Deductive reasoning, or deduction, starts with a general case and deduces specific instances.

The basic idea is that if something is true of a class of things in general, this truth applies to all legitimate members of that class. The key, then, is to be able to properly identify members of the class. Mis-categorizing will result in invalid conclusions.

Deductive reasoning, or deduction, starts with a general case and deduces specific instances.

Deduction starts with an assumed hypothesis or theory, which is why it has been called 'hypothetico-deduction'. This assumption may be well-accepted or it may be rather more shaky -- nevertheless, for the argument it is not questioned.

Deduction is used by scientists who take a general scientific law and apply it to a certain case, as they assume that the law is true. Deduction can also be used to test an induction by applying it elsewhere, although in this case the initial theory is assumed to be true only temporarily.

Every day, I get in my car to leave for work, at eight o’clock. Every day, the journey takes 45 minutes, and I arrive at work on time. If I leave for work at eight o’clock today, I will be on time.

Inductive Reasoning

Today, I left for work at eight o’clock, and was on time. Therefore, every day that I leave the house at eight o’clock, I will arrive at work on time.

The deductive statement is a perfectly logical statement, but does rely upon the initial premise being correct. Perhaps today, there are roadwork, so you will end up being late for work. This is why any hypothesis can never be completely proved, because there is always the scope for the initial premise to be wrong.

Inductive reasoning, whilst commonly used in science, is not logically valid, because it is not strictly accurate to assume that a general principle is correct.

In the above example, perhaps ‘today’ is a weekend, with less traffic. It is illogical to assume an entire premise, just because one specific data set seems to suggest it.

Deductive reasoning assumes that the basic law from which you are arguing is applicable in all cases. This can let you take a rule and apply it perhaps where it was not really meant to be applied.

Scientists will prove a general law for a particular case and then do many deductive experiments (and often get PhDs in the process) to demonstrate that the law holds true in many different circumstances.

In set theory, a deduction is a subset of the rule that is taken as the start point. If the rule is true and deduction is a true subset (not a conjunction) then the deduction is almost certainly true.

Using deductive reasoning usually is a credible and 'safe' form of reasoning, but is based on the assumed truth of the rule or law on which it is founded.

Deductive conclusions can be valid or invalid. Valid arguments obey the initial rule. For validity, the truth or falsehood of the initial rule is not considered. Thus valid conclusions need not be true, and invalid conclusions may not be false.

When a conclusion is both valid and true, it is considered to be sound. When it is valid, but untrue, then it is considered to be unsound.

All men are mortal

Socrates is a man

(Therefore,) Socrates is mortal

An argument is valid when it is impossible for both its premises to be true and its conclusion to be false. An argument can be valid even though the premises are false. Note, for example, that the conclusion of the following argument would have to be true if the premises were true, (even though they are, in fact, false):

1. Everyone who eats steak is a quarterback.

2. John eats steak.

3. [Therefore,] John is a quarterback.

The argument is valid, however, not sound. In order for a deductive argument to be sound, the premise must not only be valid, but also must be true as well.

A theory of deductive reasoning known as categorical or term logic was developed by Aristotle, but was superseded by propositional (sentential) logic and predicate logic.

Deductive reasoning is sometimes contrasted with inductive reasoning, in which one moves from a set of specific facts to a general conclusion. By thinking about phenomena such as how apples fall and how the planets move, Isaac Newton induced his theory of gravity. In the 19th century, Adams and LeVerrier applied Newton's theory (general principle) to deduce the existence, mass, position, and orbit of Neptune(specific conclusions) from perturbations in the observed orbit of Uranus (specific data).

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