Questions tagged [proof]

For questions about the correctness of a proof or the nature of proofs in general.

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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 ...
Kristian Berry's user avatar
3 votes
10 answers
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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 ...
user107952's user avatar
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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 ...
Mark's user avatar
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3 votes
9 answers
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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 ...
Meanach's user avatar
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19 answers
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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 ...
Mark's user avatar
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2 votes
2 answers
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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 ...
Zachiel's user avatar
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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, ...
Kristian Berry's user avatar
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3 answers
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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, ...
J D's user avatar
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6 answers
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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 (...
Jordan Cote's user avatar
8 votes
10 answers
4k views

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 ...
thinkingman's user avatar
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1 answer
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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?
Jérôme Verstrynge's user avatar
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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
Richard Gasquet's user avatar
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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/...
Kristian Berry's user avatar
3 votes
1 answer
182 views

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&...
Kristian Berry's user avatar
-1 votes
4 answers
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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 ...
thinkingman's user avatar
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1 answer
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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 ...
tryst with freedom's user avatar
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5 answers
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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 ...
DarkNeos's user avatar
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2 answers
58 views

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 ...
Dennis Francis Blewett's user avatar
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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 ...
Kristian Berry's user avatar
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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 ...
Kristian Berry's user avatar
-3 votes
1 answer
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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 ...
Matt K's user avatar
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4 votes
3 answers
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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 ...
Robert Columbia's user avatar
1 vote
4 answers
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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 ...
Bernardo Benini Fantin's user avatar
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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 ...
Bernard Eakins's user avatar
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0 answers
75 views

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, &...
GarretBobbyFerguson's user avatar
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1 answer
50 views

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)...
Cam J's user avatar
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2 answers
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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 ...
tryst with freedom's user avatar
5 votes
6 answers
1k views

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?
quanity's user avatar
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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 ...
AZeed's user avatar
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8 answers
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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 ...
Futilitarian's user avatar
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-2 votes
2 answers
127 views

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 ...
elemental123's user avatar
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0 answers
32 views

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 ...
J Kusin's user avatar
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1 vote
1 answer
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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?
Richard Bamford's user avatar
3 votes
6 answers
501 views

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 ...
kristian7's user avatar
1 vote
1 answer
346 views

Does Gödel’s findings boil down to part of classical mathematics (as opposed to computation) is flawed?

According to artificial intelligence researcher Joscha Bach, only classical mathematics is affected by Gödel’s incompleteness theorem however not computation where calculations are performed in a step-...
Matthias Nehlfink's user avatar
2 votes
3 answers
698 views

Gödel’s Incompleteness Theorem: How can truth go deeper than proof?

My current understanding: Math starts with a set of basic (purportedly self-evident) statements that are taken as a given without the need to prove them true, like e.g., a + b = b + a etc. Such ...
Matthias Nehlfink's user avatar
3 votes
2 answers
229 views

Reading list to build a proof-based foundation in ethic

How to tell right from wrong, good from evil, and just from unjust? How to define what is moral, what is not? Given a situation, what actions is neither animal-like nor machine-like, but human? To ...
Dick Grayson's user avatar
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1 answer
56 views

Natural Deduction Proof with double quantifier (predicate logic) [closed]

Premise: (∃y)(∀x)(Px v Py) Conclusion: ~(∀y)(~Py) I'm starting out assuming the negation, i.e., (∀y)(~Py). But then I'm unsure how to find a contradiction within that subderivation. Here's what I've ...
qwerty1005's user avatar
3 votes
1 answer
122 views

How should we characterize the logical structure of wishes?

The motivation for this question is extraordinarily stupid, but it requires just enough thought and specific knowledge of formal logic that I think it still falls within the broad scope of "...
R. Burton's user avatar
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5 votes
8 answers
308 views

Understandable definition of time

What is a thorough definition of time in terms of how it causes the universe to progress and only moves in one direction? Is something as abstract as time comprehendable to us beyond a measurement? ...
Gabash's user avatar
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1 vote
0 answers
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Nested Quantifiers Proof - Logic

When I prove this: -∃x.P(x) ⊢ ∀x.-P(x) [True] I did it like that: ∀x.-P(x) ⊢ ∀x.-P(x) because (negative ∃) -∃x.P(x) becomes ∀x.-P(x) so that we can say that it's true. However, I didn't ...
bladeavis's user avatar
-2 votes
1 answer
102 views

Proofs of propositional logic truth tree rules in natural deduction?

It is a great irony of natural deduction that some of the most seemingly obvious inferences are also some of the trickiest to prove! So far, I haven't been able to prove the following, and I'd greatly ...
Spailpín's user avatar
1 vote
2 answers
181 views

How to understand a proof by contradiction in minute detail?

I am following the course "Language, Proof, and Logic" from Stanford on EdX. I am trying to understand proof by contradiction specifically. I understand the gist of this type of proof, and I ...
xoux's user avatar
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-2 votes
1 answer
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Help me solve this predicate logic proof theory: -(∀z)(∃y)Tzy ⊢ (∃z)(∀y)-Tzy

-(∀z)(∃y)Tzy ⊢ (∃z)(∀y)-Tzy Using Lande, (AI,EI,AE,EE,->E,->I,-I,--E,^I,^E,vI,vE) I cannot figure out the proof on this sequent.
Shaffer's user avatar
-1 votes
2 answers
429 views

Predicate logic proof solve

Provide a proof for the following using FOL in forallx Use the natural deduction system and proof strategies in forallx to provide a formal proof for the following . Please provide a picture of your ...
Alexis's user avatar
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1 vote
2 answers
80 views

Does ◇◇A mean ~◻~~◻~A? If so is it by definition or it requires a proof?

In system K, ◇A is defined to mean ~◻~A. Therefore, it is very tempting to conclude ◇◇A means ~◻~~◻~A. But I am not certain whether this is valid conclusion to make, because in ◇◇A, the main operator ...
Josh's user avatar
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-2 votes
1 answer
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Proving ~◻p → ~p in System K [closed]

I am working on a proof of ~◻p → ~p in System K. It says "If it is not the case that p is necessarily true, then p is not true". I have turned all the abbreviated symbols into their ...
Josh's user avatar
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0 votes
0 answers
268 views

Proving validity/invalidity of a modal argument

□(A v B) → (□A v □B) ...(1) This symbolic argument is intuitively invalid. In (1), if we replace B with ~A, then we see that though the antecedent is necessary, the consequent is a contradiction since ...
Abdul Muhaymin's user avatar
1 vote
0 answers
115 views

How do logicians think of strength of proof systems?

I want to understand how logicians reason about strengths of proof systems and argue relative strengths of proof systems. I want to appreciate the validity of the reasoning by which we establish ...
Ajax's user avatar
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0 votes
1 answer
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What are sufficient grounds for establishing a theory?

This question delves into the definition of a theory, but somewhat into the grounds of Set Theory as well. I was wondering on what grounds is theory established and accepted. To what degree do the ...
Andrew Su's user avatar

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