Basic Logic - Inductive vs Deductive Reasoning

How Certain Are We?

If the premises are True and the argument is Valid - how certain are we of the conclusion? Remember the definition of Validity: If, for every case where the premises are True the conclusion is also True, then the argument is Valid. So - can we be certain that we have actually examined every case where the premises are True? Have we missed any?

It's sometimes said that Deductive reasoning proceeds from general rules to specific cases, and Inductive is from specific casess to general rules; however, this is not actually what it means to be either Deductive or Inductive.

Deductive Reasoning

If, when the premises are True, the conclusion is absolutely certain - then the argument is Deductive. Clearly, for this to be the case, we must have been able to observe every possible case for the premises. This is not possible in any empirical observation, therefore no empirical argument can be deductive. Deductive reasoning is the rule in areas such as geometry and algebra.

Inductive Reasoning

If, when the premises are True, the conclusion is probable - then the argument is Inductive. All empirical arguments are Inductive. This means that if empirical observations are used to create a general rule, and that rule is used to determine a specific case, the argument from general to specific is still inductive, because the basis of the rule is inductive - we cannot reach absolute certainty.

Without absolute certainty, how can we feel confident in accepting the conclusion? This is the realm of Hypothesis Testing - the ability to assign a confidence level to our acceptance of the conclusion.