Questions tagged [proof]

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

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What is the difference between "reductio ad absurdum" and "proof by contradiction"?

What is the exact difference between reductio ad absurdum and proof by contradiction? Wikipedia used to state that: Reductio ad absurdum (Latin: "reduction to the absurd") is a form of argument in ...
loudandclear's user avatar
15 votes
8 answers
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Does every truth have to be provable based on evidence?

I know the answer is "no" in general due to Gödel's Theory of Incompleteness, but I mean this question in a more real-world sense (i.e. scientific sense). In other words, I am talking about ...
Lavie's user avatar
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How to prove you are an atheist?

I've been reading a conversation between two individuals - A claiming to be atheist and B asking him to prove it, since B does not believe that A is saying the truth and can't be sure if A is really ...
easwee's user avatar
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9 answers
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Is a proof still valid if only the author understands it?

Some time ago I was reading about the recent Shinichi Mochizuki's proof for the famous ABC conjecture. It's enormous and so incredibly difficult that at that time virtually nobody was able to ...
machaerus's user avatar
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3 answers
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What is the relation between proof in mathematics and observation in physics?

Recently in his 2015 Hirzebruch Lecture in Bonn, Arthur Jaffe re-amplified his famous perspective that finding proof in mathematics is analogous to making experimental observation in physics. In ...
Urs Schreiber's user avatar
12 votes
9 answers
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How to prove (A v B), (A → C), (B → D) therefore (C v D)

Obviously since A → C and B → D then if A v B one of C or D must be true. My only idea is v must be introduced, but how would I use subproofs to show one of A /\ C or B /\ D is never false if A v B?
sumsum2's user avatar
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16 answers
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Can we logically prove that anything exists?

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 ...
user34467's user avatar
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8 answers
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How do we know if a mathematical proof is valid?

Georg Cantor has showed there are more real numbers than natural numbers in his diagonal argument. Assuming that two sets have the same size if we can make a pair up elements from set A with elements ...
user107986's user avatar
11 votes
8 answers
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Is watching an amputated limb regrow proof of the supernatural?

A typical challenge skeptics present when confronted with claims of alleged miracles is "why won't God Heal amputees?". But, would that do the job? Consider the following thought experiment: ...
user avatar
11 votes
8 answers
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What is the correct, pragmatic, reasoning response to conspiracy theories?

It's established that the burden of proof rests on the party making a claim. The problem I find, is that for any conspiracy theory - the proponent can point to a multitude of conspiracy websites or ...
dwjohnston's user avatar
11 votes
10 answers
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What makes something mathematics?

Dictionary.com definition of math: (used with a singular verb) the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. ...
Tdonut's user avatar
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Are there rules for dealing with self-reference "paradoxes" in logic?

My favorite paradox that leads to an endless regress, and also leads to a question: The sentence after this is true. The sentence before this is false. When contradictions appear in proofs, we have ...
hellyale's user avatar
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9 answers
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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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9 votes
4 answers
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Why is Modus Ponens valid?

I am having trouble understanding what defines Entailment operator. On Mathoverflow I posted this question on what I perceive to be paradox of entailment. Consider: Modus Ponens: P therefore Q P ...
Sniper Clown's user avatar
9 votes
6 answers
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Claims that we know (virtually) nothing - can they be refuted?

Here's an argument that I've heard a number of times from friends and on the Internet: "The ratio of what we know about the universe to what we have yet to discover is so small - it is therefore ...
Salim Fadhley's user avatar
9 votes
5 answers
17k views

How does one prove De Morgan's laws for quantifiers?

One of De Morgan's laws state that ¬∃x P(x) is equivalent to ∀x ¬P(x), but how would one go about formally proving this? Numerous attempts to find a solution have been futile, even proofwiki.org ...
DrDeanification's user avatar
8 votes
19 answers
7k views

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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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
8 votes
16 answers
4k views

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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2 answers
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How do proofs about logic fit into a logical framework?

I'm learning logic from Michael O'Leary's A First Course in Mathematical Logic and Set Theory. In chapter 1 he carefully explains the meaning of logical implication (p ⊨ q), logical inference (p ⟹ q), ...
WillG's user avatar
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3 answers
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In what sense are proofs just arguments that convince us, not arguments that establish truth?

In mathematics and logic, it seems that once a proof of some theorem is discovered, then it is taken to be "absolute truth" within the axiomatic system from which it was derived. My question is: are ...
neddo's user avatar
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5 answers
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What is the difference between mathematical reasoning and philosophical reasoning?

Please see question in title. Why isn't philosophy considered to be a branch of mathematics? Is study of anything not a branch of mathematics, vague and imprecise?
John Sonderson's user avatar
7 votes
3 answers
571 views

What is behind Girard's idea of distinguishing implication ( ⇒) and entailment (⊢) without separating language and meta-language?

How should we understand the distinction between ⇒ and ⊢ ? I often see that A ⇒ B lives in the object language and A ⊢ B in the meta language. But I need a different interpretation, without any ...
Boris's user avatar
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4 answers
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Are there any potential flaws in the definition of validity as "provable or falsifiable"?

I have made an argument in another thread that a proposition must be provable or falsifiable to be valid. Are there any flaws in this definition of validity? What might be a potential counter-argument ...
Chad's user avatar
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3 answers
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Are there any known deficits of "relevant logic"?

The principle of explosion is the law of classical logic and similar systems of logic, according to which any statement can be proven from a contradiction. Some early formal systems like Frege's ...
Thomas Klimpel's user avatar
6 votes
6 answers
862 views

What does it mean that a claim is a claim of nonexistence?

This question has devolved into a discussion. As I understand the discussion, everything is revolving around the veracity of statement Nonexistence can never be proven. and on what exactly ...
user avatar
6 votes
3 answers
4k views

Does predicate logic have truth tables?

As I recall in propositional logic, it was possible to draw truth tables for the arguments such as for: (P ∨ R) [I live in Paris or I live in Rome] Therefore, (~P ⊃ R) [If I don't live in Paris ...
cpx's user avatar
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In Fitch, how does one prove "(P → Q)" from the premise "(¬P ∨ Q)"?

It's all in the question really. I am working on a proof in Fitch for a class, but I am very much stuck. I am proving the tautology that "(P → Q) ↔ (¬P ∨ Q)", and I have already finished half of it, ...
Zenreon's user avatar
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Propositional Logic: How to prove the contraposition in the Fitch system?

Given that: p ⇒ q prove that: ¬q ⇒ ¬p using the Fitch system. (This being the proof of the Contraposition)
Am95's user avatar
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Given P ∨ ¬ P prove (P → Q) → ((¬ P → Q) → Q) by natural deduction

I am very new to proof and logic and I would really appreciate a rundown of this proof. I use a program called Fitch to construct my proofs. I understand there are two types of proofs. Direct ...
Metamorphosis's user avatar
6 votes
5 answers
563 views

Is it possible to prove that the universe either is or isn't a simulation? [duplicate]

Can it be philosophically proven that the Universe either is or is not a simulation? If someone was in a simulation, could they tell? What would the differences be between a simulated universe and a "...
TestinginProd's user avatar
6 votes
4 answers
946 views

Can something "count" as TRUE without support by logic and empirical data?

I was in an online debate and had this statement presented to me. I would note further that your apparent positivism rests on what is logically a faith claim - specifically, the unproveble claim ...
randomblink's user avatar
6 votes
1 answer
9k views

The difference between argument, inference, deduction and proof?

I am trying to distinguish argument, inference, deduction and proof. First, let's look at the distinction between argument and inference (if there is one). This online source states: An argument ...
EthanAlvaree's user avatar
5 votes
5 answers
1k views

Rebuttal for modus ponens

Saw this (WP:"What the Tortoise said to Achilles") on the internet. A summary is as follows. The common argument is: A: If p then q B: p C: Therefore q. This raises the following question: what if ...
lagrange103's user avatar
5 votes
6 answers
3k views

Is any aspect of the supernatural testable? What level of proof is possible for the supernatural?

Assume the supernatural does exist, and consists of beings/forces that can interact with our natural universe in ways that are contrary to the natural laws of this universe (at least as we know them). ...
LightCC's user avatar
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3 answers
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Are axioms tautologies?

My understanding is that axioms are the unprovable statements upon which systems are built. Tautologies are in essence things that can't be false. Godel's Incompleteness Theorem, though, shows that ...
Tatarize's user avatar
  • 171
5 votes
2 answers
342 views

Is there such a thing as provability of provability?

Gödel says that there are true statements that can't be proved, given a sound axiomatic system. Does anyone say anything about the provability of the provability of statements? Is it still an open ...
gurghet's user avatar
  • 151
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3 answers
3k views

What exactly are the identity rules in logic?

In first order logic, I have read that there are a couple of identity rules. If I have "a=b" does it mean that I can also write it as "b=a"? Is it true one-way or both? And if I have two ...
cpx's user avatar
  • 527
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2 answers
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How to deal with ¬∃ (negated existential quantifier) in a proof?

I need to prove that the following premises lead to a contradiction. ∀x (P(x) → Q(x)) ∃x ¬Q(x) ¬∃x (¬P(x)) A couple of things are confusing me. Does the first premise say that if x is a P then it ...
Leon's user avatar
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5 votes
5 answers
492 views

Is an argument in natural language as logically valid as in formal logic?

Is a natural language philosophical argument which is argued strictly from first principles widely considered equally as valid as a proof written in formal logic?
tom894's user avatar
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3 answers
745 views

Is it possible to show that something does not exist?

I am aware that I could show that something does not exist with the knowledge of an implication. If B is true if A exists, then A does not exist if B is false But if we have no implications, is it ...
DaPhil's user avatar
  • 259
5 votes
8 answers
307 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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5 votes
5 answers
430 views

A Paradox of Precision?

Yesterday I was talking to one of my mathematics professor regarding the notion of proof in general (whatever the word "general" means to the reader). In short my claim was, We can only be ...
user avatar
5 votes
2 answers
564 views

Philosophical interpretation of the cut rule of Sequent Calculus

It seems that the cut elimination theorem of Sequent Calculus has some interesting consequences. Quote from Alain Lecompte, La logique linéaire et la question des fondements des lois logiques (French)...
Boris's user avatar
  • 888
5 votes
2 answers
718 views

How does one prove properties of soundness and completeness for a logic using proof-theoretic semantics?

Can one prove these properties at all without relying on notions of models and interpretations? Are there other properties that proof-theorists usually prove instead? From what I've read, I've only ...
user393454's user avatar
5 votes
2 answers
557 views

What is the difference between everyday realism and metaphysical realism?

At an everyday level, we seem to subscribe to a from of strong realism which doesn't leave any room for skepticism. We are certain that individuals who hear voices in their heads or who have ...
Alexander S King's user avatar
5 votes
4 answers
1k views

What is the nature of proof in mathematics?

Preamble: I think we have this sort of questions, where we are required to find a solution for them. For example, what is the area of a circle?. I think the way to solve these problems is to try to ...
Mathnewbie's user avatar
5 votes
1 answer
214 views

Establishing Incompletenes of Modal LPC

In Hughes and Cresswell A New Introduction to Modal Logic (1996 ed.) page 271, they attempt to establish the incompleteness of the system K + G1 + BF (where K is L(P->Q)->(LP->LQ), G1 is MLP->LMP, and ...
Double AA's user avatar
  • 175
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1 answer
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Does Descartes prove that he isn't dreaming?

I had a midterm question where this was relevant, essentially it was: "Assuming you're an atheist, how would you prove to Descartes that your last vacation wasn't a dream?" I put that since ...
Tyler's user avatar
  • 204
5 votes
1 answer
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Actual and potential truth for neo-verificationists

Neo-verificationists such as Martin-Löf and Prawitz make a distinction between actual and potential truth of a proposition, roughly defined as follows: ... that a proposition A is actually true means ...
carina's user avatar
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