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Questions tagged [proof]

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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2 simple Formal Fitch Proofs Help!

I'm having difficulty proving these. I've been stuck on them for nearly 4 hours. They seem obvious, but I can't figure how to set up FORMAL proofs for them. Could anyone give me at least clues how to ...
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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 ...
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Does this symbolic logic proof work?

So, I have this proof: Let a be an open sentential variable. Let K and C be prepositional variables asserted of sentences. Given K(a) <=> C(a) & a C(a) <=> C( C(a)) C(a & b) <=> C(...
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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 ...
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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. ...
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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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Are arguments claiming the impossibility to prove or disprove anything themselves impossible to prove? [duplicate]

Could arguments claiming the impossibility to prove or disprove anything be flawed because if they were sound they would also be impossible to prove? Or could you just assume they must be right ...
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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 ...
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Proofs in math and physics

Suppose we have the case of a proof in math or physics and we want to compare the status of the derived information. I know that in math mostly all derived information or deduced details are a priori. ...
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Symbolic Logic - Negation Introduction

I am working on a problem for an online class that I'm struggling to figure out. I'm given these premises: 1. (H > (A > B)) (The > sign here represents conditional) 2. (~K & ~B) 3. (~A > K) The ...
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Prove ¬∃x ∀y (E(x, y) ↔ ¬E(y, y)) given no premises

The only way I could think of to do this problem is reductio, but since the two biconditional terms are not contradictory, I am pretty stuck.
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Natural deduction proof help!

I've gone through about 40 natural deduction proofs in the past couple days, and mostly they are no problem. For some reason, I've been stuck on 1 tedious problem for an entire day. I just can't seem ...
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Is there a way to prove the existence of choice and free will

It is practically impossible to "make" more than one decision at a point in time. Even if you "change" your mind later, it is at a later point. How do we know that those are decisions that sentient ...
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Can any correct logical reasoning in natural language sentences be translated into a formal mathematical proof?

Since natural languages (e.g. English) are prone to ambiguities and misunderstandings due to their constant evolving nature and lack of rigorous formalization, and given an arbitrary philosopher X who ...
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What's the name of this kind of fallacious proof to refute an idea?

In an argumentation where speaker A suggests an idea, we sometimes encounter this kind of fallacious proof from speaker B that speaker A's idea is bad (a very common form is known as passive-agressive ...
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Deriving “(p.q) v (p.r) from ”p.(q v r)"?

I am new to logic. and here are my tryouts for deriving deriving "(p.q) v (p.r) from "p.(q v r)", and further I want to show that ”p.(q V r)” is equivalent to ”(p.q) V (p.r)”, by using natural ...
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How to prove the tautology ¬(P↔¬P) using Fitch?

Just as the question proposes, I'm having trouble with proving this tautology. I know one should use proof by contradiction however I am currently stuck.
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Sentential logic derivation: ~(A ≡ B) ├ (~A ≡ B)

I am doing some practice sentential derivation proofs for an upcoming test and have attempt the following proof many, many times without success. ~(A ≡ B) ├ (~A ≡ B) The logic system I am using is ...
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How to prove using higher abstractions instead of diving into axioms or a little bit deeper?

Sorry for the bad formulated question, feel free to edit it. I will explain my question here. I try to reflect on my abilities of proving theorems to become better at this. That is why after reading ...
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Doubt in Searle's Mind: A Brief Introduction

I have been reading Searle's Mind: A Brief Introduction, Oxford UP (2004). In it I came across the passage in Chapter 2: if we believe that p and if we believe that if p then q we will believe ...
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Does anyone know of a philosophy which rectifies or considers the following question?

Let's imagine that I began to doubt the validity of one of my arguments, which leads me to question my ability to make rational arguments. And so begin to distrust my intuitive ideas about logic, then ...
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how to prove from ~ (A ∩ B) that ~A∩~B? [duplicate]

How to prove ~ (A ∩ B) ⊢ ~A ∩ ~B ? my thought is to assume A, find a contradiction, so I can use "~I" to get ~A; do the same for B to get ~B; then use "∩I" to get ~A ∩ ~B. But I struggle to infer ...
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In fitch, S → (R ∨ P), P → (¬R → Q) ∴ S → (Q ∨ R)

Construct a proof for the argument: S → (R ∨ P), P → (¬R → Q) ∴ S → (Q ∨ R) I have gotten to the point in the illustration, but I am unable to figure out where to go from here. I get tricked up on ...
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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 ...
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Predicate logic proofs - how to split a disjunction bound by two quantifiers

I need to complete the following proof using only primitive rules (the introduction and elimination rules for each connective and quantifier). (∃x)(∀y)(Py ∨ Qx) ⊢ (∀y)Py ∨ (∃x)Qx I've only been able ...
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Prove whether statement is true or false

Consider the following: If a |= c or b |= c, then a ∨ b |= c. Prove whether this statement is true or false. My gut instinct is to compare truth tables, but I don't think a truth table is possible ...
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Has any philosopher ever argued succesfully that anything at all does not exist?

Is it possible to prove that something does not exist? I'm asking because I find it very difficult to think of any such idea.
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Which group is bigger, the one of things we can write about, or that we can feel?

Which group is bigger, the one of things we can write about (in any language (including programing)), or the one of the things we can feel through our senses (including those things that machines help ...
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Quine - Two dogmas of empiricism - status of mathematics [duplicate]

If we do away with the analytic-synthetic distinction as per Quine, does that mean that mathematics is no more certain than empirical science? And how does mathematical proof proceed if we don't use ...
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Logic question regarding a logical truth

Is the following logically true? ∃x[Cube(x) →∀yCube(y)] I think that it is logically true. When translated into truth functional form we have: A→B. A truth table shows that it is not a tautology but ...
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Language Logic Proof Question: ¬∃x∀y[E(x,y) ↔ ¬E(y,y)]

I am wondering if I have completed this proof properly. I don't think I have it right. It's tricky! Conclusion: ¬∃x∀y[E(x,y) ↔ ¬E(y,y)] ¬¬∃x∀y[E(x,y) ↔ ¬E(y,y)] ∃x∀y[E(x,y) ↔ ¬E(y,y)] ¬E,1 ...
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How does one prove a generalised conditional statement?

I'm trying to prove a statement of the form ∀x: P(x) → Q(x). Apparently the way to do it is to prove that Q(x) for some x such that P(x), assuming that x is arbitrarily chosen, and hence ...
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What is the nature of proof in mathematics?

Preamble: I think we have this sort of questions, where we are required to find a solution for them. For example, what is the area of a circle?. I think the way to solve these problems is to try to ...
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How can you prove I'm not a dog? [closed]

This is a general question that proves you have no way of knowing anything. How can you prove, that if you see me (assume I look like you exactly, because you don't have a picture of me), I am not a ...
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Quick logic deduction question

I have to provide a natural deduction derivation for: ¬∀xFx ⊢ ∃x¬Fx That´s what I got so far: 1.¬∀xFx 2.‖ ¬∃x¬Fx (Indirect proof hypothesis) 3.‖‖ ¬¬Fy (Indirect proof hypothesis 2) 4.‖‖ Fy (...
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Use the Fitch system to prove the tautology (p ∨ ¬p)

I've scoured the math stackexchange and the philosophy one for some guidance on how to go about this while using the Fitch System. Anyone can attempt it here; http://logic.stanford.edu/intrologic/...
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Is the reasoning behind this proof that a unified scientific theory of everything in physics is impossible correct?

I wrote this originally as a response to another question posed on Quora as to whether or not but I was wondering if someone could reexamine this and find any possible possible gaps in logic or ...
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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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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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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. ...
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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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Axiomatic Proof of Symmetry and Transitivity of Identity

Given the axioms below and the rules of Modus Ponens and Universal Generalization, how can you prove that t=s → s=t for any terms s and t? Additionally, how do you prove that t = s → (s = r → t = r) ? ...
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The Principle of Explosion v. Reductio ad Absurdum

The proof for the Principle of Explosion starts by assuming a contradiction. When we use reductio ad absurdum, we establish a proof by reaching a contradictory conclusion in sub-argument and then ...
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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 ...
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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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What is the relationship between the BHK interpretation of propositional logic and Natural Deduction?

This is a repost from MathSE. I've been getting into intuitionistic logic lately, starting from propositional logic. I am interested in proof-theoretic semantics, meaning the idea that the truth of ...
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prove: ∃x ∃y (Cube(x) ∧ Cube(y) ∧ x ≠ y ∧ ∀z (Cube(z) → (z = x ∨ z = y)))

I need a formal (Fitch) first order logic proof for: ∃x ∃y (P(x) ∧ P(y) ∧ x ≠ y ∧ ∀z (P(z) → (z = x ∨ z = y))) Given ∃x ∃y (P(x) ∧ P(y) ∧ x ≠ y) ∀x ∀y ∀z ((P(x) ∧ P(y) ∧ P(z)) → (x = y ∨ x = z ∨...
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Is there something wrong in breaking the symmetry of Natural Deduction?

In the intuitionistic natural deduction (NJ) we have a nice symmetry : each logical connective has an introduction rule and an elimination rule. But when we want to switch to classical logic (NK) we ...
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Predicate logic proof

I need to prove: ∀xRax → ∀x∃yRyx My available axioms are: ϕ → (Ψ→ϕ ) (ϕ → (Ψ→χ)) → ((ϕ→Ψ) → (ϕ→χ) (~Ψ→~ϕ) → ((~Ψ→ϕ) →Ψ) ∀αϕ→ϕ(β / α) ∀α(ϕ→Ψ) → (ϕ→∀αΨ) Our rules are modus ponens and universal ...
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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 ...