What is the difference between a priori and a posteriori probability

Sunday, June 5, The Difference between a priori and a posteriori Probability. Imagine a pure thought experiment: an abstract, logical world where everything is true by definition. In this world, we have a fair dice. That probability has necessary truth — but limited only to the abstract fair game of dice one is imagining.

The analytic a priori propositions that express the imaginary world and probability in question do have necessary truth — but limited to this abstract analytic a priori system. This is the epistemological nature of a priori probabilities.

But what happens when we step into the real world? No matter how much evidence you have that the dice is fair, there is still a tiny possibility you are mistaken. There is, to be sure, a close connection between the concepts. For instance, if the truth of a certain proposition is, say, strictly a matter of the definition of its terms, knowledge of this proposition is unlikely to require experience rational reflection alone will likely suffice.

On the other hand, if the truth of a proposition depends on how the world actually is in some respect, then knowledge of it would seem to require empirical investigation.

Despite this close connection, the two distinctions are not identical. It is open to question, moreover, whether the a priori even coincides with the analytic or the a posteriori with the synthetic. First, many philosophers have thought that there are or at least might be instances of synthetic a priori justification. Consider, for example, the claim that if something is red all over then it is not green all over.

Belief in this claim is apparently justifiable independently of experience. Simply by thinking about what it is for something to be red all over, it is immediately clear that a particular object with this quality cannot, at the same time, have the quality of being green all over. But it also seems clear that the proposition in question is not analytic. Being green all over is not part of the definition of being red all over, nor is it included within the concept of being red all over.

If examples like this are to be taken at face value, it is a mistake to think that if a proposition is a priori, it must also be analytic. Second, belief in certain analytic claims is sometimes justifiable by way of testimony and hence is a posteriori. It is possible even if atypical for a person to believe that a cube has six sides because this belief was commended to him by someone he knows to be a highly reliable cognitive agent.

Such a belief would be a posteriori since it is presumably by experience that the person has received the testimony of the agent and knows it to be reliable. Thus it is also mistaken to think that if a proposition is a posteriori, it must be synthetic. Third, there is no principled reason for thinking that every proposition must be knowable. Some analytic and some synthetic propositions may simply be unknowable, at least for cognitive agents like us.

We may, for instance, simply be conceptually or constitutionally incapable of grasping the meaning of, or the supporting grounds for, certain propositions.

This raises the question of the sense in which a claim must be knowable if it is to qualify as either a priori or a posteriori. For whom must such a claim be knowable? Any rational being? Any or most rational human beings? God alone? There may be no entirely nonarbitrary way to provide a very precise answer to this question. A necessary proposition is one the truth value of which remains constant across all possible worlds.

Thus a necessarily true proposition is one that is true in every possible world, and a necessarily false proposition is one that is false in every possible world. By contrast, the truth value of contingent propositions is not fixed across all possible worlds: for any contingent proposition, there is at least one possible world in which it is true and at least one possible world in which it is false. It is reasonable to expect, for instance, that if a given claim is necessary, it must be knowable only a priori.

Sense experience can tell us only about the actual world and hence about what is the case; it can say nothing about what must or must not be the case. Contingent claims, on the other hand, would seem to be knowable only a posteriori, since it is unclear how pure thought or reason could tell us anything about the actual world as compared to other possible worlds.

While closely related, these distinctions are not equivalent. Therefore, even if the two distinctions were to coincide, they would not be identical. But there are also reasons for thinking that they do not coincide. Some philosophers have argued that there are contingent a priori truths Kripke ; Kitcher b. An example of such a truth is the proposition that the standard meter bar in Paris is one meter long.

This claim appears to be knowable a priori since the bar in question defines the length of a meter. And yet it also seems that there are possible worlds in which this claim would be false e. Comparable arguments have been offered in defense of the claim that there are necessary a posteriori truths. Take, for example, the proposition that water is H 2 O ibid. It is conceivable that this proposition is true across all possible worlds, that is, that in every possible world, water has the molecular structure H 2 O.

But it also appears that this proposition could only be known by empirical means and hence that it is a posteriori.

Finally, on the grounds already discussed, there is no obvious reason to deny that certain necessary and certain contingent claims might be unknowable in the relevant sense. If indeed such propositions exist, then the analytic does not coincide with the necessary, nor the synthetic with the contingent. In Section 1 above, it was noted that a posteriori justification is said to derive from experience and a priori justification to be independent of experience.

There is no widely accepted specific characterization of the kind of experience in question. Philosophers instead have had more to say about how not to characterize it. There is broad agreement, for instance, that experience should not be equated with sensory experience, as this would exclude from the sources of a posteriori justification such things as memory and introspection.

It would also exclude, were they to exist, cognitive phenomena like clairvoyance and mental telepathy. Such exclusions are problematic because most cases of memorial and introspective justification resemble paradigm cases of sensory justification more than they resemble paradigm cases of a priori justification.

It would be a mistake, however, to characterize experience so broadly as to include any kind of conscious mental phenomenon or process; even paradigm cases of a priori justification involve experience in this sense. This is suggested by the notion of rational insight, which many philosophers have given a central role in their accounts of a priori justification.

There is, however, at least one apparent difference between a priori and a posteriori justification that might be used to delineate the relevant conception of experience see, e. In the clearest instances of a posteriori justification, the objects of cognition are features of the actual world which may or may not be present in other possible worlds.

Moreover, the relation between these objects and the cognitive states in question is presumably causal. But neither of these conditions would appear to be satisfied in the clearest instances of a priori justification.

In such cases, the objects of cognition would appear at least at first glance to be abstract entities existing across all possible worlds e. Further, it is unclear how the relation between these objects and the cognitive states in question could be causal. While these differences may seem to point to an adequate basis for characterizing the relevant conception of experience, such a characterization would, as a matter of principle, rule out the possibility of contingent a priori and necessary a posteriori propositions.

But since many philosophers have thought that such propositions do exist or at least might exist , an alternative or revised characterization remains desirable.

It is also important to examine in more detail the way in which a priori justification is thought to be independent of experience. Here again the standard characterizations are typically negative. There are at least two ways in which a priori justification is often said not to be independent of experience. The first begins with the observation that before one can be a priori justified in believing a given claim, one must understand that claim.

The reasoning for this is that for many a priori claims experience is required to possess the concepts necessary to understand them Kant Consider again the claim that if something is red all over then it is not green all over. To understand this proposition, I must have the concepts of red and green, which in turn requires my having had prior visual experiences of these colors. It would be a mistake, however, to conclude from this that the justification in question is not essentially independent of experience.

My actual reason for thinking that the relevant claim is true does not emerge from experience, but rather from pure thought or rational reflection, or from simply thinking about the properties and relations in question. Moreover, the very notion of epistemic justification presupposes that of understanding. In considering whether a person has an epistemic reason to support one of her beliefs, it is simply taken for granted that she understands the believed proposition.

Therefore, at most, experience is sometimes a precondition for a priori justification. Second, many contemporary philosophers accept that a priori justification depends on experience in the negative sense that experience can sometimes undermine or even defeat such justification. One need not go out and check to make sure that wherever two boulders are lying next to two other boulders that there are, in fact, four boulders. Rather, such knowledge is known prior to any reference to the physical world or our five senses.

Going back as far as Plato, many philosophers have thus argued that fundamental truths, such as mathematics, logic, morality and even the existence of God can be innately known by the mind prior to any actual experience of the world. Instead of turning outward and using our 5 senses to experience the world outside, a priori knowledge turns inward and reflects upon what we seem to already know prior to any actual experience of the world. In contrast, a posteriori knowledge is gained only after sense experience has already occurred i.

One could never close their eyes, look within, and discover that the Titanic sunk on April 15, , or that water is two parts hyrdogen and one part oxygen.