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^ Yes, yes. I was assuming that the set of all truths was countably infinite. Then if you can state "A is in S" where S is some subset of the set of all truths, then S must be recursively enumerable or the complement of a recursively enumerable set and there are only countably many such sets. So, here's a nice example to show why I think is bad to stretch specific notions of math to philosophy: what is the cardinality of the set of all truths and why?

For every real number x, there exist truths "x2 is not negative", "x + 1 > x" and many more. As there are uncountably many real numbers, there's also uncountably many truths like this, and this is just a very limited class of truths. Therefore, the set of all truths clearly can't be enumerable. You can't start a mathematical proof by assuming that the thing you're trying to prove is true.

The set of all truths does not have a cardinality because it doesn't exist.

EDIT: Even in philosophical literature, concepts from mathematics are sometimes used when defining something, like in this article about defining a numerical measure for the "truthlikeness" of a statement: http://www.bbk.ac.uk/philosophy/our-staff/academics/northcott-files/SyntheseVs2011.pdf .
 
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You can't start a mathematical proof by assuming that the thing you're trying to prove is true.

I was NOT trying to 'prove' that the set of all truths was countable.

Rather, I was remarking at the fact that, if we only allow ourselves to state "A is in S" if you can define S, i.e. S is computable, then there would be only countably many such statements.

In the context of natural numbers, this is synonymous with being recursively enumerable or the complement of a r.e. set, that's why I said that.

Though, as you pointed out, if we're dealing with reals, then there are computable sets which are not enumerable/uncountable. BUT, there are still only countably many such sets; all truths you listed in your last post are computable, and indeed so would any truth one could possibly list. Therefore, again, even if the set of all truths is uncountable, with the restriction that we can only state computable statements (redundant), then the argument doesn't hold (because there will be no statement for every subset of the set of all truths, only for the computable ones and there are only countably many such sets).

In any case, you could object this restriction on the grounds of "truths exist beyond their statements". Fair enough perhaps. Should we restrict, shouldn't we? I mean, while I agree that truths exist beyond their statements, it is kinda weird to have a truth that you can't say what it is.
 
In any case, you could object this restriction on the grounds of "truths exist beyond their statements". Fair enough perhaps. Should we restrict, shouldn't we? I mean, while I agree that truths exist beyond their statements, it is kinda weird to have a truth that you can't say what it is.

Maybe it's possible to create a philosophical theory that requires that every truth must be possible to write down with a finite number of words (or logical operations). I can't immediately think of a way to show that this kind of theory would contain a logical contradiction.

I'm not trying to make you feel stupid, by the way. In some thread you said that you were starting to study mathemathics or something related in the university. Developing a mathematical intuition takes some time, at least it did for me.

Here's a paper about the axioms of real numbers and it contains solved problems: http://math.nyu.edu/~tsang/classes/precalc1/session1.pdf . This is an example of what rigorous mathematical thinking is like. The set of axioms is made to be as limited as possible. For example, they don't explicitly say that the number zero is smaller than number one - that is a theorem that has to be proved. As an example of the fact that any statement can be proved if you start by assuming something false, consider this: Let's say that numbers 0 and 1 are equal.

0 = 1 (multiply both sides by 3)
0*3 = 1*3 (do the calculation)
0 = 3 (add 2 to both sides)
2 = 5

So we've just proved that if 0 = 1, then also 2 = 5. As an exercise, you could write the proof that if 0 = 1, it means that any two numbers x and y are actually equal.
 
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Maybe it's possible to create a philosophical theory that requires that every truth must be possible to write down with a finite number of words (or logical operations). I can't immediately think of a way to show that this kind of theory would contain a logical contradiction.

Nothing fancy required for that here. "This sentence is not in the set of all truths" does it.
 
Here is 'Is Justified True Belief Knowledge?' by Edmund Gettier. This is a classic text in epistemology, published over 50 years ago. It is a very short text (3 pages) and anyone should be able to understand it relatively easily.

The counterexamples which Gettier gives are quite strange, but their general pattern can be replicated in order to formulate more plausible ones. In order to appreciate why the second counterexample works it helps to have some familiarity with formal logic, more specifically with the rule of inference known as disjunction introduction, also called or introduction. Given any true proposition it is valid to infer any disjunction which features said proposition as a disjunct, e.g. given 'A' we may validly infer 'A or B' (A v B), and it does not matter at all what 'B' is. In Gettier's example it turns out that proposition 'A' is actually false, but the key point is that because Smith believed 'A' to be true his inference that 'A or B' was justified.

Here is a short excerpt:
This is what i have in one of my notebooks: you know [p] if and only if
  1. Is True
  2. Believe that [p] is true
  3. you're justified in your belief
  4. your belief is not inferred from a false belief
  5. your belief is "sensitive"
  6. your beliefs are mostly accurate
  7. you believe [p] by a reliable method
i must have abbreviated some of the conditions because i cannot find the direct source of this at the moment.
Here is a good resource that you can use to link you to many other interesting papers.
 
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Free Will and Neuroscience: From Explaining Freedom Away to New Ways of Operationalizing and Measuring It

Introduction— Free Will as a Problem (Not Only) for Science

The concept of free will is hard to define, but crucial to both individual and social life (Kane, 2005). Free will can be the reason why someone is not sent to jail during a trial upon appealing to insanity: the subject was not “free” when they committed the crime, not because someone was pointing a gun to their head, but because a psychiatric illness prevented them from controlling their actions. According to a long-standing philosophical tradition, if someone was not “free” when they did something, they cannot be held responsible for their deed (Glannon, 2015). And the freedom in question is both “social” freedom (linked to constraints imposed by our peers or by external factors), and the one indicated by the term free will.

Free will can be defined by three conditions (Walter, 2001). The first one is the “ability to do otherwise.” This is an intuitive concept: to be free, one has to have at least two alternatives or courses of action between which to choose. If one has an involuntary spasm of the mouth, for example, one is not in the position to choose whether to twist one’s mouth or not. The second condition is the “control over one’s choices.” The person who acts must be the same who decides what to do. To be granted free will, one must be the author of one’s choices, without the interference of people and of mechanisms outside of one’s reach. This is what we call agency, that is, being and feeling like the “owner” of one’s decisions and actions. The third condition is the “responsiveness to reasons”: a decision can’t be free if it is the effect of a random choice, but it must be rationally motivated. If I roll a dice to decide whom to marry, my choice cannot be said to be free, even though I will freely choose to say “I do”. On the contrary, if I choose to marry a specific person for their ideas and my deep love for them, then my decision will be free.

Thus defined, free will is a kind of freedom that we are willing to attribute to all human beings as a default condition. Of course there are exceptions: people suffering from mental illness and people under psychotropic substances (Levy, 2013). Nevertheless, the attribution of free will as a general trend does not imply that all decisions are always taken in full freedom, as outlined by the three conditions illustrated above: “We often act on impulse, against our interests, without being fully aware of what we are doing. But this does not imply that we are not potentially able to act freely. Ethics and law have incorporated these notions, adopting the belief that usually people are free to act or not to act in a certain way and that, as a result, they are responsible for what they do, with the exceptions mentioned above” (Lavazza and Inglese, 2015).

It is commonly experienced that the conditions of “ability to do otherwise”, “control” and “responsiveness to reasons” are very rarely at work all at once. Moreover, they would require further discussion, because there is wide disagreement on those conditions as regards their definition and scope (Kane, 2016). But for the purposes of this article, this introductory treatment should suffice. In fact, the description of free will that I have sketched here is the one that dominated the theoretical discourse on, and practical applications of, the evaluation of human actions. From a philosophical point of view, however, starting with Plato, the main problem has been that of the actual existence of freedom, beyond the appearances and the insights that guide our daily life. The main challenge to free will has been determinism: the view that everything that happens (human decisions and actions included) is the consequence of sufficient conditions for its occurrence (Berofsky, 2011). More specifically, “It is the argument that all mental phenomena and actions are also, directly or indirectly, causally produced—according to the laws of nature (such as those of physics and neurobiology)—by previous events that lie beyond the control of the agents” (Lavazza and Inglese, 2015). Determinism was first a philosophical position; then, the birth of Galilean science—founded on the existence of immutable laws that are empirically verifiable—has increased its strength, giving rise to the concept of incompatibilism, namely the idea that free will and natural determinism cannot coexist. Only one of them can be true.

Throughout the centuries, despite its conceptual progress, philosophy hasn’t been able to solve this dilemma. As a result, today there are different irreconcilable positions about human free will: determinism is not absolute and free will exists; free will does not exist for a number of reasons, first of all (but not only) determinism; free will can exist even if determinism is true (Kane, 2011). A little more than 30 years ago, neuroscience and empirical psychology came into play. Although biological processes cannot be considered strictly deterministic on the observable level of brain functioning (nerve signal transmission), new methods of investigation of the brain, more and more precise, have established that the cerebral base is a necessary condition of behavior and even of mental phenomena. On the basis of these acquisitions, neuroscience has begun to provide experimental contributions to the debate on free will.

In order to better understand the neural bases of free will, provided that there are any, in this article I’ll review and integrate findings from studies in different fields (philosophy, cognitive neuroscience, experimental and clinical psychology, neuropsychology). Unlike previous reviews on free will and neuroscience (Haggard, 2008, 2009; Passingham et al., 2010; Roskies, 2010a; Brass et al., 2013), I have no claim of being exhaustive. My goal is to highlight a paradigm shift in the analysis and interpretation of the brain determinants preceding and/or causing free or voluntary action (Haggard, 2008 takes voluntary decision to be non-stimulus driven, as much as possible). Firstly, following Libet’s experiments, a widespread interpretation of the so-called readiness potential (RP) went in the direction of a deflation of freedom (Crick, 1994; Greene and Cohen, 2004; Cashmore, 2010; Harris, 2012). Indeed, the discovery of the role of the RP has been taken as evidence of the fact that free will is an illusion, since it seems that specific brain areas activate before we are aware of the onset of the movement.

However, recent studies seem to point to a different interpretation of the RP, namely that the apparent build-up of the brain activity preceding subjectively spontaneous voluntary movements (SVM) may reflect the ebb and flow of the background neuronal noise, which is triggered by many factors (Schurger et al., 2016). This interpretation seems to bridge, at least partially, the gap between the neuroscientific perspective on free will and the intuitive, commonsensical view of it (Roskies, 2010b), but many problems remain to be solved and other theoretical paths can be hypothesized. After analyzing the change of paradigm of these perspectives, I’ll propose to start from an operationalizable concept of free will (Lavazza and Inglese, 2015) to find a connection between higher order descriptions (useful for practical life) and neural bases.

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Interesting sigmond. Thanks for sharing :)
 
This article made me laugh. Its a pop science type article but the research was carried out by Keith Chen et al who a yale economist/psychologist and did a series of studies on capuchin monkeys. You can dig up the citations of their three year study if you want to learn more. In this study they taught capuchin monkeys the concept of money and observed how their behavior changed. Prostitution and stealing immediately developed as a consequence.

http://www.zmescience.com/research/...g-after-the-first-prostitute-monkey-appeared/
 
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