Goal: (A & ~B)
In other words, it looks something like this:
1 | ~(A > B)...
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How to argue statistics using logic?
Typically in logic, we have the axioms as facts which are 100% true but in statistics we have things which are true sometimes. For example, "the coin is heads" is true 50% of the time when ...
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What is the relationship between algorithms and logic?
Is an algorithm (cooking a dish, Grover's/Shor algorithm, etc.) a form of deductive reasoning or inductive reasoning, and if not what exactly is the relationship between an alogorithm and logic?
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Can we doubt all knowledge?
Can we doubt all knowledge from all sources (perception, reports, and reason)?
Regarding doubting reason, reason can't be proven, it is preceived and judged instantly by our logic, but what if our ...
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What, if any, are the most important claims to be considered proven in the absence of observation; ie: claims derived from logic alone?
This question arose during contemplation of whether free will and/or moral responsibility might ever be proven real or illusory.
Galen Strawson has proposed a proof for the impossibility of moral ...
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I've been working on this for way too long :/
I've made a lot of progress on the proof below, but I am stuck on the last steps where I need to add existential quantifiers back in: ¬∃x ∃y Smaller(x,y)
For context, I'm a logic novice, but I'm ...
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is there such a thing as non-constructive computational “proof”?
For the problems which can’t modified into a constructive proof, is there some useful notion of proving them to some computational approximation?
I’m referencing:
“Interpretations come at a cost: for ...
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What is the Difference between a Claim and a Fact? [closed]
I know for a fact a computer must somehow add numbers but since I can only suggest it, it is forced to be a claim.
In that case, how could I ever state anything as a fact if I can only ever claim it?
user50018
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Is the debate on free will over? [closed]
I've never posted on here but I am interested in philosophy. I think a lot about free will / determinism / compatibilism. I always felt like I have some degree of free will. I know free will is ...
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