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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A Paradox of Precision?

Yesterday I was talking to one of my mathematics professor regarding the notion of proof in general (whatever the word "general" means to the reader). In short my claim was, We can only be ...
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A and ~A in logical proof

In my logic class last semester, we went over proofs with the rules of induction and replacement. In a couple of the exercises, I noticed something. In each of the exercises in question, all of the ...
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What are some active areas of research in proof theory?

Is there any research activity going on in the field of proof theory today? If so, what are some of the most active areas, what types of questions do they deal with, and where can I go to find out ...
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Fitch Formal Logic Help 6.26

6.26 Premises: A v (B ^C) ~B v ~C v D Goal: A v D Prove it formally without using DeMorgan's Law
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3answers
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What is the relation between proof in mathematics and observation in physics?

Recently in his 2015 Hirzebruch Lecture in Bonn, Arthur Jaffe re-amplified his famous perspective that finding proof in mathematics is analogous to making experimental observation in physics. In ...
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1answer
56 views

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 3,...
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6answers
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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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4answers
120 views

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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0answers
53 views

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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1answer
39 views

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
3
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2answers
101 views

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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3answers
677 views

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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0answers
81 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 ...
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2answers
198 views

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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3answers
104 views

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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3answers
850 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 ...
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2answers
104 views

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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1answer
609 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
35 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 ...
3
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1answer
45 views

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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2answers
468 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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1answer
1k views

The difference between argument, inference, deduction and proof?

I am trying to sort out the difference between argument, inference, deduction and proof. First, let's look at the distinction between argument and inference (if there is one). This online source ...
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2answers
145 views

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
296 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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8answers
920 views

What qualifies something as mathematics?

What classifies something as math? Is "math" simply performing operations with a certain set of axioms in mind? Is "math" anything that involves numbers? What about mathematical logic? Google ...
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2answers
149 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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1answer
228 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, ...
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1answer
142 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
188 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
88 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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107 views

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
205 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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232 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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1answer
162 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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2answers
300 views

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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3answers
166 views

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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1answer
64 views

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 well-...
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1answer
27 views

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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110 views

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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4answers
244 views

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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2answers
326 views

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
424 views

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
186 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
140 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
324 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
322 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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499 views

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 ...