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48

I would say that generally, the burden of proof falls on whomever is making a claim, regardless of the positive or negative nature of that claim. It's fairly easy to imagine how any positive claim could be rephrased so as to be a negative one, and it's difficult to imagine that this should reasonably remove the asserter's burden of proof. Now, the problem ...


39

There are two categories of things that can be proved in philosophy: That a thinking thing exists; The trivial truths of logic. I'll cover these in order. In fact, there are philosophical arguments you'll find against them both. The basic idea that 'a thinking thing exists' comes to us via the Ancient Greeks but became widely known and was made popular by ...


20

There is some instability in the terminology here. Many authors use Reductio Ad Absurdum (RAA) as meaning the same as proof by contradiction and indirect proof. More careful authors distinguish them, taking both RAA and indirect proof to be a species of proof by contradiction. In what follows, I use P and Q for propositional meta-variables, ∧ for ...


19

Philosophy is generally predicated on, and perhaps more about, asking questions rather than finding answers. It's a search for wisdom, not truth. The only thing that all philosophers would all agree exists, besides themselves perhaps, would be questions. And sometimes, philosophers will pretend that even those don't exist. So, philosophy doesn't, as a whole, ...


15

Not necessarily tied to philosophy, but in formal debates the sides agree on a proposition to make arguments about. One side will assert the proposition and assume the burden of proof while the other side will refute the proposition. But the structure of the proposition may be anything the two sides can agree to debate. Using your example, the proposition ...


15

Let's skip straight to the end. then it is trivial that logic is circular. Correct. Logic is circular. Note that due to Agrippa's Trilemma, there are only three things logic could possibly be founded upon: unsupported axioms we take on faith, circular reasoning, or an infinite regress. Or, of course, a combination of the three. Lewis Carroll ...


11

What you're touching on, of course, is a couple of basic facts about epistemology — and how they impact the activity of proseletysation: the attempt to get someone to believe in an idea (whether religious or secular) which they not only did not know, but did not even concern themselves with, before. The short version is that because original discovery ...


11

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 standards of Euclid and Adam Riese, can one remain stuck in the common prejudice that the essence of mathematics lies in number and quantity". In antiquity mathematics was ...


10

There are a few things to unpick, here. First, there's a difference between provability in a formal system, and "truth", which is a question of the relationship between language acts and facts. The statement "Juh mapple Neele" is neither true nor false, but nonsense, unless it is recognised as a poorly pronounced version of the French phrase Je m'appele ...


10

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 Liar in two. There is no endless regress though, it ends in one step. We accept both sentences as "axioms", i.e. "true", but the second sentence implies that the ...


9

The heuristic that the burden of proof is on the affirmative side of a dispute is intended to be broader than just claims of existence and non-existence. It is also one of a number of different (sometimes conflicting) considerations that go into the determination of the burden of proof. I said 'heuristic' as there is no algorithmic way to determine where the ...


9

There are several factors at play in your question. It appears that you have (re-)discovered the distinction between implicational and non-implicational negation (also sometimes known as "choice negation" and "exclusion negation"). The literature on this topic goes back to ancient times: for example, Indian logic (both Buddhist and Nyāya) draws a ...


8

Some claims of existence are mathematical: is a given set of properties consistent? is there a number/object which satisfies a given set of constraints? Whether you set out to prove the positive or the negative, the burden is on the claimant, there's no need to worry about whether it is positive or negative existence or non-existence. There may still be an ...


8

It is a central feature of all the main formal systems that when a statement is provable, then it is provably provable. Indeed, this feature is one of the derivability conditions that is commonly used in the proof of the incompleteness theorem, and it is central to Goedel's proof of the second incompleteness theorem. But also, I might add, this principle ...


8

Some comments. 1) Well before Cantor, it was already known that we may "have troubles" in comparing infinite collections of numbers ; see at lest Galileo's paradox. 2) Of course, the purported proof : "that there are equally many reals and natural numbers, because we can make a list of natural numbers and assign a real number to each natural number" is ...


8

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. This puzzle exposes the difference between implication and inference: an implication only tells you what follows your premise, but does not tell you whether your ...


8

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 2). A heap of sand is made up of grains. So, a certain number of grains of sand comprises a heap. However, how many grains are needed to make a heap depends ...


8

"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 Logic for Mathematicians (2010), page 45 : A proof becomes a proof only after the social act of “accepting it as a proof.” This is as true for mathematics ...


7

All claimants have a burden of proof. If I were to make claim that the earth is round I would have a burden to provide proof at least if asked. In physics we spent several weeks confirming the laws of motion experimentally. When we are taught about the laws of motion it is backed up with centuries of experimental data and confirmation. The only claims ...


7

I think that we have to turn to the great philosopher Rumsfeld, who famously opined about "known knowns", "known unknowns", and "unknown unknowns." The size of what we don't know about the universe is an unknown unknown; we necessarily have no way of knowing how much (or how little) there is we don't know. So: all the more reason to examine rigorously ...


7

Does anyone know if the idea that something can't be proved, only disproved has a specific origin? It was brought to prominence in modern philosophy of science by Karl Popper, who proposed falsificationism. (I cannot recommend the latter wikipedia entry though.) I also take it that it applies to pretty much any belief, whether it's an untested hypothesis ...


7

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 philosophy is that mathematics tends to start from something like a formal system, and see how much can be proven within it. Philosophy approaches the question ...


7

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 good overview of all fields of active mathematical research can be read off from the Mathematics Subject Classification, see http://www.ams.org/msc/msc2010.html ...


7

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 ∨-elimination 7) Q --- from 6 and 3 by →-elimination 8) Q --- from 4-5 and 6-7 by ∨-elimination, discharging [c] and [d] 9) (P → Q) → ((¬ P → Q) → Q) --- from 3, 2 ...


7

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 tautology of propositional logic, like P ∨ ¬P, can produce an unlimited supply of valid predicate logic formulae through uniform substitution, i.e. by ...


7

From an neo-intuitionistic point of view based in Kant's assumption that space and time are aspects of human thought rather than reality (whether or not you follow down Brouwer directly in finding all negation questionable) mathematics is not an objective description of some conceptual world, it is an exploration of shared human intuitions and how far they ...


7

In classical and intuitionistic logic, the Principle of Explosion is often a basic law of inference. Wiki's entry deduces it from Disjunctive syllogism: Assume P as true; then (by Disjunction introduction) we have: P ∨ Q, with Q whatever. But we have also ¬P. Thus, we may conclude with Q. This is what happens in classical and intuitionsitic logic ...


7

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 theory to be closed under the relation of provability. A theory T is axiomatizable if there exists a subset of T, the axiom set, such that all of the sentences in T ...


6

Point #1 - I only know a little about the Judeo-Christian-Islamic traditions that assume a moment of divine revelation. With the exception of certain creationists, I don't know of any religious people who would deny that human beings predate the holy books. Point #3 – I don't know of any religious books or educated minds that would deny this fact or pin ...


6

Sure, there are lots of things that fit in your category-- there's nothing particularly unique about the case of God (and it's not even one case, as there are many, many different conceptions of God, each of which can be argued separately.) First of all, any ethical claims are going to be outside the bounds of "proof", because "is" does not imply "ought". ...


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