Inference: In logic, the assertion of a conclusion, in general because one has already asserted (and thus accepted) certain premisses one considers sufficient to assert the conclusion.
In natural language, there are terms that mark conclusions and inferences: "therefore", "so", "since", "because", "ergo", "it follows", "entails" and others.
One important point to be clear about is that in an inference one detaches the conclusion from the premisses, and asserts the conclusion by itself because one considers the premisses sufficient for its assertion. This is valid in deductive logic in the sense that if the premisses are all true and the conclusion follows from the premisses, then the conclusion is also true.
There are three basic kinds of inference, that cover very many specific sorts of inferences:
1. Deductions: To find conclusions that follow from given assumptions. 2. Abductions: To find assumptions from which given given conclusions follows.
3. Inductions: To confirm or infirm (support or undermine) assumptions by showing their conclusions do (not) conform to the observable facts.
Normally in reasoning all three kinds are involved: We explain supposed facts by abductions; we check the abduced assumptions by deductions of the facts they were to explain; and we test the assumptions arrived by deducing consequences and then revise by inductions the probabilities of the assumptions by probabilistic reasoning when these consequences are verified or falsified.
In ordinary life, outside mathematics, logic or science, most conclusions that are inferred are not deductively valid as they are given, either because this would be difficult or needlessly pedantic or because there is no real need for a deductively valid conclusion.
Even so, the fundamental checks of any conclusion one regards as important are (1) whether one can supply a deductively valid argument for it and (2) whether that argument has only true or only probable premisses.
The first check should guarantee one has, at least in principle, an ifthen argument with the if made up of premisses and the then the conclusion, which, since it is a deduction, has the property that if the premisses are true, then the conclusion is true. The second check should make clear, at least in principle, how good that ifthen argument is, for a deductive argument has the property that its conclusion can not be less probable than the probability of its premisses, and so the less probable the premisses one needed to deduce the conclusion, the less probable that conclusion is, from those premisses. (For if pr(QT)=1 then pr(Q&T)=pr(T), and that may be quite small and indeed must be smaller than pr(S) for any S such that pr(ST)=1.)
Charles Sanders Peirce, who first saw this threesome of kinds of inference formulated it once as follow, in terms of syllogistic logic
DEDUCTION:
Rule.All the beans from this bag are white.
Case.These beans are from this bag.
.·.Result.These beans are white.
INDUCTION:
Case.These beans are from this bag.
Result.These beans are white.
.·.Rule.All the beans from this bag are white
HYPOTHESIS [ABDUCTION]:
Rule.All the beans from this bag are white.
Result.These beans are white.
.·.Case.These beans are from this bag.
(CP 2.623, 1878)
