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Probability interpretations

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 Pierre-Simon 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 NA 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

Satisfiability

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 NP-complete problem, and is one of the most intensively studied problems in computational complexity theory.

## Satisfiability in first-order logic

Satisfiability is undecidable and indeed it isn't even a semidecidable property of formulae in first-order logic (FOL). This fact has to do with the undecidability of the validity problem for FOL. The universal validity of a formula is a semi-decidable problem. If satisfiability were also a semi-decidable problem, then the problem of the existence of counter-models would be too (a formula has counter-models iff its negation is satisfiable). So the problem of logical validity would be decidable, which contradicts the Church-Turing theorem.

Question:i have seen people use this argument to prove that there is no god. they say that god cannot be omnipotent, as he cannot create a stone that he cannot lift. if he could, there would be something he couldn't do, ie. lift the stone. thus, he cannot be omnipotent, and the Judeo-christian god cannot be. i think this is a fallacy. it has a premise which makes no sense. it is like the similar question, what happens when an unstoppable force hits an unbreakable shield? does it break (thus not unbreakable) or does the force dissipated (thus not unstoppable). this is a ridiculous question, because the premise is self contradictory. either there is an unstoppable force, so there is no shield that cannot be broken, or there is an unbreakable shield, hence no unstoppable force. the two are mutually exclusive. all the question is asking is "what happens when something impossible occurs?" this is an illogical question. the definition of impossible is that it will never occur, so asking what happens when it does is ridiculous. this obviously only applies when natural laws are followed. when they are broken, then there is no reason to worry about what happens: we cannot know without knowing which laws were broken, how, and what the effect is. thus, the question cannot be answered, as it is a fallacious question. the same goes for the original question. you are presupposing that there is such a theoretical concept as a stone God cannot lift. this is a fallacy. there is no such thing as a problem an omnipotent God cannot do: trying to think of such a concept is a waste of time. the words "a stone God cannot lift" mean nothing real. it is like asking "what happens when something both occurs and doesn't occur?" as it defies logic, the answer is not going to be logical. thus, the answer to the question "can God create a stone he cannot lift?" is that there is no question, and that the premise is fatally flawed. this is all assuming that we mean "cannot lift with no miracles." however, with a miracle (ie. a breach of natural laws) anything is possible: there are no rules, not even logical ones.

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 four-cornered 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 stone-paradox and similar arguments can't show the concept of such a being to be inconsistent.

Question:Democrats get called socialist, instead of defending themselves on why they are not socialist they simply state: "learn the definition of socialism" Are they only capable of repeating what obama whispers in their ears? well here it is for everyone The definition of socialism as Stated by the Mariam Webster dictionary "Main Entry: so cial ism Pronunciation: \ s -sh - li-z m\ Function: noun Date: 1837 1 : any of various economic and political theories advocating collective or governmental ownership and administration of the means of production and distribution of goods 2 a : a system of society or group living in which there is no private property b : a system or condition of society in which the means of production are owned and controlled by the state 3 : a stage of society in Marxist theory transitional between capitalism and communism and distinguished by unequal distribution of goods and pay according to work done" I love how a majority of the answers to this question were instead about whether or not democrats are socialist instead of if the argument they use is valid.

Question:I'm talking about the technical definition of validity, which says: An argument is valid if and only if the truth of its premises entails the truth of its conclusion. Heeltap: I meant exactly what I said. An inductive argument with true premises cannot entail the truth of its conclusion. It can only make the conclusion more probable. That is why the notion of validity doesn't apply to inductive arguments. Inductive arguments can be considered "inductively strong" or "inductively weak."

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 are--or purport to be--deductive 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 thumbs-down dwarf here? You would think that an *ignorant*dwarf would have better things to do..."

Question:A rational number is an element belonging to Q , the set of all numbers which can be represented in the form p/q where q!=0. Why do people stress on this condition so much ? p/q where q!=0 , cant they just tell fractions ? This gets me into this question , p/0 is not a fraction ? What about p/1 ? And all decimals are fractions , right ? If at all p/1 is a fraction , then it is a decimal p.000 , so does it belong to the natural/whole/integer(Z) set ? Thanks!

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 far-reaching 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.