TheInfoList

Deductive reasoning, also deductive logic, is the process of
reasoning Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: ...

from one or more
statement Statement or statements may refer to: Common uses *Statement (computer science), the smallest standalone element of an imperative programming language *Statement (logic), declarative sentence that is either true or false *Statement, a Sentence_(lin ...
s (premises) to reach a logical conclusion. Deductive reasoning goes in the same direction as that of the conditionals, and links
premise A premise or premiss is a true or false statement that helps form the body of an argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, apply ...
s with conclusions. If all premises are true, the terms are
clear Clear may refer to: Arts, entertainment, and media Music Groups * Clear (Christian band), an American CCM group from Cambridge, Minnesota * Clear (hardcore band), a vegan straight edge hardcore group from Utah Albums * Clear (Bomb the Bass album ...

, and the rules of deductive
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

are followed, then the conclusion reached is
necessarily true Logical truth is one of the most fundamental concept Concepts are defined as abstract ideas or general notions that occur in the mind, in speech, or in thought. They are understood to be the fundamental building blocks of thoughts and belief ...
. Deductive reasoning (''"top-down logic"'') contrasts with
inductive reasoning Inductive reasoning is a method of reasoning Reason is the capacity of Consciousness, consciously making sense of things, applying logic, and adapting or justifying practices, institutions, and beliefs based on new or existing information. It ...
(''"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 Uncertainty refers to Epistemology, epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially ...

. 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. The inductive reasoning is not the same as
induction Induction may refer to: Philosophy * Inductive reasoning, in logic, inferences from particular cases to the general case Biology and chemistry * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induction period, the t ...
used in mathematical proofs – mathematical induction is actually a form of deductive reasoning. Deductive reasoning differs from
abductive reasoning Abductive reasoning (also called abduction,For example: abductive inference, or retroduction) is a form of logical inference Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word ''wikt:infe ...
by the direction of the reasoning relative to the conditionals. The idea of "deduction" popularized in
Sherlock Holmes Sherlock Holmes () is a fictional detective created by British author Arthur Conan Doyle, Sir Arthur Conan Doyle. Referring to himself as a "consulting detective" in the stories, Holmes is known for his proficiency with observation, deduction, ...

stories is technically
abduction Abduction may refer to: Of a person or people * Alien abduction, memories of being taken by apparently nonhuman entities from a different planet * Bride kidnapping, a practice in which a man abducts the woman he wishes to marry * Child abducti ...
, rather than deductive reasoning. ''Deductive reasoning'' goes in the same direction as that of the conditionals, whereas ''
abductive reasoning Abductive reasoning (also called abduction,For example: abductive inference, or retroduction) is a form of logical inference Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word ''wikt:infe ...
'' goes in the direction contrary to that of the conditionals.

# 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 In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their Syntax (logic), syntax, and returns a conclusion (or multiple-conclusion logic, ...
. 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: # $P \rightarrow Q$  (First premise is a conditional statement) # $P$  (Second premise is the antecedent) # $Q$  (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 Affirming the consequent, sometimes called converse error, fallacy of the converse, or confusion of necessity and sufficiency, is a formal fallacy of taking a true conditional statement (e.g., "If the lamp were broken, then the room would be dark ...
. The following is an example of an argument using modus ponens: # 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 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 (formula) and the negation of the consequent ($\lnot Q$) and as conclusion the negation of the antecedent ($\lnot P$). In contrast to
modus ponens In propositional logic Propositional calculus is a branch of logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from ...

, reasoning with modus tollens goes in the opposite direction to that of the conditional. The general expression for modus tollens is the following: # $P \rightarrow Q$. (First premise is a conditional statement) # $\lnot Q$. (Second premise is the negation of the consequent) # $\lnot P$. (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 In philosophy, term logic, also known as traditional logic, Syllogism, syllogistic logic or Aristotelianism, Aristotelian logic, is a loose name for an approach to logic that began with Aristotle and was developed further in ancient history mostly b ...
the ''law of
syllogism A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument In logic and philosophy, an argument is a series of statements (in a natural language), called the premises or premisses (bo ...
'' 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: # $P \rightarrow Q$ # $Q \rightarrow R$ # Therefore, $P \rightarrow R$. 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: # $A = B$. # $B = C$. # Therefore, $A = C$.

# 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

Deductive arguments are evaluated in terms of their '' validity'' and ''
soundness In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...

''. An argument is “valid” if it is impossible for its
premise A premise or premiss is a true or false statement that helps form the body of an argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, apply ...
s 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 In philosophy, term logic, also known as traditional logic, Syllogism, syllogistic logic or Aristotelianism, Aristotelian logic, is a loose name for an approach to logic that began with Aristotle and was developed further in ancient history mostly b ...
– was developed by
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental quest ...

, but was superseded by propositional (sentential) logic and
predicate logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system A formal system is an used for inferring theorems from axioms according to a set of rules. These rul ...
. Deductive reasoning can be contrasted with
inductive reasoning Inductive reasoning is a method of reasoning Reason is the capacity of Consciousness, consciously making sense of things, applying logic, and adapting or justifying practices, institutions, and beliefs based on new or existing information. It ...
, 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 Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
, and Dr. Ernest W. Adams, a Professor Emeritus at
UC Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public In public relations and communication science, publics are groups of individual people, and the public (a.k.a. the general public) is the tota ...

, 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. 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 Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

and
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

, 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. 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: 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.

# History

Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental quest ...

, a
Greek philosopher Ancient Greek philosophy arose in the 6th century BC, at a time when the inhabitants of ancient Greece were struggling to repel devastating invasions from the east. Greek philosophy continued throughout the Hellenistic period The Hellenistic pe ...
, started documenting deductive reasoning in the 4th century BC.
René Descartes René Descartes ( or ; ; Latinized Latinisation or Latinization can refer to: * Latinisation of names, the practice of rendering a non-Latin name in a Latin style * Latinisation in the Soviet Union, the campaign in the USSR during the 1920s ...

, in his book
Discourse on Method ''Discourse on the Method of Rightly Conducting One's Reason and of Seeking Truth in the Sciences'' (french: Discours de la Méthode Pour bien conduire sa raison, et chercher la vérité dans les sciences) is a philosophical Philosophy (fr ...
, 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 The scientific method is an empirical Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence ...

. 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 In philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, ...
.

*
Abductive reasoning Abductive reasoning (also called abduction,For example: abductive inference, or retroduction) is a form of logical inference Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word ''wikt:infe ...
* *
Argument (logic) In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ...
*
Argumentation theory Two men argue at a political protest in New York City. Argumentation theory, or argumentation, is the interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic disciplines into one ac ...
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Correspondence theory of truth In metaphysics Metaphysics is the branch of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of ...
*
Decision making In psychology Psychology is the science of mind and behavior. Psychology includes the study of consciousness, conscious and Unconscious mind, unconscious phenomena, as well as feeling and thought. It is an academic discipline of immense s ...

*
Decision theory Decision theory (or the theory of choice not to be confused with choice theory) is the study of an agent's choices. Decision theory can be broken into two branches: normative Normative generally means relating to an evaluative standard. Normativi ...
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Defeasible reasoning In philosophical logicPhilosophical logic refers to those areas of philosophy in which recognized methods of logic have Classical logic, traditionally been used to solve or advance the discussion of philosophical problems. Among these, Sybil Wolfra ...
*
Fallacy A fallacy is the use of invalid or otherwise faulty reason Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of ...
*
Fault tree analysis Image:Fault tree.svg, A fault tree diagram Fault tree analysis (FTA) is a top-down, Deductive reasoning, deductive failure analysis in which an undesired state of a system is analyzed using Boolean logic to combine a series of lower-level events. ...
*
Geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

* Hypothetico-deductive method *
Inference Inferences are steps in reasoning Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic ...

*
Inquiry An inquiry (also spelled as enquiry in British English British English (BrE) is the standard dialect of the English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon E ...

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Legal syllogismLegal syllogism is a legal concept concerning the law and its application, specifically a form of argument based on deductive reasoning Deductive reasoning, also deductive logic, is the process of reasoning from one or more argument (logic), statemen ...
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Logic and rationality As the study of argument is of clear importance to the reasons that we hold things to be true, logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argum ...
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Logical consequence Logical consequence (also entailment) is a fundamental concept Concepts are defined as abstract ideas A mental representation (or cognitive representation), in philosophy of mind Philosophy of mind is a branch of philosophy that studies ...
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Logical reasoning Two kinds of logical reasoning are often distinguished in addition to formal deduction: induction and abduction. Given a precondition or '' premise'', a conclusion or ''logical consequence Logical consequence (also entailment) is a fundamental ...
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Mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal sys ...
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Natural deductionIn logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, acc ...
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Peirce's theory of deductive reasoning In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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Propositional calculus Propositional calculus is a branch of logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fal ...
* Retroductive reasoning *
Scientific method The scientific method is an empirical Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence ...

*
Subjective logic Subjective logic is a type of probabilistic logicThe aim of a probabilistic logic (also probability logic and probabilistic reasoning) is to combine the capacity of probability theory to handle uncertainty with the capacity of deductive logic to ex ...
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Theory of justification Justification (also called epistemic justification) is a concept in epistemology Epistemology (; ) is the Outline of philosophy, branch of philosophy concerned with knowledge. Epistemologists study the nature, origin, and scope of knowledge, ...