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The word probabilityhas been used in a variety of ways since it was first coined in relation togames of chance. Does probability measure the real, physical tendency of something to occur, or is it just a measure of how strongly one believes it will occur? In answering such questions, we interpret the probability values of probability theory.
There are two broad categories of probability interpretations which can be called 'physical' and 'evidential' probabilities. Physical probabilities, which are also called objective or frequency probabilities, are associated with random physical systems such as roulette wheels, rolling dice and radioactive atoms. In such systems, a given type of event (such as the dice yielding a six) tends to occur at a persistent rate, or 'relative frequency', in a long run of trials. Physical probabilities either explain, or are invoked to explain, these stable frequencies. Thus talk about physical probability makes sense only when dealing with well defined random experiments. The two main kinds of theory of physical probability are frequentist accounts (such as those of Venn, Reichenbach and von Mises) and propensity accounts (such as those of Popper, Miller, Giere and Fetzer).
Evidential probability, also called Bayesian probability (or subjectivist probability), can be assigned to any statement whatsoever, even when no random process is involved, as a way to represent its subjective plausibility, or the degree to which the statement is supported by the available evidence. On most accounts, evidential probabilities are considered to be degrees of belief, defined in terms of dispositions to gamble at certain odds. The four main evidential interpretations are the classical (e.g. Laplace's) interpretation, the subjective interpretation (de Finetti and Savage), the epistemic or inductive interpretation (Ramsey, Cox) and the logical interpretation (Keynes and Carnap).
Some interpretations of probability are associated with approaches to statistical inference, including theories of estimation and hypothesis testing. The physical interpretation, for example, is taken by followers of "frequentist" statistical methods, such as R. A. Fisher, Jerzy Neyman and Egon Pearson. Statisticians of the opposing Bayesian school typically accept the existence and importance of physical probabilities, but also consider the calculation of evidential probabilities to be both valid and necessary in statistics. This article, however, focuses on the interpretations of probability rather than theories of statistical inference.
The terminology of this topic is rather confusing, in part because probabilities are studied within so many different academic fields. The word "frequentist" is especially tricky. To philosophers it refers to a particular theory of physical probability, one that has more or less been abandoned. To scientists, on the other hand, "frequentist probability" is just what philosophers call physical (or objective) probability. Those who promote Bayesian inference view "frequentist statistics" as an approach to statistical inference that recognises only physical probabilities. Also the word "objective", as applied to probability, sometimes means exactly what "physical" means here, but is also used of evidential probabilities that are fixed by rational constraints, such as logical and epistemic probabilities.
Classical definition
The first attempt at mathematical rigour in the field of probability, championed by PierreSimon Laplace, is now known as the classical definition. Developed from studies of games of chance (such as rolling dice) it states that probability is shared equally between all the possible outcomes, provided these outcomes can be deemed equally likely .
This can be represented mathematically as follows: If a random experiment can result in N mutually exclusive and equally likely outcomes and if N_{A} of these outcomes result in the occurrence of the event A, the probability of A is defined by P(A) = {N_A \over N} .
There are two clear limitations to the classical definition. Firstly, it is applicable only to situations in which there is only a 'finite' number of possible outcomes. But some important random experiments, such as tossing a coin until it rises heads, give rise to an infinite set of outcomes. And secondly, you need to determine in advance that all the possible outcomes are equally likely without relying on the notion of probability to avoid circularityâ€”for instance, by symmetry considerations.
Frequentism
Frequentists posit that the probability of an event is its relative frequency over time, i.e., its relative frequency of occurrence after repeating a process a large number of times under similar conditions. This is also known as aleatory probability. The events are assumed to be governed by some random physical phenomena, which are either phenomena that are predictable, in principle, with sufficient information (see Determinism); or phenomena which are essentially unpredictable. Examples of the first kind include tossing dice or spinning a roulette wheel; an example of the second kind is radioactive decay. In the case of tossing a fair coin, frequentists say that the probability of getting a heads is 1/2, not because there are two equally likely outcomes but because repeated series of large numbers of trials demonstrate that the empirical frequency converges to the limit 1/2 as the number of trials goes to infinity.
If we denote by \textstyle n_a the number of occurrences of an event \mathcal{A} in \textstyle n trials, then if \lim_{n \to \infty}{n_a \over n}=p we say that \textstyle P(\mathcal{A})=p
The frequentist view has its own problems. It is of course impossible to actually perform an infinity of repetitions of a random experiment to determ
In mathematical logic, satisfiability and validityare elementary concepts concerninginterpretation. A formula is satisfiable with respect to a class of interpretations if it is possible to find an interpretation that makes the formula true. A formula is valid if all such interpretations make the formula true. These notions can be relativised to satisfiability and validity within an axiomatic theory, where we count only interpretations that make all axioms of that theory true.
The opposites of these concepts are unsatisfiability and invalidity, that is, a formula is unsatisfiable if none of the interpretations make the formula true, and invalid if some such interpretation makes the formula false.
These four concepts are related to each other in a manner exactly analogous to Aristotle's square of opposition.
The four concepts can be raised to apply to whole theories: a theory is satisfiable (valid) if one (all) of the interpretations make(s) each of the axioms of the theory true, and a theory is unsatisfiable (invalid) if all (one) of the interpretations make(s) each of the axioms of the theory false.
Reduction of validity to satisfiability
For classical logics, it is generally possible to reexpress the question of the validity of a formula to one involving satisfiability, because of the relationships between the concepts expressed in the above square of opposition. In particular Ï† is valid if and only if Â¬Ï† is unsatisfiable, which is to say it is not true that Â¬Ï† is satisfiable.
Propositional satisfiability
In the case of classical propositional logic, satisfiability is decidable for propositional formulae. In particular, satisfiability is an NPcomplete problem, and is one of the most intensively studied problems in computational complexity theory.
Satisfiability in firstorder logic
Satisfiability is undecidable and indeed it isn't even a semidecidable property of formulae in firstorder logic (FOL). This fact has to do with the undecidability of the validity problem for FOL. The universal validity of a formula is a semidecidable problem. If satisfiability were also a semidecidable problem, then the problem of the existence of countermodels would be too (a formula has countermodels iff its negation is satisfiable). So the problem of logical validity would be decidable, which contradicts the ChurchTuring theorem.
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Answers:"There are no rules, not even logical ones." is itself a rule. So your proposed solution defeats itself. A retreat into irrationality is not the proper response to this sort of fallacy since rationality and logic can deal with it perfectly well. The problem is two things. Firstly, that you think there's actually a stone. And secondly, that you think that it's a limitation on power to be unable to do things with nothing. But neither of these are true. Elaborating on this we can say: 1) The phrase "a stone created by an omnipotent being that is so heavy that that being can't lift it" has no referent in logical space. That is, it picks out no possible thing. When we make the sound "the stone" followed by a certain description we are inclined to think, because we can form clear mental images of various sorts of stones, that the description must actually be of some sort of possible stone or another ... but it's not. It's not a stone and neither is it anything else. It's utterly and absolutely nothing whatsoever ... not anywhere or anytime or anyhow. It is no more real than round squares or fourcornered triangles. and 2) A being's potency is not diminished by being unable to do things with impossibilia (necessary nonexistents), only with possibilia. That is, the space of possible actions is the only space relevant to determining how powerful a being is; not the null space of impossible actions. Power is about efficacious actions, and a being is powerful exactly insofar as possible actions are available to them, not insofar as nonactions are available. Action on nothing is a nonaction. So, in summary, no such stone exists anywhere or anytime to be or have been either created or lifted or for any other positively specified thing whatsoever to be done to it or be true of it. If there are any created things, then an omnipotent being can create them. If there are any lifted things, then an omnipotent being can lift them. But for things that are neither stones nor any other thing at all then neither creating nor lifting nor anything else is done with or to them by anything at all, omnipotent or otherwise. None of the above constitutes a proof of an omnipotent being of any sort. It only shows that the stoneparadox and similar arguments can't show the concept of such a being to be inconsistent.
Answers:Yes, isn't it funny that they spend all day saying "You don't know what socialism is," then they NEVER demonstrate that they know, either. Most internet discussion forums seem to be heavily populated with individuals who go to great lengths to extol the virtures of socialism. The problem is, none of them seem to agree about what socialism actually is. See the answers to my question: What IS socialism? An escape from poverty, or a luxury for prosperous exporter nations? I have asked this question several times. Each time, I get answers from around a dozen persons. And no two of them give the same answer. But each person is supremely confident that his or her definition is the right one, and that this is the authoritative definition that we should all view as the true meaning of socialism. Even more amusing is, even though the other answers are plainly visible to them, each person seems blissfully aware that nobody else agrees with their chosen definition. Less amusingly, even though they consistently demonstrate a total lack of consensus about what socialism is, they frequently mock and berate opponents of socialism for "not understanding" what socialism is, then they invariably fail to demonstrate that they understand, either. Their answers tend to fall into four general categories: The purist view: Socialism has never existed anywhere. No nation has ever established a pure socialist state according to the original definition of socialism. Subgroups of this view include those who declare that socialism is a stepping stone to communism, which is correct according to the original definition; others dutifully recite the definition from the dictionary. The indiscriminate vew: Socialism already exists almost everywhere. All forms of public works and charity are socialist, because they are done to benefit all of society. Roads, bridges, schools, police/fire departments, national defense, and charities to help the poor are all forms of socialism. And almost all nations have at least some of these, therefore all nations are socialist. The progressive view: Socialism exists in certain "progressive" nations, such as Canada and many countries in Europe. Some individuals count Japan as a "progressive" socialist nation. The individuals who hold to this view seem unaware that most of these so called "progressive" nations are highly capitalistic exporter nations, which seems to contradict the idea of having some form of Marxism. And the "progressive" nations that are not strong exporters are debtor nations. See this website to get the economic statstics for these nations: http://www.tradingeconomics.com/Economics/BalanceOfTrade.aspx?Symbol=NOK The remnant of communism view: Socialism exists in formerly communist nations that have not yet fully abandoned Marxism, and they are making a gradual transition to capitalism. This view is rarely cited, but I include it here for completeness. Of course, there are also various answers from those who strenuously oppose socialism. I am focused here on the answers from those who advocte socialism, or who give a neutral but objective answer. Tne only thing that seems to be the common thread of socialist advocacy is: taking money from those who have it, and giving money to those who don't. And we're all supposed to believe that the advocates have no motive other than pure unaduterated altruism in their hearts. So the fundamental question is: how can we regard socialism as a viable political cause when even its most ardent advocates can't agree on what it is? And when there is scant actual evidence that it exists or ever has existed, other than as a remnant of a failed communist state?
Answers:This is a constructive comment to your Q and is intended to add to it's educational value. I'm sure you meant to say: "A deductive argument is valid if and only if the truth of its premises entails the truth of its conclusion. Not so, if we refer to *inductive* arguments where relevant premises do not entail, but only provide a degree of support for the conclusion. For more understanding see: http://www.jimpryor.net/teaching/vocab/validity.html "Most of the arguments philosophers concern themselves with areor purport to bedeductive arguments. Mathematical proofs are a good example of deductive argument. Most of the arguments we employ in everyday life are not deductive arguments but rather inductive arguments. Inductive arguments are arguments which do not attempt to establish a thesis conclusively. Rather, they cite evidence which makes the conclusion somewhat reasonable to believe. The methods Sherlock Holmes employed to catch criminals (and which Holmes misleadingly called "deduction") were examples of inductive argument. Other examples of inductive argument include: concluding that it won't snow on June 1st this year, because it hasn't snowed on June 1st for any of the last 100 years; concluding that your friend is jealous because that's the best explanation you can come up with of his behavior, and so on. It's a controversial and difficult question what qualities make an argument a good inductive argument. Fortunately, we don't need to concern ourselves with that question here. In this class, we're concerned only with deductive arguments. Philosophers use the following words to describe the qualities that make an argument a good deductive argument: Valid Arguments We call an argument deductively valid (or, for short, just "valid") when the conclusion is entailed by, or logically follows from, the premises. Validity is a property of the argument's form. It doesn't matter what the premises and the conclusion actually say. It just matters whether the argument has the right form. So, in particular, a valid argument need not have true premises, nor need it have a true conclusion. The following is a valid argument: All cats are reptiles. Bugs Bunny is a cat. So Bugs Bunny is a reptile. Neither of the premises of this argument is true. Nor is the conclusion. But the premises are of such a form that if they were both true, then the conclusion would also have to be true. Hence the argument is valid. To tell whether an argument is valid, figure out what the form of the argument is, and then try to think of some other argument of that same form and having true premises but a false conclusion. If you succeed, then every argument of that form must be invalid. A valid form of argument can never lead you from true premises to a false conclusion. " nb: I too would like to ask: "Why is the thumbsdown dwarf here? You would think that an *ignorant*dwarf would have better things to do..."
Answers:We use the concepts of ratios and rational numbers in cases where we're not exactly talking about fractions, although there is a mathematical equivalence. We could refer to any portion less than the whole of something as a "fraction" of it, but some fractions (by that definition) are not rational numbers. Between 0 and 1, there are actually an infinite number of rational numbers, but there are also an infinite number of irrational ones. The rational and irrational numbers together make up the real numbers. Back to this in a moment, but first let me deal with specific cases you brought up. x/0 is not a fraction for any real number x, nor is it a rational or irrational number. Its value is undefined. p/1 is a rational number for any integer p. The integers are a subset of the rational numbers. All decimals are fractions. Decimals are not a separate type of number, but just a particular way of writing numbers. But decimals only represent rational numbers if they meet one of two conditions: (1) they have a finite length, or (2) at some point, they repeat a sequence infinitely. For example, 2.1234 represents the ratio 21,234 / 100,000 and is therefore rational. 3.63636363.... (infinitely repeating "63"s forever) represents the ratio 40/11 and is therefore rational. But there are other numbers which, if converted to decimal form, would continue infinitely and never repeat. These include the square root of 3 or PI, the ratio of a circle's circumference to its diameter. These are irrational. They are real numbers, but they cannot be represented as a ratio of integers. So the concept of rationality turns out to be a bit more farreaching than the notion of "fractions." That's why the emphasis. Just as the set of natural numbers was basic knowledge for a lot of math, the set of rationals becomes the basis for a lot more.
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