# Questions tagged [proof]

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

187 questions
2answers
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### Solving a proof in which the goal is the negation of a variable in Fitch

I'm working on an assignment and I'm stuck on this proof. I feel like I'm on the right track but I can't find the way to prove the goal. A ^ B (A ^ ~C) --> ~D A -> ~C (B ^ E) --> (C v D) ~E I ...
1answer
246 views

### Clear and canonical examples of analytical proofs in philosophy

I am used to the proof methods of mathematics, and I have formally studied formal logic and mathematical logic. I like philosophy but have never followed a rigorous course in analytical philosophy. ...
2answers
146 views

### Do α and β entail each other?

Show whether the following is true or false: α |= β or β |= α, for any two formulas α and β I'm assuming here that α and β are formulas, not a set of formulas. My thought is that I can prove that ...
2answers
60 views

### How to find a stance towards a controversial topic

When is a stance towards a topic "proven"? To create this example I will take the anti vax topic. My first impulse is: anti vax people are stupid. They ignore basic science. I myself would (...
3answers
173 views

### New riddle of induction; does the observer know the arbitrary time t?

Wikipedia, in "New riddle of induction", sets out Nelson Goodman's paradox as follows: Goodman defined grue relative to an arbitrary but fixed time t as follows: An object is grue if and only if ...
1answer
163 views

### Using predicate logic, how to solve symmetric and anti reflexive

The networks is: A->B->C->D The channels used by the network are: lo, med, hi h-hi, l-lo, m-med i) A network uses one, and only one channel. ii) Networks within close proximity cannot both use the ...
1answer
246 views

### How does a truth tree provide positive and negative effect tests for implication?

I'm trying to prove that the truth-tree method can be used to give a positive effect test for implication, and a negative effect test for non-implication. I've been given the fact that The truth-tree ...
1answer
79 views

### Correct Way of Handling A Corollary of A Corollary?

I have a conclusion S that is moderately interesting. While the corollary of S is more interesting, the corollary of the corollary of S is extremely interesting. Should I just label them corollary 1 ...
2answers
47 views

### Structure of an if and only if proof

I am trying to get this proof to work out and so far I feel like I have the first part right but I'm stuck on how to get the A→B part.
15answers
14k 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 ...
3answers
270 views

### Fitch Proof Question

I'm having trouble with a proof and I'm not sure if it's valid or not. If it appears to be invalid, we are supposed to assign names to the letters in the proof and check it in a World, but when I do ...
9answers
1k views

### 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 ...
3answers
641 views

### How to find redundant premises?

Suppose I have this simple argument: ( P ⊃ Q ) R P Therefore, Q Here's the truth table I made in an excel worksheet: As you can see this comes out valid but in reality it isn't valid ...
2answers
168 views

### Where can I learn about the philosophy behind mathematical and logical proofs?

I'm looking for something that dives into the philosophical idea of a "proof," and explains how the subjects of mathematics and logic deal with it. Does anyone have any book or article recommendations ...
4answers
3k 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 ...
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 ...
1answer
110 views

### Why cannot the following theory be refuted by logic but is rejected because of lack of empirical support?

The following statements are taken from a book: The man in the street, and also the philosopher K. Marbe, believe that after a run of seventeen heads tail becomes more probable. This argument has ...
1answer
254 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 ...
2answers
982 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 ...
11answers
1k 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 ...
2answers
59 views

### Logical equivalence proofs

Trying to master logical equivalence proofs out of a textbook is proving to be difficult. I’m hung up on these four problems. I can make some progress, but usually get stuck towards the very end. Any ...
1answer
85 views

### De Morgan for Quantifiers Formal Proof: Inhabitance Question

I have a problem reading existing question's answers. I've successfully encoded the proof of one of De Morgan's Laws, ¬∃x P(x) → ∀x ¬P(x), for Quantifiers formally via cubicaltt type-checker via Curry-...
3answers
146 views

### Is anything not proven impossible therefore possible?

Is it a truism that, except for that which is proven impossible, everything is or must be considered possible? If so, why? It seems to me to be an argument from ignorance to say that just because we ...
1answer
133 views

### De Morgan for Quantifiers Formal Proof: ∀∃-intro and -elim Questions

I have a problem reading existing question's answers. I've successfully encoded the proof of one of De Morgan's Laws, ¬∃x P(x) → ∀x ¬P(x), for Quantifiers formally via cubicaltt type-checker via Curry-...
3answers
146 views

### 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 (...
1answer
261 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 ...
1answer
82 views

### How to prove or disprove validity of ◻◻p → ◻p in the frame (Q,<) of the rational numbers with the usual less-than ordering?

I was wondering if someone can perhaps help with this proving. I am not sure how to handle a temporal frame that is not a set of only natural numbers. Any suggestions? Thank you.
2answers
72 views

### I need some help determining the validity of the following argument

“I got the highest grade on the last test and I have perfect attendance. If I get a cold, then I miss at least one class. I came down with a cold. Therefore, if I missed at least one class, then I ...
2answers
40 views

### Proof with conditional introduction

Below is a screen-cap of part of a video where a proof using conditional introduction is shown, which is proving under certain assumptions that given A is true, then the adjacent sentence is also true....
5answers
192 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?
2answers
127 views

### Prove or disprove ~◇◻p → ◇◇~p in system K

How to start with the following proof? Any help would be appreciated. I have tried by assuming the left side is true, however, I get confused with the negation. ~◇◻p → ◇◇~p
3answers
8k views

### 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 ...
3answers
157 views

### Modal validity & vagueness

(2nd version to make explicit my implicit assumptions about A, B and C, and the definitions of the non-logical constants "⊂" and "≡".) Intuitively, the following modal argument seems valid to me and ...
4answers
281 views

### Is the Kalam cosmological argument scientifically provable?

Kalam Cosmological Argument: (1) Everything that has a beginning of its existence has a cause of its existence. (2) The universe has a beginning of its existence. Therefore: (3) The universe has a ...
1answer
349 views

### If there is anything that could behave like a god, is it then a real god? [closed]

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 ...
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, ...
2answers
83 views

### How to get the refutation of (OP ⊃ OQ) ∴ O(P ⊃ Q) in Deontic Logic

In Deontic Logic, one could easily infer "If it is obligatory that P, then it is Obligatory that Q", from "It is obligatory that if P then Q" O(P ⊃ Q) ∴ (OP ⊃ OQ) Where the ⊃ is an implication (...
2answers
470 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 ...
1answer
84 views

### Language Proof and logic Chapter 13 problem 31

I have been working on this problem for over an hour and I think I have simply missed something. I need some help. I don't see how this is supposed to work out Here are the premises: ∀x ∀y[Likes(x,...
2answers
121 views

### Language proof and logic Chapter 15 question 21 how?

I'm really not understanding the set up of how to go about solving this problem any help is welcome
2answers
145 views

### Language proof and logic Chapter 15 question 16 help

I'm trying to go about solving this problem but I'm having problems even knowing how to approach it. Can someone help me to set it up? Here is the premise: ∀x∀y(x ⊆ y ↔️ ∀z(z ∈ x ⟶ z ∈ y) Here is ...
2answers
114 views

### If I saw UFOs, and I was of sound mind and body, does that give the right to say that it is true? [closed]

Around a year ago I saw some spectacular things in the skies above me, on three separate occasions. I believe I was of sound mind and what I saw really did exist. Given that what I saw was so out-...
2answers
135 views

### language proof and logic chapter 13 question 49 Help

Premises: ∃xP(x) ∀x∀y((P(x)∧P(y)) → x = y) Prove: ∃x(P(x)∧∀y(P(y) → y = x)) I've started it but the end is starting to get super muddy and not work out and I don't know where I went wrong.
3answers
939 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 ...
2answers
121 views

### In Fitch, how does one prove “P” from the premise “(¬P ∨ Q)→P”?

I can't figure out how to prove that formally. Please, help!!
4answers
1k views

### How to prove ‘∃xP(x)’ from ‘¬∀x(P(x)→Q(x))’

What would a formal Fitch proof for this look like? I am given ¬∀x(P(x)→Q(x)), and need to derive ∃xP(x) from it. I started with this, but I don't know if I am doing the right thing, and where to go ...
3answers
310 views

### 2 simple Formal Fitch Proofs

I'm having difficulty proving these. They seem obvious, but I can't figure how to set up formal proofs for them. Could anyone give me clues on how to start them? ¬(P∧¬Q) from the premise P→Q; ¬Q→(R→P)...
2answers
231 views

### Fitch Proof by Contradiction help

Hi, I'm pretty new to writing formal proofs and I was wondering if I could get some help solving this question. I've set up the problem and I was thinking of perhaps proving it by contradiction that ...
1answer
84 views

### Is Frankenstein's monster tantamount to positive proof in Science?

Specifically Biology presents some problems for me. For instance, now that we have Evolution we know what to look for. Thus we are bound to observe some adaptations, and over time even new species. ...
2answers
75 views

### Can results be predicted?

I wanted to know that, what can we assume as the result of some experiment which we have not conducted on the basis of mathematical proofs? I mean, in general, equations are created after analyzing ...