Proof is a chain of reasoning using rules of inference, ultimately based on a set of axioms, that lead to a conclusion.

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If supposing that a statement is false gives rise to a paradox, does this prove that the statement is true?

If supposing that a statement is false gives rise to a paradox, does this prove that the statement is true? Let me attempt to be a little more precise: Suppose you have a proposition. Furthermore, ...
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175 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 ...
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38 views

help with deductive proof

∀x (Fx ∨ x=c), ¬Fb ∧ Gb |- ¬Fa → Ga So far I don't understand how to switch variables around to make it true. I've got a subproof set up assuming ¬Fa and concluding Ga In that proof I make F(b) ∨ b ...
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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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1answer
47 views

If many people show positivity, is that irrefutable proof of no negativity?

I want to know how we can analyze the number of preferences versus lack of preferences, or dislikes versus likes, and determine if the balance or lack thereof signifies a truly positive affirmation or ...
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1answer
49 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
46 views

Help with a modal Hilbert-style proof of (□(a>b)&◊(a&c))>◊(b&c)

Can't grasp how it can be proved. To proof just propositional calculus formula (without modal operators) at first seems rather natural to me. Tried the law of importation scheme but it didn't work ...
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1answer
73 views

showing c=a v b=c from a=c v b=c?

I am currently using the book Language, Proof and Logic So far I have 1. a=c v b=c subproof 2. a=c =intro 3 c=c =elim 1,2 4. c=a =elim 2,3 end subproof new subproof 5. b=c 6. c=c =intro 7. c=b ...
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78 views

The Customer is not Always Right [closed]

This issue has always been a philosophic issue to me, sense I have worked at McDonalds for the last six years of my life. On minute one of my first day of training, I was told "the customer is always ...
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789 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 ...
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178 views

If there is anything that could behave like a god, is it then a real god?

Let's assume that the our universe's physics allows time travel. This is my main assumption! Let's further assume that there is an quite intelligent, invulnerable being, that found a way to achieve ...
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62 views

Showing b=a from a=b in Fitch Style System

I'm in the middle of completing a proof, and need to show that b=a. I already have a=b. I think this should be trivial, however, I cannot find a way to do it inside the system. Using the programs ...
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2answers
62 views

How are 'proof of inexistence' and 'proof of impossibility of existence' different?

What is the difference between proof of inexistence (A) and proof of impossibility of existence (B)? Does A implies B? Does B implies A? I know that there is a scientific axiom that says 'proof lies ...
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2answers
155 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 says anything about the provability of the provability of statements? Is it still an open ...
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232 views

Evaluating proof of conspiracy theories is time consuming.

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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1answer
105 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 ...
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1answer
156 views

Prooving ( ∃x Fx ∨ ∃x ~Fx) from no premises

I have a proof problem which I am stuck on. { } ⊢ ( ∃x Fx ∨ ∃x ~Fx) I already figured it out in order to get a∨b, I will have to do ~a→b or ~b→a. However, I have absolutely no idea how to get to ...
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2answers
165 views

Prove P v ~P using most basic rules?

Is there a way to prove P v ~P in basic inference rules? I can't think of where to start because nothing applies to this. I was thinking about usinig Conditional proof, but I don't know what should I ...
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What are Takeuti oridnal diagrams and how do they describe numbers?

I understand chiefly from writings of Solomon Feferman that Gaisi Takeuti invented ordinal diagrams for This is the following one line answer from Wikipedia: Takeuti (ordinal diagrams): Takeuti ...
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3answers
99 views

Can we add to PA a new predicate T such that for every sentence A of the old vocabulary the new theory proves T(Godel numeric number of A) iff A

I am new to logic but I believe this is not a difficult problem, yet I am still soo confused, and the reason for that is because there are so many gaps in my knowledge or maybe I have overlooked so ...
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3answers
140 views

I've confounded myself with this formal logic “proof”

Let "r" mean "it's raining" and "s" mean "it's snowing." "->" is "implies"; "V" is "or" (inclusive); "~" is "not"; "^" is "and" Here's the "proof": (1) (r->s)V(s->r) is true because it's a ...
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What can suffice as a scientific proof for God? to what domain can such a proof belong to? [closed]

"Scientific" theories require proof, and there are certain guidelines and standards for the proofs to be acceptable to the "scientific" community in that domain (Algebra, Computer Science, etc.). ...
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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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174 views

Negative facts and truths

I know Russell and Wittgenstein argued about negative truths. It is easy to prove the existence of some property provided there is considerable empirical evidence for its existence, but what if we are ...
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357 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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274 views

How to check if this proof is valid?

I'm having some doubt if this proof is valid or invalid, especially regarding the line 4 derived from line 2. Do I need to change the letter in there? 1. (z)~Fz ∴ ~(z)Fz Using Indirect proof ...
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1answer
414 views

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 ...
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806 views

On validity of Modus Ponens

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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823 views

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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365 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 ...
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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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180 views

Underlying logical structure behind Descartes' Ontological argument?

Quoting SEP: Version A: Whatever I clearly and distinctly perceive to be contained in the idea of something is true of that thing. I clearly and distinctly perceive that necessary ...
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Is there something in the real world that cannot be proved nor disproved? [closed]

It is often said that you cannot prove nor disprove God. People who bring forward this kind of reasoning often try to persuade you that there is a kind of balance, a truce; you can't prove your point ...
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310 views

Does the law of excluded middle follow the law of excluded middle?

Example is that the word unknown is not unknown so it doesn't define itself. Similarly French is not French, it's in English, and "long" is not long and only 4 letters short. That's by example my ...
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Is illogical = not logical?

I think law of excluded middle makes sense to mean that a statement should be either logical or illogical but in this case I don't assume "not logical" = "illogical" since the author didn't say ...
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6answers
373 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 ...
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14answers
4k views

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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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 says: Reductio ad absurdum (Latin: "reduction to the absurd") is a form of argument in which a ...
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393 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 ...
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203 views

Are the limitations of language proof against the divinity of holy books? [closed]

Many religions like Islam and Hinduism have holy books(the Quran and the Vedas, respectively) which claim some kind of superhuman origin. However, are the limitations of languages then counter to ...
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9answers
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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 ...