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26

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, ...


23

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 ...


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 ...


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Both definitions are outdated.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 ...


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 =...


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 ...


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

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

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

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

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 ...


7

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 ...


6

As written, the B's argument seems an invalid red herring Assuming the question is: Is atheism true or false? whether either party believes the premise that it is true is simply irrelevant to the argument. There are many things that are true, but which nobody believes. For instance, it's very likely a supernova has occurred which is not yet visible on ...


6

I'm not sure I follow the details of your question: the second argument schema you present is, of course, valid. We might have given that schema the name modus ponens. What would follow exactly? The fact remains that modus ponens (the first schema) is also valid. In any event, and re your broader worry, logic is rock bottom. That is, there is no non-logical ...


6

That we can very reliably make predictions on the basis of what we do know (or, rather that in the past we have been able to) is the best counterargument I know of against that argument. Although this appears in various guises everywhere from Popper to coherentism, the simple observation that we routinely do not walk into walls (and manage to build fairly ...


6

A brief answer - It is possible to prove that something "can't exist", which in turn implies that it doesn't exist. But trying to prove that something doesn't exist directly is a futile effort. iphigenie's links will give you more detail than I have time to re-hash at the moment though :)


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"....


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 ...


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