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Deductive reasoning, also deductive logic, is the process of reasoning
from one or more statements (premises) to reach a logical conclusion.[1]
Deductive reasoning goes in the same direction as that of the conditionals, and
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 deductive reasoning,
a conclusion is reached reductively by applying general rules which hold over
the entirety of a closed domain of discourse, narrowing the range under
consideration until only the conclusion(s) remains. In deductive reasoning
there is no uncertainty.[2]
In inductive reasoning, the conclusion is reached by generalizing or
extrapolating from specific cases to general rules resulting in a conclusion
that has epistemic uncertainty.[2] The inductive reasoning is not the same as
induction used in mathematical proofs – mathematical induction is actually
a form of deductive reasoning. Deductive reasoning differs from abductive
reasoning by the direction of the reasoning relative to the conditionals. The
idea of "deduction" popularized in Sherlock Holmes stories is
technically abduction, rather than deductive reasoning. Deductive reasoning
goes in the same direction as that of the conditionals, whereas abductive
reasoning goes in the direction contrary to that of the conditionals.
Reasoning with modus ponens, modus tollens, and the law of syllogism:
Modus ponens:
Main article: Modus ponens:
Modus ponens (also known as "affirming the antecedent" or "the
law of detachment") is the primary deductive rule of inference. It applies
to arguments that have as first premise a conditional statement ( P ? Q
{\displaystyle P\rightarrow Q} P\rightarrow Q) and as second premise the
antecedent ( P {\displaystyle P} P) of the conditional statement. It obtains
the consequent ( Q {\displaystyle Q} Q) of the conditional statement as its
conclusion. The argument form is listed below: P ? Q {\displaystyle
P\rightarrow Q} P\rightarrow Q (First premise is a conditional statement) P
{\displaystyle P} P (Second premise is the antecedent) Q {\displaystyle Q} Q
(Conclusion deduced is the consequent) In this form of deductive reasoning, the
consequent ( Q {\displaystyle Q} Q) obtains as the conclusion from the premises
of a conditional statement ( P ? Q {\displaystyle P\rightarrow Q} P\rightarrow
Q) and its antecedent ( P {\displaystyle P} P). However, the antecedent ( P
{\displaystyle P} P) cannot be similarly obtained as the conclusion from the
premises of the conditional statement ( P ? Q {\displaystyle P\rightarrow Q}
P\rightarrow Q) and the consequent ( Q {\displaystyle Q} Q). Such an argument
commits the logical fallacy of affirming the consequent. The following is an
example of an argument using modus ponens: If an angle satisfies 90° <
A {\displaystyle A} A < 180°, then A {\displaystyle A} A is an obtuse
angle. A {\displaystyle A} A=120°. A {\displaystyle A} A is an obtuse
angle. Since the measurement of angle A {\displaystyle A} A is greater than
90° and less than 180°, we can deduce from the conditional (if-then)
statement that A {\displaystyle A} A is an obtuse angle. However, if we are
given that A {\displaystyle A} A is an obtuse angle, we cannot deduce from the
conditional statement that 90° < A {\displaystyle A} A < 180°.
It might be true that other angles outside this range are also obtuse.
Modus tollens:
Main article: Modus tollens:
Modus tollens (also known as "the law of contrapositive") is a
deductive rule of inference. It validates an argument that has as premises a
conditional statement (formula) and the negation of the consequent ( ¬ Q
{\displaystyle \lnot Q} {\displaystyle \lnot Q}) and as conclusion the negation
of the antecedent ( ¬ P {\displaystyle \lnot P} \lnot P). In contrast to
modus ponens, reasoning with modus tollens goes in the opposite direction to
that of the conditional. The general expression for modus tollens is the
following: P ? Q {\displaystyle P\rightarrow Q} P\rightarrow Q. (First premise
is a conditional statement) ¬ Q {\displaystyle \lnot Q} {\displaystyle
\lnot Q}. (Second premise is the negation of the consequent) ¬ P
{\displaystyle \lnot P} \lnot P. (Conclusion deduced is the negation of the
antecedent) The following is an example of an argument using modus tollens: If
it is raining, then there are clouds in the sky. There are no clouds in the
sky. Thus, it is not raining.
Law of syllogism:
In term logic 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 {\displaystyle P\rightarrow Q}
P\rightarrow Q Q ? R {\displaystyle Q\rightarrow R} {\displaystyle Q\rightarrow
R} Therefore, P ? R {\displaystyle P\rightarrow R} {\displaystyle P\rightarrow
R}. The following is an example: If the animal is a Yorkie, then it's a dog. If
the animal is a dog, then it's a mammal. Therefore, if the animal is a Yorkie,
then it's a mammal. 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. Another example is the transitive property
of equality which can be stated in this form: A=B {\displaystyle A=B} A=B. B=C
{\displaystyle B=C} {\displaystyle B=C}. Therefore, A=C {\displaystyle A=C}
A=C.
Simple example:
An example of an argument using deductive reasoning: All men are mortal. (First
premise) Socrates is a man. (Second premise) Therefore, Socrates is mortal.
(Conclusion) 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."
Validity and soundness:
Argument terminology 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”: 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 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. 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.[citation needed]
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).
Probability of Conclusion:
The probability of the conclusion of a deductive argument cannot be calculated
by figuring out the cumulative probability of the argument’s premises. Dr.
Timothy McGrew, a specialist in the applications of probability theory, and Dr.
Ernest W. Adams, a Professor Emeritus at UC Berkeley, pointed out that the
theorem on the accumulation of uncertainty designates only a lower limit on the
probability of the conclusion. So the probability of the conjunction of the
argument’s premises sets only a minimum probability of the conclusion. The
probability of the argument’s conclusion cannot be any lower than the
probability of the conjunction of the argument’s premises. For example, if
the probability of a deductive argument’s four premises is ~0.43, then it
is assured that the probability of the argument’s conclusion is no less
than ~0.43. It could be much higher, but it cannot drop under that lower
limit.[3][4] There can be examples in which each single premise is more likely
true than not and yet it would be unreasonable to accept the conjunction of the
premises. Professor Henry Kyburg, who was known for his work in probability and
logic, clarified that the issue here is one of closure – specifically,
closure under conjunction. There are examples where it is reasonable to accept
P and reasonable to accept Q without its being reasonable to accept the
conjunction (P&Q). Lotteries serve as very intuitive examples of this,
because in a basic non-discriminatory finite lottery with only a single winner
to be drawn, it is sound to think that ticket 1 is a loser, sound to think that
ticket 2 is a loser,...all the way up to the final number. However, clearly it
is irrational to accept the conjunction of these statements; the conjunction
would deny the very terms of the lottery because (taken with the background
knowledge) it would entail that there is no winner.[5][4] Dr. McGrew further
adds that the sole method to ensure that a conclusion deductively drawn from a
group of premises is more probable than not is to use premises the conjunction
of which is more probable than not. This point is slightly tricky, because it
can lead to a possible misunderstanding. What is being searched for is a
general principle that specifies factors under which, for any logical
consequence C of the group of premises, C is more probable than not. Particular
consequences will differ in their probability. However, the goal is to state a
condition under which this attribute is ensured, regardless of which
consequence one draws, and fulfilment of that condition is required to complete
the task. This principle can be demonstrated in a moderately clear way.
Suppose, for instance, the following group of premises: {P, Q, R} Suppose that
the conjunction ((P & Q) & R) fails to be more probable than not. Then
there is at least one logical consequence of the group that fails to be more
probable than not – namely, that very conjunction. So it is an essential
factor for the argument to “preserve plausibility” (Dr. McGrew coins
this phrase to mean “guarantee, from information about the plausibility of
the premises alone, that any conclusion drawn from those premises by deductive
inference is itself more plausible than not”) that the conjunction of the
premises be more probable than not.[4]
History:
Aristotle, a Greek philosopher, started documenting deductive reasoning in the
4th century BC.[6]
René Descartes, in his book Discourse on Method, refined the idea for
the Scientific Revolution. Developing four rules to follow for proving an idea
deductively, Decartes laid the foundation for the deductive portion of the
scientific method. Decartes' background in geometry and mathematics influenced
his ideas on the truth and reasoning, causing him to develop a system of
general reasoning now used for most mathematical reasoning. Similar to
postulates, Decartes believed that ideas could be self-evident and that
reasoning alone must prove that observations are reliable. These ideas also lay
the foundations for the ideas of rationalism.[7]
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