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Questions tagged [proof]

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

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12answers
14k views

Can you prove anything in philosophy?

I don't understand philosophy very well, and so I am wondering whether you can "prove" anything in philosophy. It always seems you can go a layer down, and find another question, almost endlessly ...
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13answers
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Does a negative claimant have a burden of proof?

I have often heard it said that the burden of proof is on the positive claimant but not on the one making a negative claim. A person claiming, "God exists" has a burden of proof but not a person ...
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2answers
7k views

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 ...
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3answers
407 views

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 ...
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14answers
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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 ...
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11answers
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What makes something mathematics?

What classifies something as math? Is "math" simply performing operations with a certain set of axioms in mind? Is "math" anything that involves numbers? What about mathematical logic? Google ...
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10answers
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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 ...
11
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8answers
1k views

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 ...
11
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11answers
932 views

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 ...
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3answers
1k views

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, ...
9
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4answers
2k views

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 ...
9
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8answers
826 views

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 ...
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6answers
788 views

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 ...
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4answers
488 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 ...
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3answers
1k views

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 ...
7
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3answers
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Proving 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 does ...
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5answers
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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?
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4answers
747 views

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 ...
7
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2answers
426 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 ...
7
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2answers
344 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 ...
7
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3answers
682 views

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 ...
6
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6answers
496 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 ...
6
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2answers
1k views

Mill's Proof and Nozick's Experience Machine

I'm trying to grasp how Mill's claim that the only good is happiness/pleasure is able to respond to Nozick's though experiment. Humans strive for virtue and other goods only if they are associated ...
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3answers
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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 ...
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5answers
502 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 "...
6
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2answers
977 views

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 ...
6
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1answer
168 views

Is it provable that epistemically possible (possible for all I know) does not imply possible?

Here is an argument that it is not. Let's start with some equivalences: X is epistemically possible iff X is possible for all I know iff not (X is impossible given what I know) iff X is not ...
6
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1answer
226 views

Does anyone have a proof checker they prefer using for modal logic?

I am looking for a proof checker for modal logic using natural deduction or sequent calculus. I am trying to learn Isabelle, but I think there should be a simpler solution. Although I can rely on ...
5
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5answers
715 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 ...
5
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5answers
378 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 ...
5
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2answers
236 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 ...
5
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3answers
888 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 ...
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2answers
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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 ...
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2answers
1k views

Origin of the idea that something can't be proved, only disproved

Does anyone know if the idea that something can't be proved, only disproved has a specific origin? I often hear it and would like to make a reference to it in a term paper I'm writing. Is it from the ...
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3answers
502 views

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, ...
5
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2answers
438 views

Can there be true conclusions without assumptions?

I was thinking of the sentence "I think therefore I am", which I had for a long time considered indisputable because it's self-evident. Then I considered the hypothetical situation where my ...
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7answers
278 views

Is there any difference of terminology between a person who would never worship God and one who will upon proof?

What I mean is that I've seen atheists being asked what would it take for them to worship God. Ultimately their answers amount to, "nothing will make me worship God - even if God exists - because I ...
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1answer
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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 ...
5
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1answer
175 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 ...
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1answer
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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 ...
5
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2answers
108 views

Multiple universal quantifiers in an argument

Consider the argument ∀x∀y((S(x,a)∧ S(a,y))→S(x,y)), ∀x¬S(x,x) ├ ∀x(S(x,a)→¬S(a,x)) My approach to formally proving this was to first eliminate ∀x and use x0 as the free variable. Then afterwards ...
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6answers
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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). ...
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2answers
769 views

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 ...
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3answers
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Do picture proofs of the Pythagorean theorem make it empirical?

As I understand it, the Pythagorean Theorem, which defines the metric for Euclidean space, is said to be strictly mathematical in the sense that it is derived from a set of purely theoretical axioms (...
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3answers
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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)
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1answer
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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 ...
4
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1answer
235 views

How to Prove P(a) → ∀x(P(x) ∨ ¬(x = a)) using Natural Deduction

How would a formal Fitch proof look like. I am given P(a) → ∀x(P(x) ∨ ¬(x = a)) to prove using Natural Deduction of predicate logic. I am confused on how to proceed with the proof. Please advice me ...
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5answers
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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?
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3answers
474 views

Help with simple deductive proof

I am taking a class on natural deduction for the first time and we are currently on deductive proofs, I am having trouble with this one: Premise: A Premise: [(A&B) or (C&D)] Conclusion: not (...
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1answer
229 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)...