Deductive reasoning

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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]

Logic study of inference and demonstration

Logic, is the systematic study of the form of valid inference, and the most general laws of truth. A valid inference is one where there is a specific relation of logical support between the assumptions of the inference and its conclusion. In ordinary discourse, inferences may be signified by words such as therefore, hence, ergo, and so on.

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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.

A premise or premiss is a statement that an argument claims will induce or justify a conclusion. In other words, a premise is an assumption that something is true.

A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then". In an implication, if P implies Q, then P is called the antecedent and Q is called the consequent. In some contexts, the consequent is called the apodosis.

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.

Inductive reasoning is a method of reasoning in which the premises are viewed as supplying some evidence for the truth of the conclusion. 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.

Reductionism

Reductionism is any of several related philosophical ideas regarding the associations between phenomena which can be described in terms of other simpler or more fundamental phenomena.

The closed-world assumption (CWA), in a formal system of logic used for knowledge representation, is the presumption that a statement that is true is also known to be true. Therefore, conversely, what is not currently known to be true, is false. The same name also refers to a logical formalization of this assumption by Raymond Reiter. The opposite of the closed-world assumption is the open-world assumption (OWA), stating that lack of knowledge does not imply falsity. Decisions on CWA vs. OWA determine the understanding of the actual semantics of a conceptual expression with the same notations of concepts. A successful formalization of natural language semantics usually cannot avoid an explicit revelation of whether the implicit logical backgrounds are based on CWA or OWA.

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.

Abductive reasoning is a form of logical inference which starts with an observation or set of observations then seeks to find the simplest and most likely explanation for the observations. This process, unlike deductive reasoning, yields a plausible conclusion but does not positively verify it. Abductive conclusions are thus qualified as having a remnant of uncertainty or doubt, which is expressed in retreat terms such as "best available" or "most likely". One can understand abductive reasoning as inference to the best explanation, although not all uses of the terms abduction and inference to the best explanation are exactly equivalent.

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:

In logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion. For example, the rule of inference called modus ponens takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is valid with respect to the semantics of classical logic, in the sense that if the premises are true, then so is the conclusion.

Material conditional logical connective often symbolized by a forward arrow

The material conditional is a logical connective that is often symbolized by a forward arrow "→". The material conditional is used to form statements of the form p → q which is read as "if p then q". Unlike the English construction "if... then...", the material conditional statement p → q does not specify a causal relationship between p and q. It merely means "if p is true, then q is also true" such that the statement p → q is false only when p is true and q is false. In the truth table of p → q , If p is false, then p → q is true regardless of whether q is true or false since (1) p → q does not take care of cases with false p and (2) this truth table is useful to prove some mathematical theorems.

  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.

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 and invalidly inferring its converse even though the converse may not be true. This arises when a consequent has one or more other antecedents.

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 Argument terminology used in logic.png
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. [ 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 started documenting deductive reasoning in the 4th century BC. [2]

See also

Related Research Articles

In classical logic, disjunctive syllogism is a valid argument form which is a syllogism having a disjunctive statement for one of its premises.

In propositional logic, modus ponens is a rule of inference. It can be summarized as "P implies Q and P is asserted to be true, therefore Q must be true."

In propositional logic, modus tollens is a valid argument form and a rule of inference. It is an application of the general truth that if a statement is true, then so is its contra-positive.

Denying the antecedent, sometimes also called inverse error or fallacy of the inverse, is a formal fallacy of inferring the inverse from the original statement. It is committed by reasoning in the form:

In classical logic, hypothetical syllogism is a valid argument form which is a syllogism having a conditional statement for one or both of its premises.

In natural languages, an indicative conditional is the logical operation given by statements of the form "If A then B". Unlike the material conditional, an indicative conditional does not have a stipulated definition. The philosophical literature on this operation is broad, and no clear consensus has been reached.

Logical biconditional

In logic and mathematics, the logical biconditional is the logical connective of two statements asserting " if and only if ", where is an antecedent and is a consequent. This is often abbreviated " iff ". The operator is denoted using a doubleheaded arrow (↔), a prefixed E "Epq", an equality sign (=), an equivalence sign (≡), or EQV. It is logically equivalent to and to , meaning "both or neither".

Sequent calculus is, in essence, a style of formal logical argumentation where every line of a proof is a conditional tautology instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal argument according to rules and procedures of inference, giving a better approximation to the style of natural deduction used by mathematicians than David Hilbert's earlier style of formal logic where every line was an unconditional tautology. There may be more subtle distinctions to be made; for example, there may be non-logical axioms upon which all propositions are implicitly dependent. Then sequents signify conditional theorems in a first-order language rather than conditional tautologies.

Inferences are steps in reasoning, moving from premises to logical consequences. Charles Sanders Peirce divided inference into three kinds: deduction, induction, and abduction. Deduction is inference deriving logical conclusions from premises known or assumed to be true, with the laws of valid inference being studied in logic. Induction is inference from particular premises to a universal conclusion. Abduction is inference to the best explanation.

In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the truth of "A implies B" the truth of "Not-B implies not-A", and conversely. It is very closely related to the rule of inference modus tollens. It is the rule that:

Logical form form for logical arguments, obtained by abstracting from the subject matter of its content terms

In philosophy and mathematics, a logical form of a syntactic expression is a precisely-specified semantic version of that expression in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal formal language, the meaning of a logical form can be determined unambiguously from syntax alone. Logical forms are semantic, not syntactic constructs; therefore, there may be more than one string that represents the same logical form in a given language.

In philosophy, a formal fallacy, deductive fallacy, logical fallacy or non sequitur 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.

In logic, contraposition is an inference that says that a conditional statement is logically equivalent to its contrapositive. The contrapositive of the statement has its antecedent and consequent inverted and flipped: the contrapositive of is thus . For instance, the proposition "All cats are mammals" can be restated as the conditional "If something is a cat, then it is a mammal". The law of contraposition says that statement is identical to the contrapositive "If something is not a mammal, then it is not a cat."

The paradoxes of material implication are a group of formulae that are truths of classical logic but are intuitively problematic.

In logic and philosophy, an argument is a series of statements, called the premises or premisses, intended to determine the degree of truth of another statement, the conclusion. The logical form of an argument in a natural language can be represented in a symbolic formal language, and independently of natural language formally defined "arguments" can be made in math and computer science.

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 is valid if and only if every argument of that logical form is valid.

Logical consequence 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.

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