23
Aristotle's solution was largely accepted until the end of 19th century when Cantor and Dedekind formalized the notion of continuum in terms of set theory. Under their interpretation time is in fact composed of indivisible nows, just like a line is composed of points, and any other magnitude is composed of indivisible elements as well. It does not mean that ...
18
In Hartry Field's Saving Truth from Paradox (2009), he splits the resolution of the Liar paradox into two broadly distinct strategies. Either we can accept Classical logic, but need to restrict the class of propositions over which Truth can meaningfully operate, or we can weaken logical inference to block either the deduction of a contradiction from the ...
15
A simple model of infinitely deep infinite time
Here's a model for the ordering of events in time such that you can have three different objects, each infinitely old, and each infinitely older than the last. Informally, it involves time not only having an infinite past, but a "very very" infinite past.
We typically represent time by real numbers, possibly ...
13
Sophie's World: A Novel About the History of Philosophy (Worldcat link) might be better for an 11 or 12 year-old, but is worth mentioning. It follows a 14 year-old girl who starts wondering about philosophical questions and engages with a philosophy teacher to discuss in an accessible way ideas from early modern philosophy.
12
Your proposed solution does not solve the paradox.
The whole point of the paradox is that the term 'pile' is vague. That is, given an object (e.g. a collection of grains of sand) it is indeterminate whether the term applies to this object or not. It is indeterminate since it's not clear just how many grains constitute a heap (for any number n, you can ...
12
Russell's paradox arises within naïve set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself. Hence the paradox.
The "root" of the paradox is the so-called unrestriceted Comprehension Principle of naïve set theory:
for every property φ(x) ...
11
When I was young I really enjoyed the books of logician Raymond Smullyan, who wrote several books of logic puzzles held together with minimal but amusing narratives, including The Lady or the Tiger? and To Mock a Mockingbird. They are very accessible, even to a young audience, but cover some surprisingly sophisticated and advanced concepts. Lewis Carroll ...
10
I believe it is impossible. I recommend you read (if you haven't already) Descartes' meditations where he famously concludes I think therefore I am - http://www.sacred-texts.com/phi/desc/med.txt:
Archimedes, in order that he might draw the terrestrial globe out of
its place, and transport it elsewhere, demanded only that one point
should be fixed and ...
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
A good paper to read on this subject is an old classic: Gilbert Ryle's Systematically Misleading Expressions. (Proceedings of the Aristotelian Society, 32: 139-170 (1932). Also in his Collected Papers, vol 2.)
Ryle's view is that ordinary non-philosophical use of language frequently contains "improper" usages, by which he means usages that, while having a ...
9
You might find Graham Priests book The Limits of Thought helpful in refining your question. Priest argues that
thought runs into true contradictions when it runs up against its own limits
This was noted by Kant - his famous antinomies - which motivated his critical project; however Priest credits Hegel for deciding the contradictions are unavoidable, ...
9
This is all about the difference between natural language and formal language. In formal language, a term cannot be used unless it's well-defined according to the standards of the language. In natural language, on the other hand, well-defined terms are the exception rather than the rule.
The Sorites paradox forces us to us to recognize that a term like "...
9
You've stumbled upon an old problem in philosophy, The Paradox of Inquiry, first formulated in Plato's Meno.
The problem can be reformulated as follows:
Either you know the answer to a question, or you don't. If you do, then there is no point searching for it. If you don't, then you will not know what to search for.
The short answer is that you can ...
9
Technically speaking, a paradox isn't a problem to be solved in and of itself. A paradox points at a weakness, misconception, or internal contradiction of the philosophical/analytical structures we are using. When and if we fix the philosophical underpinnings, the paradox will simply become irrelevant.
Wittgenstein talked about this when he developed his '...
8
Here is another motivation for dialethism - inconsistent set theory:
It allows for a formalisation of naive set theory with the naive expectation that any predicate determines a set. That is, it's another solution to Russell's paradox apart from the theory of types or ZFC.
So it has a universal set, and Cantor's paradox is now a theorem.
This theory proves ...
8
The previous answers betray a lack of familiarity with the literature. Your solution, using the least number principle, essentially works. It is a known argument for epistemism about vagueness, the position that vague properties have sharp unknowable boundaries. If I recall, it is discussed at the beginning of the last chapter of Timothy Williamson's ...
7
Since R's 'objecthood' is primary, why doesn't it make sense to say that R can neither have the attributes is a member of R nor not-is a member of R correctly attributed to it? If this is the case then Russell's Paradox is dissolved, since it is the assumption that R must satisfy either is a member of R or not-'is a member of R that seemingly gets us into ...
7
This reminds me of the older question Was Wittgenstein anticipating Gödel? There is more to it in the case of Kant than there was in the case of Wittgenstein though, at least in spirit. One could say that Kant pioneered in epistemology the stratification into levels of discourse, which Gödel later applied to formal semantics.
When the Gödel theorem ...
7
There is a long controversy as to what should count as the "size" of an infinite set, and there provably does not exist a notion that satisfies both the bijectivity principle, a.k.a. Hume's principle (bijective sets have equal size), and the part-whole principle (whole is greater than its part). So any notion of size for infinities will be counterintuitive ...
7
Answer 1 - 'yes'
Suppose I ask, 'Can Tom walk ?' I am asking about the truth-value of the proposition, 'Tom can walk'. I expect the answer 'Yes' or 'No'. The answer, 'Yes', is right if Tom can walk and wrong if he can't - in which case the right answer is 'No'.
There are, of course, since we are using a natural language, indefinitely many possibilities ...
7
It can. Ramsey put it nicely in his "last papers" written around 1929 under the influence of Peirce's pragmatism (quoted from Marion, Wittgenstein, Ramsey and British Pragmatism):
"We want our beliefs to be consistent not only with one another but also with the facts: nor is it even clear that consistency is always advantageous; it may well be better to ...
6
I am a mathematician that found this page by accident, so I can't help you with the Zeno's paradox part (I think that was solved by calculus hundreds of years ago). But I would like to clarify some misconceptions.
The thermal time hypothesis is not directly related to loop quantum gravity. It is instead a mathematical result from the theory of von Neumann ...
6
The prisoner's reasoning is forced to be paradoxical. The chain of days is just a distraction.
The judge tells the prisoner: you will die today unless you are sure of the date of your death.
The prisoner thinks: "Wait, that's ridiculous, he just told me I'd die today, and I know the date. So of course I am sure of the date. That means I can't be ...
6
Paradoxes are indeed invalid arguments, but what makes them special is that they rest on seemingly unproblematic assumptions. We know, for example, that Achilles will in fact outrun the tortoise (Zeno's Paradoxes), we know that the surprise exam will take place (Surprise Exam Paradox), and so on. Because we know the conclusions of those paradoxes are false, ...
6
I think Orwell's Animal Farm can also be a good read for 9 year old. Its not strictly philosophical but still worth a read for every smart kid ( and adult ), I think.
6
I agree with some of the other answers that you're going to have a very difficult time spelling out exactly what the decision procedure involved would be. If the decision procedure boils down to you just stipulating that x is a "pile" iff x is composed of 47526 grains of sand (say), then you've haven't solved the problem but only transferred it to a new ...
6
I think you're misunderstanding the idea behind "Last Thursdayism" on two fronts.
First, as can be seen from the selection of "Thursday", the main point of the posit is to point out a problem in proving things that we can only observe indirectly by effects. Or to word it another way, the observer only has access to what they are observing and everything ...
6
If you take a really silly literal view of the sentence the statement is a performative contradiction but not equivalent to the liar's paradox.
Let's say being intolerant of something is along the lines of forbidding it. This is plausible. So saying
I am intolerant of intolerance
means
Forbidding things is forbidden!
If we agree that forbidding ...
5
Several answers have pointed out the affirmative reasons why dialetheism might be worth considering, implying that the chief motivation of dialetheism lies in the applicability to certain situations (whether they be logical or material) where the only correct description involves a dialetheia and which is otherwise intractable or has to be side-stepped and ...
5
In set theory, the distinction you are asking about translates to the question of whether the domain of a model must be a set, or whether it is allowed to be a proper class. This is an important distinction giving rise to many subtle issues. In many mathematical contexts, we are tempted to allow a structure whose domain is a proper class, and the question is ...
Only top voted, non community-wiki answers of a minimum length are eligible
