Inductionis a term usually used in cases concerning probability, in which the
conclusion can be false even when the premises are true. In contrast to deduction, the
premise provides grounds for the conclusion, but it does not necessitate it. Inductive
logic generates “correct” conclusions based on observation or data. (But note that not
all inductive logic leads to correct generalizations, making the validity of many such
arguments probabilistic or “iffy” by nature.)
One can see how both these processes work in the scientific world, especially in the
scientific method in which general principles are inferred from certain facts. For exam-
ple, by observation of events (induction) and from principles already developed (deduc-
tion), new hypotheses are formulated. Hypotheses are then tested by applications; and
as the results satisfy the conditions of the hypotheses, laws are developed by induction.
Future laws are then often developed, many of them determined by deduction.
What is a conclusionin logic?
A conclusion is a statement (proposition) found by applying a set of logical rules (syl-
logisms) to a set of premises. In addition, the final statement of a proof is called the
proof’s conclusion. For example, in a statement that includes “if ... then,” the result
following the “then” in the statement is called the conclusion. 119
FOUNDATIONS OF MATHEMATICS
What is modus ponens?
T
he Latin term modus ponensmeans “mode that affirms,” or in the case of
logic, stands for the rule of detachment. This rule (also known as a rule of
inference) pertains to the “if ... then” statement and forms the basis of most
proofs: “If pthen q,” or if pis true, then the conclusion qis true. It is often seen
as the following:If p, then q.
p. Therefore, q.
To see this another way:
p &q: “If it is raining, then there are clouds in the sky.”
p: “It is raining.”
q: “There are clouds in the sky.”
There are several ways to break down the modus ponens. The argument
form has two premises: The “if-then” (or conditional claim), or namely that p
implies q; and that p (called the antecedent of the conditional claim) is true.
From these two premises it can be logically concluded that q (called the conse-
quent of the conditional claim) must be true as well; in other words, if the
antecedent of a conditional is true, then the consequent must be true.