Science of Deduction
Aristotle started documenting deductive reasoning in the 4th century BC.
Deductive reasoning, also deductive logic, logical deduction or, informally, "top-down" logic, is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion.
Deductive reasoning links premises with conclusions.
If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true.
1. All men are mortal. 2. Socrates is a man. 3. Therefore, Socrates is mortal.
The first premise states that all objects classified as "men" have the attribute "mortal".
The second premise states that "Socrates" is classified as a "man" – a member of the set "men".
The conclusion then states that "Socrates" must be "mortal" because he inherits this attribute from his classification as a "man".
I. Law of detachment:
The law of detachment is the first form of deductive reasoning.
A single conditional statement is made, and a hypothesis (P) is stated.
The conclusion (Q) is then deduced from the statement and the hypothesis.
The most basic form is listed below:
1. P → Q (conditional statement)
2. P (hypothesis stated)
3. Q (conclusion deduced)
In deductive reasoning, we can conclude Q from P by using the law of detachment.
However, if the conclusion (Q) is given instead of the hypothesis (P) then there is no definitive conclusion.
The following is an example of an argument using the law of detachment in the form of an if-then statement:
1. If an angle satisfies 90° < A < 180°, then A is an obtuse angle.
2. A = 120°.
3. A is an obtuse angle.
Since the measurement of angle A is greater than 90° and less than 180°, we can deduce that A is an obtuse angle.
If however, we are given the conclusion that A is an obtuse angle we cannot deduce the premise that A = 120°.
II. Law of syllogism:
The law of syllogism takes two conditional statements and forms a conclusion by combining the hypothesis of one statement with the conclusion of another.
Here is the general form:
1. P → Q
2. Q → R
3. Therefore, P → R.
The following is an example:
1. If Larry is sick, then he will be absent.
2. If Larry is absent, then he will miss his classwork.
3. Therefore, if Larry is sick, then he will miss his classwork.
We deduced the final statement by combining the hypothesis of the first statement with the conclusion of the second statement.
We also allow that this could be a false statement.
This is an example of the transitive property in mathematics.
The transitive property is sometimes phrased in this form:
1. A = B.
2. B = C.
3. Therefore, A = C.
III. Law of contrapositive:
The law of contrapositive states that, in a conditional, if the conclusion is false, then the hypothesis must be false also.
The general form is the following:
1. P → Q.
3. Therefore, we can conclude ~P.
The following are examples:
1. If it is raining, then there are clouds in the sky.
2. There are no clouds in the sky.
3. Thus, it is not raining.
IV. Validity and Soundness:
Deductive arguments are evaluated in terms of their validity and soundness.
An argument is “valid” if it is impossible for its premises to be true while its conclusion is false.
In other words, the conclusion must be true if the premises are true.
An argument can be “valid” even if one or more of its premises are false.
An argument is “sound” if it is valid and the premises are true.
It is possible to have a deductive argument that is logically valid but is not sound.
Fallacious arguments often take that form.
The following is an example of an argument that is “valid”, but not “sound”:
1. Everyone who eats carrots is a quarterback.
2. John eats carrots.
3. Therefore, John is a quarterback.
The example’s first premise is false – there are people who eat carrots who are not quarterbacks – but the conclusion would necessarily be true, if the premises were true.
In other words, it is impossible for the premises to be true and the conclusion false.
Therefore, the argument is “valid”, but not “sound”.
False generalizations – such as “Everyone who eats carrots is a quarterback” – are often used to make unsound arguments.
The fact that there are some people who eat carrots but are not quarterbacks proves the flaw of the argument.
Deductive reasoning is generally considered to be a skill that develops without any formal teaching or training.
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