Questions tagged [proof]

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

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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 ...
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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 ...
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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?
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6 answers
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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 ...
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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-...
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3 answers
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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 ...
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3 votes
2 answers
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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 ...
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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 ...
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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 "...
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8 answers
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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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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 ...
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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 ...
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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 ...
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What are examples of proving post-analytical hypotheses and what is the post-analytical meaning of truth?

I've enjoyed Cornel West's MasterClass lectures -especially the lecture about The Impact of Music on Community- and it made me wonder about verification of post-analytic statements of truths. Before I ...
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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4 votes
2 answers
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Is the Categorical Imperative Simply Bad Math? :)

The title is clickbait, but the question is not. First, The Categorical Imperative: Act only according to that maxim whereby you can, at the same time, will that it should become a universal law. ...
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2 votes
10 answers
658 views

Proof for the absence of free will?

There are a number of arguments which aim to prove the impossibility of free will. The Standard Argument (incorporating the Determinism Objection and the Randomness Objection) is well known and ...
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Help with Sentential Logic Proof

everyone. I'm running into real trouble figuring out the following sentential logic problem. (S v T) ⊃ (S ⊃ ~T) (S ⊃ ~T) ⊃ (T ⊃ K) S v T What we're supposed to get: S v K Any help that anyone might ...
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Is the following derivation of predicate logic correct?

I recently discovered https://proofs.openlogicproject.org/, except I'm still figuring out the appropriate rules that are needed for the system to check my proof. I thought it'd be quicker to turn to ...
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2 votes
5 answers
112 views

Why are there problems we don’t know how to solve?

For example, there is a simple deterministic algorithm for determining whether any given number is a perfect square. But why don’t we know how to solve things like the p vs np problem or the collatz ...
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5 answers
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When can we have certainty in what we claim to know?

This is a question that my friends asked me a few times. I am confused, because I have been trying to answer this question myself, but I can't seem to come up with any valid answer. For example, what ...
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Fitch proofs help?

I'm new to logic and can see how to write these out informally, but need some help seeing how they should be translated into formal proofs in Fitch.
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1 answer
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Why is the use of the ND rule ∃E not correct in this proof?

Is there anyone who could explain to me why these errors occur? It seems to me the rule was used properly.
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2 votes
3 answers
118 views

Prove the following theorem: Need hints to finish it

This is not homework. I do it for fun and learning. I use the Logic Book. Problem has to be done in SD+. How to prove the following argument : |- [~A =>(~B=>C)]=>[(A v B) v (~~B v C )] I ...
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What makes an argument objectively more "compelling"?

If person A gives an argument to person B in order to convince them about the truth of claim X, how can B determine how compelling A's argument is in a way that is as objective as possible (i.e. in a ...
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-1 votes
1 answer
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Help with Fitch formal proof?

I'm having trouble solving this formal proof in Fitch. I've put together most of it, but I think I need to use disjunction elim(?) at some point and am having trouble doing that.
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2 votes
1 answer
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Is there anyway to prove things happen/exist if I'm not aware of them?

I don't even know how to properly ask this, but how can one prove things happen without them knowing? Things only exist for me when I'm aware of them, either by direct contact (I see it, I feel it etc....
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Proof truth is absolute and not relative

First of all, I want to point out I am not any expert in philosophy, so this question is not based in no further readings, but only my own knowledge in logic. Imagine I want to proof the truth is ...
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Is a game player winning against very long odds being a cheat, an example of the logical fallacy of personal incredulity?

There's this Minecraft speed-runner called Dream who has been accused of cheating due to his drop luck (ie receiving useful things in the game). A moderator team has calculated that the p-value of the ...
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1 vote
0 answers
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Looking for the correct english term for [closed]

I just remembered a conversation I had (in german) about 18 years ago and the other person used the (german) term "All Theorie", which in this context was not related to the theory of ...
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4 votes
1 answer
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Does software exist to automatically validate an argument?

I'd like to Derive logical arguments from English, and test their validity using a program. Does software exist for step 2? It would be fine if it were to give up because The validity isn't ...
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2 answers
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How do I prove ∀x(A(x) -> B(x)) from ~∃x(A(x) ^ ~B(x)) using a fitch proof? [closed]

What would the formal fitch proof for this be? This question came up in my practice problems and I'm really stuck on how to proceed. I'm assuming that you start with an assumption, but I can't figure ...
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-2 votes
2 answers
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Is there any proof of assertion that "assertion can be proved"?

Here's some reason to doubt any proof: Dissent – The uncertainty demonstrated by the differences of opinions among philosophers and people in general. Progress ad infinitum – All proof rests on ...
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2 votes
1 answer
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Soundness and Completeness of Tableaux

Tableaux to my knowledge are both sound and complete. The statement: "If P is valid then tableau for -P eventually closes". Does this statement prove that tableau is sound and complete or ...
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Is the included derivation feasible? If so, would my proof be correct?

The simple derivation seems correct and intuitive, and yet I feel as if something is off. I would greatly appreciate it if someone could double-check the provided formal proof. Thank you in advance ...
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1 answer
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How do philosophers answer a question like 'How do you know something exists?'

I recently watched a video from Rationality Rules titled "The Argument from Personal Experience - Debunked (Why Personal Experiences are NOT Proof)". As the title reveals, the video's goal ...
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-1 votes
2 answers
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How would an philosopher and scientist solve the following kidnapping - scenario?

I would like to hear your opinion as philosophers and scientists regarding how you would solve the problem of proof in the following scenario: "Plato" who has dementia and a damaged left ...
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1 answer
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Rudimentary Proof in SL

So I'm currently being introduced to SL and asked to prove a statement. A simple question with the premise L & W and L ⇒ ¬ F. I am asked to prove W & ¬ F. Could someone verify my solution? So ...
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2 answers
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Prove that the following is a logical truth (tautology) using a natural deduction derivation: (B → C) ˅ (¬B → C) [closed]

Prove that the following is a logical truth (tautology) using a natural deduction derivation: (B → C) ˅ (¬B → C) How do I prove this using statement logic? I know I need to start with a supposition ...
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2 answers
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What is the difference between these two types of proofs?

While employing induction method for proving, is deriving the string(formula) "Fn → Fn+1 " any different from showing that if Fn holds true, then so does Fn+1 ? By showing I mean that we use ...
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Functional Abbreviation for Inst Expression in Turing's Paper [closed]

In Turing's 1936 paper On Computable Numbers Page 30-31, and its Correction Page 1-2 For a Turing Machine M, Inst(i,j,k,LEFT,l) means that if M scans symbol j under m-configuration i, then the symbol ...
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How does one prove (A->B)vC from the premise ~A? [closed]

Is the premise really enough to prove this?
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1 answer
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How to prove the following arguments [closed]

I'm trying to do a bunch of proofs to get better at them but it seems like I need some help with negation. Can anyone who has time prove the following arguments? I would really appreciate it! ¬(P ∧ ¬Q)...
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Validity of the Definiton of the Conditional [closed]

Can a proof for Premise (P→Q) ... Goal (¬P∨Q) be derived using only the following rules? Conjunction Introduction Conjunction Elimination Left Conjunction Elimination Right Disjunction Introduction ...
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