# استدلال استنتاجی

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استدلال کل به جزء یا استدلال قیاسی (deductive reasoning) یکی از دو استدلال معروف در منطق (استدلال کل به جزء و استدلال جزء به کل {the deductive and inductive approaches}) می‌باشد . استدلال کل به جزء هنگامی است که در استدلال از یک نظریه (تیوری) کلی استفاده کنیم و به فرضیه یا فرضیه‌های جزئی برسیم. در پژوهش ها، زمانی که پژوهشگر از نظریه استفاده می‌کند و فرضیه می‌سازد و برای ازمون فرضیه‌ها داده جمع‌آوری می‌کند و نتیجه می‌گیرد می گوییم پژوهشگر از روش استدلال کل به جزء استفاده کرده‌است. روش استدلال جزء به کل (inductive approaches) بر عکس این روش می‌باشد یعنی پژوهشگر داده جمع‌آوری می‌کند و الگو بین داده‌ها و متغیرها کشف می‌کند سپس فرضیه می‌سازد و فرضیه را آزمون می‌کند و در نهایت نظریه می‌دهد. روش‌های جز به کل و کل به جز از دو روش مهم در پژوهش می‌باشد (the deductive and inductive methods).

در منطق بدست آوردن یک گزاره از دنباله‌ای از یک مجموعه گزاره‌ها است. دنبالهٔ گزاره استفاده شده مفروضات و گزارهٔ بدست آمده نتیجه نامیده می‌شود. استدلال یا گواه آوردن قیاسی، منطق قیاسی نیز نامیده می‌شود. این روش استدلال کردن یا گواه آوردن از بحث‌های قیاسی به دست می‌آید. در این گونه بحث، تلاش می‌شود تا نشان داده شود که نتیجه به‌طور بایسته و ضروری، از مجموعه‌ای از پیش فرض‌ها یا فرضیه‌ها به دست می‌آید. بحث قیاسی هنگامی معتبر است که نتیجه به‌طور بایسته و ضروری، از پیش فرض و فرضیه به دست آید. گواه آوری یا استدلال قیاسی در کنار گواه آوری استقرایی (inductive reasoning)، یکی از دو روش رایج در شناخت و رسیدن به دانایی یا معرفت است.

## یک مثال ساده

مثال زیر، ویژگی این روش را نشان می‌دهد:

- همه انسان‌ها می‌میرند
- سقراط انسان است
- بنابراین: سقراط خواهد مرد.

پیش فرض نخست بیان می‌کند که همه موجودات قرار گرفته زیر نام و عنوان «انسان» یک روز خواهند مرد. عبارت دوم بیان می‌کند: سقراط هم زیر عنوان یک «انسان» قرار دارد. در نتیجه سقراط بالاخره می میرد، زیرا او نیز به عنوان یک انسان خواهد مرد،چون که عنوان «انسان» به او نسبت داده شده و این ویژگی او را نیز شامل می‌شود.

## تاریخچه

ارسطو در قرن چهارم پیش از میلاد اقدام به مستندسازی و ثبت استدلال استنتاجی کرد.

## آموزش

کارشناسان بر این عقیده‌اند که رشد «توانایی استدلال» الزاماً به آموزش و یادگیری نیاز ندارد و با توجه به اینکه در مقاطع تحصیلات آکادمیک دانش‌پژوه نیاز به بهره‌برداری از این قوه دارد، آموزش استدلال استنتاجی به دوره‌های تحصیلات متوسطه واگذار می‌شود تا در مباحث اثبات‌های ریاضیاتی به آن پرداخته شود.

## منابع

1. «Social Research Methods - Knowledge Base - Deduction & Induction». www.socialresearchmethods.net. دریافت‌شده در ۲۰۱۸-۱۱-۲۰.

Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion.

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 the following way; 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) is left (there is no epistemic uncertainty; i.e. unrecognized parts of the currently available set; all parts of the currently available set are available and recognized). In inductive reasoning, the conclusion is reached by generalizing or extrapolating from specific cases to general rules, i.e., there is epistemic uncertainty (unrecognized parts of the currently available set). However, the inductive reasoning mentioned here 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. Deductive reasoning goes in the same direction as that of the conditionals, whereas abductive reasoning goes in the opposite direction to that of the conditionals.

## Simple example

An example of an argument using deductive reasoning:

1. All men are mortal. (First premise)
2. Socrates is a man. (Second premise)
3. 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."

## Reasoning with modus ponens, modus tollens, and the law of syllogism

### 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 ($P\rightarrow Q$ ) and as second premise the antecedent ($P$ ) of the conditional statement. It obtains the consequent ($Q$ ) of the conditional statement as its conclusion. The argument form is listed below:

1. $P\rightarrow Q$ (First premise is a conditional statement)
2. $P$ (Second premise is the antecedent)
3. $Q$ (Conclusion deduced is the consequent)

In this form of deductive reasoning, the consequent ($Q$ ) obtains as the conclusion from the premises of a conditional statement ($P\rightarrow Q$ ) and its antecedent ($P$ ). However, the antecedent ($P$ ) cannot be similarly obtained as the conclusion from the premises of the conditional statement ($P\rightarrow Q$ ) and the consequent ($Q$ ). Such an argument commits the logical fallacy of affirming the consequent.

The following is an example of an argument using modus ponens:

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 from the conditional (if-then) statement that $A$ is an obtuse angle. However, if we are given that $A$ is an obtuse angle, we cannot deduce from the conditional statement that 90° < $A$ < 180°. It might be true that other angles outside this range are also obtuse.

### 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 ($P\rightarrow Q$ ) and the negation of the consequent ($\lnot Q$ ) and as conclusion the negation of the antecedent ($\lnot P$ ). 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:

1. $P\rightarrow Q$ . (First premise is a conditional statement)
2. $\lnot Q$ . (Second premise is the negation of the consequent)
3. $\lnot P$ . (Conclusion deduced is the negation of the antecedent)

The following is an example of an argument using modus tollens:

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.

### Law of syllogism

In proposition 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:

1. $P\rightarrow Q$ 2. $Q\rightarrow R$ 3. Therefore, $P\rightarrow R$ .

The following is an example:

1. If the animal is a Yorkie, then it's a dog.
2. If the animal is a dog, then it's a mammal.
3. 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:

1. $A=B$ .
2. $B=C$ .
3. Therefore, $A=C$ .

## 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.

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).

## History

Aristotle, a Greek philosopher, started documenting deductive reasoning in the 4th century BC. 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 lay 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.