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 ...
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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 ...
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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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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 ...
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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 ...
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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: ...
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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?
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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 ...
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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 ...
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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.
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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 ...
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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 ...
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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 ...
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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 ...
user66156
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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 ...
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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
...
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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 ...
user62907
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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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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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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 ...
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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), ...
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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 ...
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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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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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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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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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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 ...
user409
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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, ...
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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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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 ...
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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 ...
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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 "...
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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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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 ...
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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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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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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 ...
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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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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 ...
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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 ...
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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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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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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 ...
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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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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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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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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? ...
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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 ...
user13627
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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)...
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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 ...
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