Deduction CIS308 Dr Harry Erwin

Syllogism A syllogism consists of three parts: the major premise, the minor premise, and the conclusion. In Aristotle, each of the premises is in the form: "Some/all A belong to B," where "Some/All A' is one term and "belong to B" is another, but more modern logicians allow some variation. Each of the premises has one term in common with the conclusion: in a major premise, this is the major term (i.e., the predicate) of the conclusion; in a minor premise, it is the minor term (the subject) of the conclusion. For example: –Major premise: All humans are mortal. –Minor premise: Socrates is human. –Conclusion: Socrates is mortal.

Deductive reasoning Deductive reasoning is the kind of reasoning where the conclusion is necessitated by previously known premises. If the premises are true then the conclusion must be true. For instance, beginning with the premises "sharks are fish" and "all fish have fins", you may conclude that "sharks have fins".

Certainty This is distinguished from inductive reasoning and abductive reasoning where inferences can be made with some likelihood but never with complete certainty.

Deductive reasoning Deductive reasoning is dependent on its premises. That is, a false premise can possibly lead to a false result, and inconclusive premises will also yield an inconclusive conclusion.

Examples An example of deductive reasoning is the following: –All men are mortal (major premise), –Socrates is a man (minor premise), –Therefore Socrates is mortal. Note that replacing "mortal" with any nonsensical property will not affect the validity of the argument: –All men are purple-skinned, –Socrates is a man, –Therefore Socrates is purple-skinned. Intuitively, one might deny the major premise or the conclusion; yet anyone accepting the premises must accept the conclusion.

Popular misuses of the term It is occasionally taught that deductive reasoning proceeds from the general to the particular, while inductive reasoning proceeds from the particular to the general. This is false - or at least, it is not the way logicians use these terms. There are deductively valid arguments that proceed from the particular to the general (Oscar is grouchy, therefore something is grouchy) and inductive arguments that proceed from the general to the particular (most University students are smart, therefore this particular University student is smart).

Sherlock Holmes Sherlock Holmes frequently describes his methods as involving deductive reasoning in the various stories about the character. However, most of his "deductions" in fact used inductive or abductive reasoning; very few were actually deductive in nature. There was nearly always some conceivable, if vanishingly unlikely, way his conclusions could have turned out to be incorrect, a fact exploited by many parodies of the Sherlock Holmes stories.

Inference rules The following table (next slide) lists some inference rules of propositional calculus. The table makes use of mathematical notation. The following symbols occur in the table: