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25

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 their existence. Anti-realists, on the other hand, hold that unverifiable in principle statements have no truth values. So if no trace of dinosaurs remains, ...


24

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 disjunction elimination initially although disjunction introduction may be needed later. The OP notes the following: Obviously since A → C and B → D then if A v B one ...


17

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 exist"). But in practice that has two problems: Either, if science demonstrates God's existence, in what sense is He then "super"natural and not ...


13

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 logic. The tableau calculus is complete for first-order validity, meaning that every valid inference will be detected as such. But first-order logic is not co-...


12

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Both definitions are outdated.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 ...


11

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 observed systematically is ipso facto subject to the natural sciences, so the definition itself precludes the existence of miracles. It's a neat little Catch-22 ...


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 ...


10

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 ‘false in Russell’s system’ means: the opposite has been proved in Russell’s system. Lampert claims Wittgenstein is assuming what needs to be proven: Whether P =...


10

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 places the agents outside of the known laws of the universe. The claim that gods and magical beings are real is essentially the assertion that it is possible to fit ...


9

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 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 ...


9

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 ...


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

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

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 ...


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 ...


8

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 ...


8

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) ~A (from 3, p2, modus tollens) B (from 5, p1, disjunctive syllogism) D (from 6, p3, modus ponens) D & ~D (4, 7) Since D & ~D is a contradiction, our ...


8

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". In a sense that can be specified rigorously, system F must have the capabilities to manufacture the provability predicate for F, i.e. a suitable formula PrF(...


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

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 the premises are true, then the conclusion cannot be false. From the detail in your question I assume you are aware of this much. Having redundant premises is ...


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

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 sciences. However, you are probably wasting your time on the various hobgoblins and eerie powers you list. We do not see such phenomena, werewolves, resurrections, ...


7

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 assumptions. The basics of "deductive sciences" are well-known since Aristotle and the same A discussed the issue of infinite regress in the foundations of ...


6

The answer depends highly on what position you hold in regard to evaluation of conspiracy theories' claims. Let me present the following example. Science was making huge strides in the late 19th century, including revolutionary discoveries by Louis Pasteur and his followers. That opened an opportunity for numerous hustlers to claim discoveries of magic ...


6

Actually, You Can Prove A Negative Sometimes In general, sure, it can be difficult to disprove the existence of something on a universal level, because theoretically you would need total, infallible awareness of the entire universe to prove it with complete certainty. The problem here lies in that the scope of the concern and requirement of proof is ...


6

You can assume the negation of this sentence and easily derive a contradiction from it. By DeMorgan's laws it becomes the conjunction of two negations. Each of these negations is a negated existential sentence which is the equivalent of a universally quantified negation. That is, you have the conjunction of "everything is not F" and "everything is not not F"....


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