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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Logic proof on biconditional [closed]

(P → Q) ↔ ( ¬P ∨ Q) is the goal, there's no premises I start with 2.|_ P -> Q.................. 3.||_ P...................... 4., _ ~P.................... 5., |....................RULE: | INTRO ...
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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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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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Using the conception of 'reliable, unchanging' does 'truth' exist?

An 'archaic' definition for TRUE,TRUTH implies constancy, reliability, unchanging, fidelity. Using this concept of TRUTH is the following valid? There exists either that which is TRUE or that which ...
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Prove (P → Q) ↔ (¬Q → ¬P) using conditional elimination and negation introduction. [closed]

I'm trying to prove that (P → Q) ↔ (¬Q → ¬P) using Fitch. I know I have to prove two subproofs. 1) P → Q 2)¬Q → ¬P
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What exactly are the identity rules in logic?

In first order logic, I have read there are a couple of identity rules and I have a question for example: If you have "a=b" does it mean that I can also write it as "b=a"? Is it true the one-way or ...
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Does the famous Descartes quote “dubito, ergo cogito, ergo sum” suggests secure knowledge of ones existence?

After a discussion about the "difficulties to distinguish knowledge from faith" someone replied to me that the quote implies faith because it uses the word "think". But as it is generally understood: ...
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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 ...
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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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How does this propositional proof make sense?

How does the following proof of argument is valid and makes sense? 1. (R • C) [It is raining and It is cloudy] 2. ~R [It is not raining] Therefore, S [It is snowing] According to proof by ...
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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 ...
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Subjective Proofs

I've been having thoughts for a while about what constitutes a proof. Formal logic usually consists of incredibly detailed steps and, as such, is usually not utilized that often in everyday life. ...
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489 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 ...
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1answer
31 views

Multiple universal quantifiers in an argument

Consider the argument ∀x∀y((S(x,a)∧ S(a,y))→S(x,y)), ∀x¬S(x,x) ├ ∀x(S(x,a) → ¬S(a,x)) My approach to formally proving this was to first eliminate ∀x and use x0 as the free variable. Then ...
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Proving the negation of universal quantification

Consider the following argument ∀(R(x) ∨ S(x)), ∃x(¬R(x)) ⊦ ¬∀x(¬S(x)) My strategy is to try and proof that ∀x(¬S(x)) is a contradiction, and therefore ¬∀x(¬S(x)) must be true. My solution so far ...
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426 views

Why is this argument valid?

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 ...
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2answers
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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 ...
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1answer
183 views

Logic – Deduction in Tarski's World (Fitch/LPL 13.36)

I am working on proving the following question: | ∀x [Dodec(x) → LeftOf(x, a)] | ∀x [Tet(x) → RightOf(x, a)] |––– | ∀x [SameCol(x, a) → Cube(x)] The question has the following rules: […] give a ...
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2answers
120 views

What's the difference (if any) between demonstration and description?

How do philosophers of various schools* explain the difference (if any) between demonstration and mere description? Are they synonymous, or are they different? How so? My first impressions: To ...
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Handling “paradoxes” in logic: The sentence after this is true. The sentence before this is false

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, ...
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1answer
133 views

Is a proof still valid if many people say that is true? [duplicate]

A proof is some explanation to convincing others that a statement is true (or false in case of a counterexample). As Yuri Manin once wrote: "A proof becomes a proof only after the social act of ...
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1answer
140 views

Definition of “proof”

I read a book of logical puzzles once and there was a little story in it (fictional) about some boy who received an F on his geometry test because his professor said his proof that all angles of a ...
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1answer
70 views

Modal Logic Proof, a formula is valid just if relations have a property

So I'm trying to prove that, in a language with two diamonds <1>,<2>, the formula p -> [2]<1> p is valid just in case for any x,y and R2(x,y) then R1(y,x). I have the "if" direction easily, ...
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Does scientific and logical evidence have a supremacy over emperical evidence?

Emperical evidence comes from observations and experience but not from theoretical proofs. Science heavily relies on both, yet empirical evidences are outnumbered in many fields such as mathematics, ...
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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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6answers
187 views

Is logic built on assumptions?

I'm sorry if this sounds like a stupid question, but how can we know that our logical approach to ideas is not in itself based on assumptions. For example, how can we know that the workings of the ...
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221 views

If you're the smartest person on earth, how do you know if you're making logic errors? [closed]

In any logical argument, there is the practical step of verifying that it is sound. When there are experts in that particular area, they can check the argument for soundness. For two examples: A ...
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1answer
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What is it mean by “to be meet with in space” by Moore?

I was reading the Moore's proof of external world, and I am completely stuck with an idea/phrase of "to be meet with in space." It is on page 130 on the following pdf philosophical papers, collier ...
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132 views

Truth Tables and Venn Diagrams: Prove Contingent statement are consistent [closed]

I am trying to prove how consistent statements can be contigent. The problem i am having how to use a truth table to prove this.
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LPL Q6.26 without --> or material implication

This is a repeated question: Language Logic and Proof Q. 6.26 "Using the natural deduction rules, give a formal proof of A ∨ D from the premises A ∨ (B ∧ C) (¬ B ∨ ¬ C) ∨ D" The LPL textbook ...
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Help with a proof in CSL

Premise: (A implies B) implies C Conclusion: (C implies A) implies A I had a logic exam a few hours ago, and this was one of the problems, but I really didn't know where to start. Since the ...
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Transitive Incompleteness of Logical WFF's Due to Godel's Incompleteness Theorem

If a set of theorems, or wff's, are used in conjunction with one another, does this have an impact on their completeness in terms of soundness? For example, I have five theorems of logic, or ...
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1answer
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Are analogical middle terms sufficient for a valid demonstration?

William A. Wallace, O.P., in “Thomism and the Quantum Enigma,” The Thomist 61 (1997): 455–468, claims that analogical middle terms are sufficient for a valid demonstration and that this is a ...
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Mathematical versus philosophical reasoning (and the mathematics of philosophical arguments)?

What is the difference between mathematical reasoning and philosophical reasoning and why isn't philosophy just considered to be a branch of mathematics? Is any study not a branch of mathematics ...
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Burden of proof

Who has the burden of proof when trying to prove or disprove someones religious beliefs? I have always believed it belongs to the person who is making a claim but can there be exceptions?
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Does a statement exist which is its own proof?

In fake/pseudo mathematical notation: does there exist a statement S such that S = Proof(S)?
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An argument is valid if the premises CANNOT all be true without the conclusion being true as well

"An argument is valid if the premises CANNOT all be true without the conclusion being true as well." Argument_1: P or Q. not Q. Therefore, P. Argument_1 is valid. Argument_2: B or M. M or C. ...
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1answer
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How to discharge an assumption?

I am proving the follow argument [(A ↔ B ) → C] ⊢ [ - ( A ^ B) V C ] Using the following set of rules I, ^E, vI, vE, →I, →E, ↔I, ↔E, --E, -I Here are my steps but I got to the point where I ...
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1answer
168 views

Help with a proof for the following arguments

I want to proof the following argument from both sides using the following rules ^I, ^E, vI, vE, →I, →E, ↔I, ↔E, --E, -I Argument [(A ↔ B ) → C] ⊢ [ - ( A ^ B) V C ] Please let me know ...
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1answer
126 views

Proof validity methods other than truth tables

I have read or heard some time ago that truth tables cannot be used to validate arguments which involves the use of quantifiers i.e. in predicate or quantificational logic where you can find ...
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7answers
282 views

The blur between proof and evidence [closed]

Consider this: Evidence is the foundations of proof. So, enough evidence creates proof. However, how much evidence is needed to make proof depends on the concerning persons' circumstances, ...
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3answers
296 views

How does one contradiction in argument makes the argument valid?

For any argument such as the following that we solve by "Proof by contradiction" method: (A v B) (A ⊃ C) (B ⊃ D) [ Therefore, (C v D) Assumption: ~(C v D) Therefore, ~C {from 4} Therefore, ~D {from ...
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Testing the validity of syllogism argument

I came across a validation method for testing the validity of a syllogistic argument which seems quite easier to grasp: For example: To test the argument: no P is B some C is B Therefore, some ...
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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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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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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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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
135 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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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 ...