Deductive reasoning also deductive logic or logical deduction or informally topdown logic is the process of reasoning from one or more statements premises to reach a logically certain conclusion It differs from ID: 258105
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Deductive reasoningSlide2
Deductive reasoning
, also
deductive logic
or
logical deduction
or, informally,
"
top-down
" logic
, is the process of
reasoning
from one or more
statements
(premises) to reach a logically certain conclusion. It differs from
inductive reasoning
or
abductive
reasoning
.
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
.
Deductive reasoning (top-down logic) contrasts with
inductive reasoning
(bottom-up logic) in the following way: In deductive reasoning, a conclusion is reached
reductively
by applying general rules that hold over the entirety of a
closed domain of discourse
, narrowing the range under consideration until only the conclusion is left. In inductive reasoning, the conclusion is reached by generalizing or extrapolating from initial information. As a result, induction can be used even in an
open domain
, one where there is
epistemic uncertainty
. Note, however, that the inductive reasoning mentioned here is not the same as
induction
used in mathematical proofs –
mathematical induction
is actually a form of deductive reasoning.Slide3
Simple example
An example of a deductive argument:
All men are mortal.
Socrates is a man.
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".Slide4
Law of detachment
Modus ponens
The law of detachment (also known as
affirming the antecedent
and
Modus ponens
) 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:
P → Q (conditional statement)
P (hypothesis stated)
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:
If an angle satisfies 90° < A < 180°, then A is an obtuse angle.
A = 120°.
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.Slide5
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:
P → Q
Q → R
Therefore, P → R.
The following is an example:
If Larry is sick, then he will be absent.
If Larry is absent, then he will miss his classwork.
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:
A = B.
B = C.
Therefore A = C.Slide6
Law of contrapositive
Modus
tollens
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:
P → Q.
~Q.
Therefore we can conclude ~P.
The following are examples:
If it is raining, then there are clouds in the sky.
There are no clouds in the sky.
Thus, it is not raining.Slide7
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 though the 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:
Everyone who eats carrots is a quarterback.
John eats carrots.
Therefore, John is a quarterback.
The example's first premise is false – there are people who eat carrots and are not quarterbacks – but the conclusion must be true, so long as the premises are true (i.e. it is impossible for the premises to be true and the conclusion false). Therefore the argument is valid, but not sound. Generalizations are often used to make invalid arguments, such as "everyone who eats carrots is a quarterback." Not everyone who eats carrots is a quarterback, thus proving the flaw of such arguments.
In this example, the first statement uses
categorical reasoning
, saying that all carrot-eaters are definitely quarterbacks. This theory of deductive reasoning – also known as
term logic
– was developed by
Aristotle
, but was superseded by
propositional (sentential) logic
and
predicate logic
.
Deductive reasoning can be contrasted with
inductive reasoning
, in regards to validity and soundness. In cases of inductive reasoning, even though the premises are true and the argument is "valid", it is possible for the conclusion to be false (determined to be false with a counterexample or other means).Slide8
Education
Deductive reasoning is generally thought of as a skill that develops without any formal teaching or training. As a result of this belief, deductive reasoning skills are not taught in secondary schools, where students are expected to use reasoning more often and at a higher level. It is in high school, for example, that students have an abrupt introduction to
mathematical proofs
– which rely heavily on deductive reasoning.