INFERENCE

1. A Primitive Concept

The concept of inference is primitive, meaning it can only be defined based on equally complex concepts or through an example of inference from the field of logic. For instance, Brody (1967, pp. 66–67) defines inference as « the derivation of a proposition (the conclusion) from a set of other propositions (the premises). » Inference establishes new truths based on known truths.
In an extended sense, the term « inference » refers to any derivation of an accepted proposition based on the prior acceptance of other propositions.

There are two types of inference, inference strictly speaking and immediate inference.
– In immediate inference, the conclusion is derived from a single proposition, see proposition, §4.
– Strict, or direct inference is based on two propositions, its premises, see syllogism.

2. Deductive and Inductive Inferences in Traditional Logic

Traditional logic distinguishes between deductive inferences (deduction) and inductive inferences (induction). The valid conclusion from true premises in a syllogistic deduction  is necessarily true (« apodictic »),  while an inductive conclusion is only probable.
Analogical inference is accepted only as a heuristic tool, with no evidential value, see analogy.

Generalization / Restriction – Deduction and induction are considered two complementary processes.
– Induction goes from the particular to the general, or from the general to the most general, « This Syldavian is redheaded, therefore, Syldavians are redheaded. »
– Deduction, on the other hand, goes from the most general to the least general, « All men are mortal, therefore Athenians are mortal,” and then to the individual, « Socrates is mortal. »
However, syllogistic deduction can also generalize:
« All horses are mammals, all mammals are vertebrates, therefore all horses are vertebrates. »

Other forms of reasoning – According to Aristotle’s view of rhetoric, the enthymeme is the argumentative counterpart of deductive inference, and the example is the counterpart of inductive inference.
Wellman (1971) considers conduction to be a special kind of inference on a par with deduction and induction.
Since Toulmin (1958), the study of everyday reasoning has developed on the basis of an open number of reasoning schemes (argument schemes).

3. Direct Inference and Analytic Statements

An analytic proposition is a proposition that is true “by definition”, i.e., by virtue of its meaning. Good definitions are analytically true, “a single person is an adult unmarried adult.”

Direct logical inference is based on quantifiers or “empty words.”
Immediate analytical inference operates on the meaning of the “full words” of the basic proposition, « He is single, so he is not married. »

In arguments such as, “This is our duty, therefore we must do it,” the proposition “we must do it,” introduced by therefore,  is semantically contained in the argument “it is our duty.” By definition a duty is something that people must do. This conclusion, if it is a conclusion at all, is direct.
More generally, an analytic inference is one in which the conclusion is embedded in the argument, and the conclusion merely develops the argument’s semantic content. For example, If I’m told that my colleague recently “quit smoking” I can analytically infer that he or she smoked in the past, see. presupposition.

Consider the following example:

Talking about the birth of the gods implies that at one time the gods did not exist. Talking about the death of the gods is just as impious as talking about their birth. For this, your colleague was recently sentenced to death.

Birth is defined as the “beginning of life.” However, this definition does not directly imply the threatening conclusion; an additional step is required to explicitly define “beginning”, which establishes an equivalence between the time after death and the time before birth. For this reason, the conclusion does not seem as obvious as in the previous cases.

3. Pragmatic Inference

This concept explains how utterances are interpreted in discourse. In the dialogue:

S1    — Did you see anyone I know at the party?
?S2    — Oh yes, Peter, Paul, and Mary.

From S2’s answer, S1 will infer that S2 did not meet anyone else whom they both know. This inference is based on the maxim of quantity: “When asked a question, provide the most accurate information possible, both quantitatively and qualitatively”.
Therefore, if S2 met Bruno at the party–a person known to S1–then S2 can be said to have lied to S1 by omission, see cooperation.