Deductive reasoning

Deductive reasoning, also deductive logic, logical deduction is the process of reasoning from one or more statements (premises) to reach a logically certain 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 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. In inductive reasoning, the conclusion is reached by generalizing or extrapolating from specific cases to general rules, i.e., there is epistemic uncertainty. 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 () and as second premise the antecedent () of the conditional statement. It obtains the consequent () of the conditional statement as its conclusion. The argument form is listed below:

  1.   (First premise is a conditional statement)
  2.   (Second premise is the antecedent)
  3.   (Conclusion deduced is the consequent)

In this form of deductive reasoning, the consequent () obtains as the conclusion from the premises of a conditional statement () and its antecedent (). However, the antecedent () cannot be similarly obtained as the conclusion from the premises of the conditional statement () and the consequent (). 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° < < 180°, then is an obtuse angle.
  2. = 120°.
  3. is an obtuse angle.

Since the measurement of angle is greater than 90° and less than 180°, we can deduce from the conditional (if-then) statement that is an obtuse angle. However, if we are given that is an obtuse angle, we cannot deduce from the conditional statement that 90° < < 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 () and the negation of the consequent () and as conclusion the negation of the antecedent (). 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. . (First premise is a conditional statement)
  2. . (Second premise is the negation of the consequent)
  3. . (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. Therefore, .

The following is an example:

  1. If Larry is sick, then he will be absent.
  2. If Larry is absent, then he will miss his classwork.
  3. 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. Another example is the transitive property of equality which can be stated in this form:

  1. .
  2. .
  3. Therefore, .

Validity and soundness

Argument terminology used in logic
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”:

  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.

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 started documenting deductive reasoning in the 4th century BC.[2]

See also

References

  1. ^ Sternberg, R. J. (2009). Cognitive Psychology. Belmont, CA: Wadsworth. p. 578. ISBN 978-0-495-50629-4.
  2. ^ Evans, Jonathan St. B. T.; Newstead, Stephen E.; Byrne, Ruth M. J., eds. (1993). Human Reasoning: The Psychology of Deduction (Reprint ed.). Psychology Press. p. 4. ISBN 9780863773136. Retrieved 2015-01-26. In one sense [...] one can see the psychology of deductive reasoning as being as old as the study of logic, which originated in the writings of Aristotle.

Further reading

External links

A priori

A priori may refer to:

A priori and a posteriori

A priori and a posteriori languages

A priori estimate, a type of estimate for the size of a solution of a differential equation

A priori probability, a type of probability derived by deductive reasoning, with application in statistical mechanics

Apriori algorithm, an algorithm used with databases

Argument-deduction-proof distinctions

Argument-deduction-proof distinctions originated with logic itself. Naturally, the terminology evolved.

Bulverism

Bulverism is a logical fallacy. The method of Bulverism is to "assume that your opponent is wrong, and explain his error." The Bulverist assumes a speaker's argument is invalid or false and then explains why the speaker came to make that mistake, attacking the speaker or the speaker's motive. The term Bulverism was coined by C. S. Lewis to poke fun at a very serious error in thinking that, he alleges, recurs often in a variety of religious, political, and philosophical debates.

Similar to Antony Flew's "subject/motive shift", Bulverism is a fallacy of irrelevance. One accuses an argument of being wrong on the basis of the arguer's identity or motive, but these are strictly speaking irrelevant to the argument's validity or truth.

Corollary

A corollary ( KORR-ə-lerr-ee, UK: korr-OL-ər-ee) is a statement that follows readily from a previous statement.

Deduction board game

Deduction board games are a genre of board game in which the players must use deductive reasoning and logic in order to win the game. While many games, such as bridge or poker require the use of deductive reasoning to some degree, deduction board games feature deductive reasoning as their central mechanic.

Deduction board games typically fall into two broad categories; abstract and investigation games.

Deductive closure

Deductive closure is a property of a set of objects (usually the objects in question are statements). A set of objects, O, is said to exhibit closure or to be closed under a given operation, R, provided that for every object, x, if x is a member of O and x is R-related to any object, y, then y is a member of O. In the context of statements, a deductive closure is the set of all the statements that can be deduced from a given set of statements.

In propositional logic, the set of all true propositions exhibits deductive closure: if set O is the set of true propositions, and operation R is logical consequence (“”), then provided that proposition p is a member of O and p is R-related to q (i.e., p  q), q is also a member of O.

Formal fallacy

In philosophy, a formal fallacy, deductive fallacy, logical fallacy or non sequitur (Latin for "it does not follow") is a pattern of reasoning rendered invalid by a flaw in its logical structure that can neatly be expressed in a standard logic system, for example propositional logic. It is defined as a deductive argument that is invalid. The argument itself could have true premises, but still have a false conclusion. Thus, a formal fallacy is a fallacy where deduction goes wrong, and is no longer a logical process. However, this may not affect the truth of the conclusion since validity and truth are separate in formal logic.

While a logical argument is a non sequitur if, and only if, it is invalid, the term "non sequitur" typically refers to those types of invalid arguments which do not constitute formal fallacies covered by particular terms (e.g. affirming the consequent). In other words, in practice, "non sequitur" refers to an unnamed formal fallacy.

A special case is a mathematical fallacy, an intentionally invalid mathematical proof, often with the error subtle and somehow concealed. Mathematical fallacies are typically crafted and exhibited for educational purposes, usually taking the form of spurious proofs of obvious contradictions.

A formal fallacy is contrasted with an informal fallacy, which may have a valid logical form and yet be unsound because one or more premises are false.

Good and necessary consequence

The phrase good and necessary consequence was used more commonly several centuries ago to express the idea which we would place today under the general heading of logic; that is, to reason validly by logical deduction or better, deductive reasoning.

Even more particularly, it would be understood in terms of term logic, also known as traditional logic, or as many today would also consider it to be part of formal logic, which deals with the form (or logical form) of arguments as to which are valid or invalid.

In this context, we may better understand the word "good" in the phrase "good and necessary consequence" more technically as intending a "valid argument form".

One of the best recognized articulations of the authoritative and morally binding use of good and necessary consequence to make deductions from Scripture can be readily found in probably the most famous of Protestant Confessions of faith, the Westminster Confession of Faith (1646), Chapter 1, sec. 6, as well as in others, including the Heidelberg Catechism, and the Belgic Confession.

Hypothetico-deductive model

The hypothetico-deductive model or method is a proposed description of scientific method. According to it, scientific inquiry proceeds by formulating a hypothesis in a form that can be falsifiable, using a test on observable data where the outcome is not yet known. A test outcome that could have and does run contrary to predictions of the hypothesis is taken as a falsification of the hypothesis. A test outcome that could have, but does not run contrary to the hypothesis corroborates the theory. It is then proposed to compare the explanatory value of competing hypotheses by testing how stringently they are corroborated by their predictions.

Inductive reasoning

Inductive reasoning is a method of reasoning in which the premises are viewed as supplying some evidence for the truth of the conclusion; this is in contrast to deductive reasoning. While the conclusion of a deductive argument is certain, the truth of the conclusion of an inductive argument may be probable, based upon the evidence given.Many dictionaries define inductive reasoning as the derivation of general principles from specific observations, though some sources find this usage "outdated".

Logical consequence

Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises? All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth.

Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. A sentence is said to be a logical consequence of a set of sentences, for a given language, if and only if, using only logic (i.e. without regard to any personal interpretations of the sentences) the sentence must be true if every sentence in the set is true.

Logicians make precise accounts of logical consequence regarding a given language , either by constructing a deductive system for or by formal intended semantics for language . The Polish logician Alfred Tarski identified three features of an adequate characterization of entailment: (1) The logical consequence relation relies on the logical form of the sentences, (2) The relation is a priori, i.e. it can be determined with or without regard to empirical evidence (sense experience), and (3) The logical consequence relation has a modal component.

Principle

A principle is a proposition or value that is a guide for behavior or evaluation. In law, it is a rule that has to be or usually is to be followed, or can be desirably followed, or is an inevitable consequence of something, such as the laws observed in nature or the way that a system is constructed. The principles of such a system are understood by its users as the essential characteristics of the system, or reflecting system's designed purpose, and the effective operation or use of which would be impossible if any one of the principles was to be ignored. A system may be explicitly based on and implemented from a document of principles as was done in IBM's 360/370 Principles of Operation.

Examples of principles are, entropy in a number of fields, least action in physics, those in descriptive comprehensive and fundamental law: doctrines or assumptions forming normative rules of conduct, separation of church and state in statecraft, the central dogma of molecular biology, fairness in ethics, etc.

In common English, it is a substantive and collective term referring to rule governance, the absence of which, being "unprincipled", is considered a character defect. It may also be used to declare that a reality has diverged from some ideal or norm as when something is said to be true only "in principle" but not in fact.

Probabilistic argumentation

Probabilistic argumentation refers to different formal frameworks in the literature. All share the idea that qualitative aspects can be captured by an underlying logic, while quantitative aspects of uncertainty can be accounted for by probabilistic measures.

Reason

Reason is the capacity of consciously making sense of things, establishing and verifying facts, applying logic, and changing or justifying practices, institutions, and beliefs based on new or existing information. It is closely associated with such characteristically human activities as philosophy, science, language, mathematics and art, and is normally considered to be a distinguishing ability possessed by humans.

Reason, or an aspect of it, is sometimes referred to as rationality.

Reasoning is associated with thinking, cognition, and intellect. The philosophical field of logic studies ways in which humans reason formally through argument. Reasoning may be subdivided into forms of logical reasoning (forms associated with the strict sense): deductive reasoning, inductive reasoning, abductive reasoning; and other modes of reasoning considered more informal, such as intuitive reasoning and verbal reasoning. Along these lines, a distinction is often drawn between logical, discursive reasoning (reason proper), and intuitive reasoning, in which the reasoning process through intuition—however valid—may tend toward the personal and the subjectively opaque. In some social and political settings logical and intuitive modes of reasoning may clash, while in other contexts intuition and formal reason are seen as complementary rather than adversarial. For example, in mathematics, intuition is often necessary for the creative processes involved with arriving at a formal proof, arguably the most difficult of formal reasoning tasks.

Reasoning, like habit or intuition, is one of the ways by which thinking moves from one idea to a related idea. For example, reasoning is the means by which rational individuals understand sensory information from their environments, or conceptualize abstract dichotomies such as cause and effect, truth and falsehood, or ideas regarding notions of good or evil. Reasoning, as a part of executive decision making, is also closely identified with the ability to self-consciously change, in terms of goals, beliefs, attitudes, traditions, and institutions, and therefore with the capacity for freedom and self-determination. In contrast to the use of "reason" as an abstract noun, a reason is a consideration given which either explains or justifies events, phenomena, or behavior. Reasons justify decisions, reasons support explanations of natural phenomena; reasons can be given to explain the actions (conduct) of individuals.

Using reason, or reasoning, can also be described more plainly as providing good, or the best, reasons. For example, when evaluating a moral decision, "morality is, at the very least, the effort to guide one's conduct by reason—that is, doing what there are the best reasons for doing—while giving equal [and impartial] weight to the interests of all those affected by what one does."Psychologists and cognitive scientists have attempted to study and explain how people reason, e.g. which cognitive and neural processes are engaged, and how cultural factors affect the inferences that people draw. The field of automated reasoning studies how reasoning may or may not be modeled computationally. Animal psychology considers the question of whether animals other than humans can reason.

Sophistical Refutations

Sophistical Refutations (Greek: Σοφιστικοὶ Ἔλεγχοι; Latin: De Sophisticis Elenchis) is a text in Aristotle's Organon in which he identified thirteen fallacies. According to Aristotle, this is the first work to treat the subject of deductive reasoning. (Soph. Ref., 34, 183b34 ff.). The fallacies Aristotle identifies are the following:

Fallacies in the language (in dictione)Equivocation

Amphibology

Composition

Division

Accent

Figure of speech or form of expressionFallacies not in the language (extra dictionem)

Soundness

In mathematical logic, a logical system has the soundness property if and only if every formula that can be proved in the system is logically valid with respect to the semantics of the system.

The converse of soundness is known as completeness. In most cases, this comes down to its rules having the property of preserving truth.

Speculative reason

Speculative reason, sometimes called theoretical reason or pure reason, is theoretical (or logical, deductive) thought, as opposed to practical (active, willing) thought. The distinction between the two goes at least as far back as the ancient Greek philosophers, such as Plato and Aristotle, who distinguished between theory (theoria, or a wide, bird's eye view of a topic, or clear vision of its structure) and practice (praxis), as well as techne.

Speculative reason is contemplative, detached, and certain, whereas practical reason is engaged, involved, active, and dependent upon the specifics of the situation. Speculative reason provides the universal, necessary principles of logic, such as the principle of non-contradiction, which must apply everywhere, regardless of the specifics of the situation.

On the other hand, practical reason is the power of the mind engaged in deciding what to do. It is also referred to as moral reason, because it involves action, decision, and particulars. Though many other thinkers have erected systems based on the distinction, two important later thinkers who have done so are Aquinas (who follows Aristotle in many respects) and Immanuel Kant.

Turnstile (symbol)

In mathematical logic and computer science the symbol has taken the name turnstile because of its resemblance to a typical turnstile if viewed from above. It is also referred to as tee and is often read as "yields", "proves", "satisfies" or "entails". The symbol was first used by Gottlob Frege in his 1879 book on logic, Begriffsschrift.

In TeX, the turnstile symbol is obtained from the command \vdash. In Unicode, the turnstile symbol () is called right tack and is at code point U+22A2. (Code point U+22A6 is named assertion sign ().) On a typewriter, a turnstile can be composed from a vertical bar (|) and a dash (–). In LaTeX there is a turnstile package which issues this sign in many ways, and is capable of putting labels below or above it, in the correct places.

Validity (logic)

In logic, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. It is not required for a valid argument to have premises that are actually true, but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. A formula is valid if and only if it is true under every interpretation, and an argument form (or schema) is valid if and only if every argument of that logical form is valid.

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