Deductive reasoning

# Deductive reasoning

Overview
Deductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis. A deductive argument is valid
Validity
In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....

if the conclusion does follow necessarily from the premises, i.e., if the conclusion must be true provided that the premises are true.
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Encyclopedia
Deductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis. A deductive argument is valid
Validity
In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....

if the conclusion does follow necessarily from the premises, i.e., if the conclusion must be true provided that the premises are true. A deductive argument is sound
Soundness
In mathematical logic, a logical system has the soundness property if and only if its inference rules prove only formulas that are valid with respect to its semantics. In most cases, this comes down to its rules having the property of preserving truth, but this is not the case in general. The word...

if it is valid and its premises are true. Deductive arguments are valid or invalid, sound or unsound. Deductive reasoning is a method of gaining knowledge. An example of a deductive argument:
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 states that "Socrates" must be mortal because he inherits this attribute from his classification as a man.

## Law of detachment

The law of detachment is the first form of deductive reasoning. A single conditional statement
Conditional statement
In computer science, conditional statements, conditional expressions and conditional constructs are features of a programming language which perform different computations or actions depending on whether a programmer-specified boolean condition evaluates to true or false...

is made, and then a hypothesis (P) is stated. The conclusion (Q) is deduced from the hypothesis and the statement. The most basic form is listed below:
1. P→Q
2. P (Hypothesis stated)
3. Q (Conclusion given)

We can conclude Q from P by using the law of detachment from deductive reasoning. However, if the conclusion (Q) is given instead of the hypothesis (P) then there is no valid conclusion.

The following is an example of an argument using the law of detachment in the form of an If-then statement:
1. If m∠A>90°, then ∠A is an obtuse angle.
2. m∠A=120°.
3. ∠A is an obtuse angle.

Since the measurement of angle A is greater than 90°, we can deduce that A is an obtuse angle.

## Law of syllogism

The law of syllogism
Syllogism
A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...

takes two conditional statements and forms a conclusion by combining the hypothesis of one statement with the conclusion of another. The following is an example:
1. If Larry is sick, then he will be absent from school.
2. If Larry is absent, then he will miss his classwork.
3. If Larry is sick, then he will miss his classwork.

We deduced the solution by combining the hypothesis of the first problem with the conclusion of the second statement.
We also conclude that this could be a false statement.

## Deductive logic

Deductive arguments are generally 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, whatever they may be, 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.

The following is an example of an argument that is valid, but not sound; a premise is false:
1. Everyone who eats steak is a quarterback.
2. John eats steak.
3. Therefore, John is a quarterback.

The example's first premise is false (there are people who eat steak that 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
Validity
In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....

, but not sound
Soundness
In mathematical logic, a logical system has the soundness property if and only if its inference rules prove only formulas that are valid with respect to its semantics. In most cases, this comes down to its rules having the property of preserving truth, but this is not the case in general. The word...

.

The theory of deductive reasoning known as categorical or term logic
Term logic
In philosophy, term logic, also known as traditional logic or aristotelian logic, is a loose name for the way of doing logic that began with Aristotle and that was dominant until the advent of modern predicate logic in the late nineteenth century...

was developed by Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...

, but was superseded by propositional (sentential) logic
Propositional calculus
In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...

and predicate logic
Predicate logic
In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulae contain variables which can be quantified...

.

Deductive reasoning can be contrasted with inductive reasoning
Inductive reasoning
Inductive reasoning, also known as induction or inductive logic, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances...

. 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).

## Hume's skepticism

Philosopher David Hume
David Hume
David Hume was a Scottish philosopher, historian, economist, and essayist, known especially for his philosophical empiricism and skepticism. He was one of the most important figures in the history of Western philosophy and the Scottish Enlightenment...

presented grounds to doubt deduction by questioning induction
Inductive reasoning
Inductive reasoning, also known as induction or inductive logic, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances...

. Hume's problem of induction
Problem of induction
The problem of induction is the philosophical question of whether inductive reasoning leads to knowledge. That is, what is the justification for either:...

starts by suggesting that the use of even the simplest forms of induction simply cannot be justified
Theory of justification
Theory of justification is a part of epistemology that attempts to understand the justification of propositions and beliefs. Epistemologists are concerned with various epistemic features of belief, which include the ideas of justification, warrant, rationality, and probability...

by inductive reasoning itself. Moreover, induction cannot be justified by deduction either. Therefore, induction cannot be justified rationally. Consequentially, if induction is not yet justified, then deduction seems to be left to rationally justify itself – an objectionable conclusion to Hume.

Hume did not provide a strictly rational solution per se. He simply explained that we do induce, and that it is useful that we do so, but not necessarily justified. Certainly we must appeal to first principles of some kind, including laws of thought.

• Argument (logic)
• Logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

• Mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

• Abductive reasoning
Abductive reasoning
Abduction is a kind of logical inference described by Charles Sanders Peirce as "guessing". The term refers to the process of arriving at an explanatory hypothesis. Peirce said that to abduce a hypothetical explanation a from an observed surprising circumstance b is to surmise that a may be true...

• Analogical reasoning
Analogy
Analogy is a cognitive process of transferring information or meaning from a particular subject to another particular subject , and a linguistic expression corresponding to such a process...

• Correspondence theory of truth
Correspondence theory of truth
The correspondence theory of truth states that the truth or falsity of a statement is determined only by how it relates to the world, and whether it accurately describes that world...

• Defeasible reasoning
Defeasible reasoning
Defeasible reasoning is a kind of reasoning that is based on reasons that are defeasible, as opposed to the indefeasible reasons of deductive logic...

• Decision making
Decision making
Decision making can be regarded as the mental processes resulting in the selection of a course of action among several alternative scenarios. Every decision making process produces a final choice. The output can be an action or an opinion of choice.- Overview :Human performance in decision terms...

• Decision theory
Decision theory
Decision theory in economics, psychology, philosophy, mathematics, and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision...

• Fallacy
Fallacy
In logic and rhetoric, a fallacy is usually an incorrect argumentation in reasoning resulting in a misconception or presumption. By accident or design, fallacies may exploit emotional triggers in the listener or interlocutor , or take advantage of social relationships between people...

• Geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

• Hypothetico-deductive method

• Inquiry
Inquiry
An inquiry is any process that has the aim of augmenting knowledge, resolving doubt, or solving a problem. A theory of inquiry is an account of the various types of inquiry and a treatment of the ways that each type of inquiry achieves its aim.-Deduction:...

• Inductive reasoning
Inductive reasoning
Inductive reasoning, also known as induction or inductive logic, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances...

• Inference
Inference
Inference is the act or process of deriving logical conclusions from premises known or assumed to be true. The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic.Human inference Inference is the act or process of deriving logical conclusions...

• Logical consequence
• Natural deduction
Natural deduction
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning...

• Propositional calculus
Propositional calculus
In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...

• Retroductive reasoning
• Scientific method
Scientific method
Scientific method refers to a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering empirical and measurable evidence subject to specific principles of...

• Soundness
Soundness
In mathematical logic, a logical system has the soundness property if and only if its inference rules prove only formulas that are valid with respect to its semantics. In most cases, this comes down to its rules having the property of preserving truth, but this is not the case in general. The word...

• Syllogism
Syllogism
A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...

• Vincent F. Hendricks
Vincent F. Hendricks
Vincent Fella Hendricks , is a Danish philosopher and logician. He holds two doctoral degrees in philosophy and is Professor of Formal Philosophy at University of Copenhagen, Denmark. He was previously Professor of Formal Philosophy at Roskilde University, Denmark. He is member of IIP, the...

, Thought 2 Talk: A Crash Course in Reflection and Expression, New York: Automatic Press / VIP, 2005, ISBN 87-991013-7-8
• Philip Johnson-Laird
Philip Johnson-Laird
Philip Johnson-Laird is a professor at Princeton University's Department of Psychology and author of several notable books on human cognition and the psychology of reasoning....

, Ruth M. J. Byrne
Ruth M. J. Byrne
Ruth M.J.Byrne FTCD, MRIA, was born in Dublin, Ireland. She is a cognitive scientist and author of several books on human reasoning, including Deduction , Human Reasoning , and .She is currently Professor of Cognitive Science, in the Institute of...

, Deduction, Psychology Press 1991, ISBN 9780863771491jiii
• Zarefsky, David, Argumentation: The Study of Effective Reasoning Parts I and II, The Teaching Company 2002