Probability Theory: the logic of science
Plausible reasoning
The actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore the true logic for this world is the calculus of Probabilities, which takes account of the magnitude of the probability which is, or ought to be, in a reasonable man's mind.
–Maxwell
Aristotle deductive reasoning(apodeixis)
strong syllogism
- if A is true, then B is true; now A is true, therefore B is true
- if A is true, then B is true; now B is false, therefore A is false
Plausible reasoning
weak syllogism(epagoge)
- if A is true, then B is true; now B is true, therefore A becomes more plausible.
- if A is true, then B is true, now A is false, therefore B becomes less plausible.
weaker syllogism
- if A is true, then B becomes more plausible; now B is true, therefore, A becomes more plausible.
In science, though scientists try hard to invent theory which uses only deductive reasoning, palusible reasoning always exits in the conjectures suggested by analogies.
Boolean algebra
- logical product(conjunction): $AB$
- logical sum(disjunction): $A+B= B+A$
- Idempotence: $AA=A,A+A=A$
- Commutativity:$AB=BA,A+B=B+A$
- Associativity: $A(BC)=(AB)C=ABC,A+B+C=(A+B)+C=A+(B+C)$
- Distributivity:$A(B+C)=AB+AC,A+(BC)=(A+B)(A+C)$
- Duality: $if\ C=AB, then\ \overline C=\overline A+\overline B$,$if\ D=A+B, then\ \overline D=\overline A \overline B$
Proposition
$A \Rightarrow B$
A implies B, it means $A\overline B$ or $\overline A + B$ or $A=AB$
Adequate sets of operations
- 原文作者:mlyixi
- 原文链接:https://mlyixi.github.io/post/math/Probability-Theory-the-logic-of-science/
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