To infer is to pass from premises to a conclusion by a rule; the rule is judged on its form, not on the content.

Inference

Definition

To infer is to pass from accepted propositions — the premises — to a new proposition — the conclusion — by following a rule.

The rule does not look at what is being talked about. It looks only at the form of the linkage.

Valid is not true

These are two distinct judgements, bearing on two distinct objects.

	Bears on	Question asked
Validity	The form of the argument	Does the conclusion follow from the premises?
Truth	The content of the propositions	Do the premises describe the world?

"All cats fly; Felix is a cat; therefore Felix flies" is valid and its conclusion false. The form is beyond reproach, the first premise is not.

An argument both valid and with true premises is called sound. Its conclusion is then guaranteed true — that is the whole point of deduction.

Two rules, and their two counterfeits

Take the implication "if $P$ then $Q$ ".

Form	Premises	Conclusion	Status
Modus ponens	$P \Rightarrow Q$ , and $P$	$Q$	Valid
Modus tollens	$P \Rightarrow Q$ , and $\neg Q$	$\neg P$	Valid
Affirming the consequent	$P \Rightarrow Q$ , and $Q$	$P$	Invalid
Denying the antecedent	$P \Rightarrow Q$ , and $\neg P$	$\neg Q$	Invalid

The last two resemble the first two and are worth nothing. "If it rains, the street is wet; the street is wet; therefore it rains" forgets the sprinkler: an implication does not forbid other causes.

Three ways of inferring

Type	Movement	Guarantee
Deduction	From general to particular	Preserves truth
Induction	From particular to general	None — supports without proving
Abduction	From effect to its best cause	None — proposes, does not prove

Only deduction transmits the truth of the premises to the conclusion. The other two extend it beyond what the premises contain: they produce hypotheses, not theorems.

This difference is not a defect of the latter two. No new knowledge would ever come out of deduction alone, which merely makes explicit what the premises already carried.

What a counterexample refutes

To show that a form is invalid, a single interpretation suffices in which the premises are true and the conclusion false.

It is the same gesture as negating a universal statement: one case is enough, and it bears on the form, never on a particular example of content.

Summary

Validity judges the form, truth judges the content; both are needed to conclude.