reading notes of《Probability Theory - The Logic of Science》

文章目录

1.Plausible reasoning
- 1.1.Deductive and plausible reasoning
- 1.2.Analogies with physical theories
- 1.3.The thinking computer
- 1.4.Introducing the robot
- 1.5.Boolean algebra
- 1.6.Adequate sets of operations
- 1.7.The basic desiderata
- 1.8.Comments
- - 1.8.1.Common language vs. formal logic
  - 1.8.2.Nitpicking

1.Plausible reasoning

Suppose some dark night a policeman walks down a street, apparently deserted. Suddenly he hears a burglar alarm, looks across the street, and sees a jewelry store with a broken window. Then a gentleman wearing a mask comes crawling out through the broken window, carrying a bag which turns out to be full of expensive jewelry. The policeman doesn’t hesitate at all in deciding that this gentleman is dishonest. But by what reasoning process does he arrive at this conclusion? Let us first take a leisurely look at the general nature of such problems.

1.1.Deductive and plausible reasoning

The evidence did not make the gentleman’s dishonesty certain, but it did make it extremely plausible.
As is generally credited to the Organon of Aristotle (fourth century bc) deductive reasoning (apodeixis) can be analyzed ultimately into the repeated application of two strong syllogisms:
- if A is true, then B is true - Aistruetherefore,Bistrue\frac{A\ is\ true}{therefore,\ B\ is \ true}therefore, B is trueA is true
- if A is true, then B is true - Bisfalsetherefore,Aisfalse\frac{B\ is \ false}{therefore,\ A \ is\ false}therefore, A is falseB is false
In almost all situations confronting us we do not have the right kind of information to allow this kind of reasoning. We fall back on weaker syllogisms (epagoge):
- if A is true, then B is true - Bistruetherefore,Abecomesmoreplausible\frac{B\ is\ true}{therefore,\ A\ becomes \ more \ plausible}therefore, A becomes more plausibleB is true
- if A is true, then B is true - Aisfalsetherefore,Bbecomeslessplausible\frac{A\ is\ false}{therefore,\ B\ becomes \ less\ plausible}therefore, B becomes less plausibleA is false
The reasoning of our policeman was not even of the above types. It is best described by a still weaker syllogism: if A is true, then B becomes more plausible - Bistruetherefore,Abecomesmoreplausible\frac{B\ is\ true}{therefore,\ A\ becomes \ more \ plausible}therefore, A becomes more plausibleB is true. But in spite of the apparent weakness of this argument, the policeman’s conclusion has a very strong convincing power.
This show that the brain, in doing plausible reasoning, not only decides whether something becomes more plausible or less plausible, but that it evaluates the degree of plausibility in some way. And the brain also makes use of old information (prior information) as well as the specific new data of the problem.
The same idea is expressed in a remark of S. Banach (quoted by S. Ulam, 1957): Good mathematicians see analogies between theorems; great mathematicians see analogies between analogies.

1.2.Analogies with physical theories

In physics, we learn quickly that the world is too complicated for us to analyze it all at once. We can make progress only if we dissect it into little pieces and study them separately. Sometimes, we can invent a mathematical model which reproduces several features of one of these pieces (prior information), and whenever this happens we feel that progress has been made. These models are called physical theories. As knowledge advances (specific new data of the problem), we are able to invent better and better models (more accurate conclusions), which reproduce more and more features of the real world, more and more accurately.
The analogy with physical theories is deeper than a mere analogy of method. Often, the things which are most familiar to us turn out to be the hardest to understand.

1.3.The thinking computer

Just as von Neumann indicated, the only real limitations on making ‘machines which think’ are our own limitations in not knowing exactly what “thinking” consists of.
Every time we can construct a mathematical model which reproduces a part of common sense by prescribing a definite set of operations, this shows us how to ‘build a machine’, (i.e. write a computer program) which operates on incomplete information and, by applying quantitative versions of the above weak syllogisms, does plausible reasoning instead of deductive reasoning.

1.4.Introducing the robot

In order to direct attention to constructive things and away from controversial irrelevancies, we shall invent an imaginary being. Its brain is to be designed by us, so that it reasons according to certain definite rules.
Our robot is going to reason about propositions. We do not require that the truth or falsity of such an ‘Aristotelian proposition’ be ascertainable by any feasible investigation; indeed, our inability to do this is usually just the reason why we need the robot’s help.

1.5.Boolean algebra

The symbol AB called the logical product or the conjunction, the expression A+B called the logical sum or disjunction.
Two propositions with the same truth value are equally plausible.
The theory of plausible reasoning based on weak syllogisms is not a ‘weakened’ form of logic; it is an extension of logic with new content not present at all in conventional deductive logic. It will be clear that our rules include deductive logic as a special case.
Note carefully that in ordinary language one would take ‘A implies B’ to mean that B is logically deducible from A. But, in formal logic, ‘A implies B’ means only that the propositions A and AB have the same truth value. (A is false ⇒\Rightarrow⇒ B is true)
This great difference in the meaning of the word ‘implies’ in ordinary language and in formal logic is a tricky point that can lead to serious error if it is not properly understood.

1.6.Adequate sets of operations

An expression B = f(A1,…,An) involving n propositions is a logic function on a space S of M = 2ⁿ points; and there are exactly 2^M such functions.
Three operations {conjunction, disjunction, negation} i.e. {AND, OR, NOT}, suffice to generate all possible logic functions; or, more concisely, they form an adequate set. For disjunction of A,B is the same as denying that they are both false: A+B=(AˉBˉ)‾A+B=\overline{(\bar A \bar B)}A+B=(AˉBˉ), Therefore, the two operations {AND, NOT} already constitute an adequate set for deductive logic (Does it follow that these two commends are the only ones needed to write any computer program?)
The ‘NAND’ is defined as the negation of ‘AND’: A↑B≡AB‾=Aˉ+BˉA\uparrow B \equiv \overline{AB}=\bar A+\bar BA↑B≡AB=Aˉ+Bˉ, then we have at once
- Aˉ=A↑A\bar A= A\uparrow AAˉ=A↑A
- AB=(A↑B)↑(A↑B)AB=(A\uparrow B)\uparrow (A \uparrow B)AB=(A↑B)↑(A↑B)
- A+B=(A↑A)↑(B↑B)A+B=(A\uparrow A)\uparrow (B \uparrow B)A+B=(A↑A)↑(B↑B)
Therefore, every logic function can be constructed with NAND alone. (likewise the operation NOR)

1.7.The basic desiderata

To each proposition about which it reasons, our robot must assign some degree of plausibility, based on the evidence we have given it; and whenever it receives new evidence it must revise these assignments to take that new evidence into account.
We adopt a natural but nonessential convention: that a greater plausibility shall correspond to a greater number. For example, A∣B>C∣BA|B>C|BA∣B>C∣B says that, given B, A is more plausible than C.
It will also be convenient to assume a continuity property, which is hard to state precisely at this stage; to say it intuitively: an infinitesimally greater plausibility ought to correspond only to an infinitesimally greater number.
We make no attempt to define A|BC when B and C are mutually contradictory. Whenever such a symbol appears, it is understood that **B and C are compatible propositions. **
we want to give our robot another desirable property that it always reasons consistently.

1.8.Comments

Quite generally, the situations of everyday life are those involving many coordinates. It is just for this reason, we suggest, that the most familiar examples of mental activity are often the most difficult to reproduce by a model.
We stress that we are in no way asserting that degrees of plausibility in actual human minds have a unique numerical measure. Our job is not to postulate – or indeed to conjecture about – any such thing; it is to investigate whether it is possible, in our robot, to set up such a correspondence without contradictions.

1.8.1.Common language vs. formal logic

It appears to us that ordinary language, carefully used, need not be less precise than formal logic; but ordinary language is more complicated in its rules and has consequently richer possibilities of expression than we allow ourselves in formal logic.
Our aim is to avoid this by developing the general principles of inference once and for all, directly from the requirement of consistency, and in a form applicable to any problem of plausible inference that is formulated in a sufficiently unambiguous way.

1.8.2.Nitpicking

if a new logic was found to conflict with Aristotelian logic in an area where Aristotelian logic is applicable, we would consider that a fatal objection to the new logic.
An n-valued logic applied to one set of propositions is either equivalent to a two-valued logic applied to an enlarged set, or else it contains internal inconsistencies.

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概率沉思录：1.Plausible reasoning

文章目录

1.Plausible reasoning

1.1.Deductive and plausible reasoning

1.2.Analogies with physical theories

1.3.The thinking computer

1.4.Introducing the robot

1.5.Boolean algebra

1.6.Adequate sets of operations

1.7.The basic desiderata

1.8.Comments

1.8.1.Common language vs. formal logic

1.8.2.Nitpicking

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