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25 votes
Accepted

Does every truth have to be provable based on evidence?

The answer is a point of contention between realism and anti-realism. Truths that "do not have evidence" are termed verification-transcendent truths (coined by Dummett), and realists are committed to ...
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24 votes

How to prove (A v B), (A → C), (B → D) therefore (C v D)

Here is part of the question: My only idea is v must be introduced, but how would I use subproofs to show one of A/\C or B/\D is never false if A v B? It might be best to think of using ...
  • 18.9k
17 votes

Is watching an amputated limb regrow proof of the supernatural?

Logically, if we could prove that God healed amputees then it would as a corollary prove the existence of God. (it is simply the argument that; "if X is specifically observed to do Y, then X must ...
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14 votes

Can we doubt all knowledge?

To answer the question in the title: Yes. That's a key trait of any good scientist. To answer your last question in the body: Because we have no better option to depend on or behave according to.
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13 votes

What makes something mathematics?

Both definitions are outdated. As Husserl put it already back in 1901: "Only if one is ignorant of the modern science of mathematics, particularly of formal mathematics, and measures it by ...
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13 votes

Does software exist to automatically validate an argument?

You are looking for a so-called automated theorem prover. See e.g. pyPL or Tree Proof Generator for two implementations of the calculus of analytic tableaux for classical propositional and first-order ...
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11 votes

Is watching an amputated limb regrow proof of the supernatural?

The term 'supernatural' is generally used by modern skeptics in the sense: "That which cannot be explained by natural processes using the natural sciences." However, any event that can be ...
  • 14.3k
11 votes

Can we doubt all knowledge?

You can of course do anything you want... But to doubt all knowledge is to indulge in radical skepticism, is it not? If we were all radical skeptics, then we'd be living in a world in which knowledge ...
10 votes
Accepted

Are there rules for dealing with self-reference "paradoxes" in logic?

The same effect can be achieved with a single sentence:"This sentence is false". It is known as the Liar paradox and goes back to an ancient sophist Epimenides. Your two sentences simply split the ...
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10 votes

How is Wittgenstein’s “notorious paragraph” about the Gödel's Theorem not obviously correct?

Timm Lampert, cited by the OP, quotes Wittgenstein (§8 of Remarks on the Foundations of Mathematics, Appendix 3): ‘True in Russell’s system’ means, as was said: proved in Russell’s system; and ‘...
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10 votes
Accepted

Is watching an amputated limb regrow proof of the supernatural?

Short Answer As an athiest who advocates for philosophy, I would suggest there would be many rational bases for attacking your attribution of the regrowth to the supernatural which by definition ...
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9 votes

How do we know if a mathematical proof is valid?

Some comments. 1) Well before Cantor, it was already known that we may "have troubles" in comparing infinite collections of numbers ; see, at least, Galileo's paradox. 2) Of course, the ...
9 votes

What is the difference between mathematical reasoning and philosophical reasoning?

Philosophical arguments are made mathematical all the time. Its why you will see First Order Logic symbols thrown around on this Stack Exchange. I think the big difference between mathematics and ...
  • 17.2k
9 votes
Accepted

Does predicate logic have truth tables?

NO, because validity for predicate logic means true in all interpretations, and thus we have to take into account also interpretations with infinite domains, like the set N of natural numbers. Every ...
8 votes
Accepted

Rebuttal for modus ponens

Lewis Carroll's puzzle first appeared in the April 1895 issue of Mind. It directly influenced the formulation of the first primitive proposition of Whitehead & Russell's Principia Mathematica. ...
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8 votes
Accepted

The blur between proof and evidence

One possible objection is that you're claiming something doesn't exist merely because people have varying abilities for recognizing (or not) said candidate existant (which you seem to posit in premise ...
8 votes

Is a proof still valid if only the author understands it?

"A language that I don't understand is no language." (Wittgenstein, MS 109) Is a proof still valid if only the author understands it? I do not think so. See Yuri Manin, A Course in Mathematical ...
8 votes
Accepted

Are axioms tautologies?

It is worth separating the logic from the epistemology. Let's start with the logic. A (first order) theory is a set of sentences. Usually we are interested in deductive systems, so we require a ...
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8 votes

How to prove (A v B), (A → C), (B → D) therefore (C v D)

You can use proof by contradiction: p1: A v B p2: A -> C p3: B -> D assume ~(C v D) ~C & ~D (from 1, De Morgan's law) ~C (from 2, conjunction elimination) ~D (from 2, conjunction elimination) ~...
  • 81
8 votes
Accepted

How to show (in a hand waving manner) that the Godel sentence is true

Gödel’s Incompleteness Theorem is a result about formal systems. Its proof requires certain assumptions about the properties of specific formal system F: basically, about its "expressive capabilities"...
8 votes

Proof for the absence of free will?

Seems like no one brought up Frankfurt and hierarchical compatabilism. First-order desires: desires that are directed to objects or states of affairs. We desire things like being healthy, being well-...
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7 votes
Accepted

How to find redundant premises?

Regarding the statement from your question: "it isn't valid" By definition, an argument is valid if the premises and our accepted working of logical rules create a situation such that if all of ...
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7 votes

Given P ∨ ¬ P prove (P → Q) → ((¬ P → Q) → Q) by natural deduction

1) P ∨ ¬ P --- premise 2) (P → Q) --- assumed [a] 3) (¬ P → Q) --- assumed [b] 4) P --- assumed [c] for ∨-elimination 5) Q --- from 4 and 2 by →-elimination 6) ¬ P --- assumed [d] for ∨-...
7 votes

Is any aspect of the supernatural testable? What level of proof is possible for the supernatural?

Yes, of course. You can *scientifically prove** things deemed supernatural. But once you do, they are no longer supernatural. They are "natural," as demonstrated by the methods of the natural ...
7 votes

Can we logically prove that anything exists?

IMO, there are two related but different issues here. We prove statements : in math and logic we prove a theorem from axioms. There is no way of proving a statement "from scratch", i.e. without ...
6 votes

What makes something mathematics?

From a modern point of view mathematics is considered the science of formal structures. Simple examples of such structures are topological spaces, groups, vector spaces, differentiable manifolds. A ...
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6 votes
Accepted

The difference between argument, inference, deduction and proof?

One issue is that different authors use "argument" and "inference" in ways different from each other, and from the colloquial meaning. For example, your source interprets "argument" as just the list ...
  • 41.2k
6 votes

What is the relation between proof in mathematics and observation in physics?

This point of view is better reflected if we change "observation" to "experiment" in the title, mere observation is more analogous to conjecture, so it may be somewhat misleading. This is how Jaffe ...
  • 41.2k
6 votes
Accepted

How do you prove A <-> C given the following premises?

given that the two main things you need to prove are A -> C and C -> A. As a general strategy, it is often the easiest to do so with conditional proofs. Given the three assumptions you've been given, ...
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