Modus tollens

"Indirect proof" redirects here. For proof by contradiction, see Reductio ad absurdum. Reductio ad absurdum ( Latin for "reduction to the absurd" also known as an apagogical argument, reductio ad impossibile

In logic, modus tollendo tollens^[1] (Latin for "the way that denies by denying")^[2] is the formal name for indirect proof or proof by contraposition (contrapositive inference), often abbreviated to MT or modus tollens. Logic is the study of the principles of valid demonstration and Inference. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. For contraposition in the field of traditional logic see Contraposition (traditional logic. ^[3]^[4] It can also be referred to as denying the consequent, and is a valid form of argument (unlike similarly-named but invalid arguments such as affirming the consequent or denying the antecedent). The term validity (also called logical truth, analytic truth, or necessary truth) as it occurs in Logic refers generally to a property of In Logic, an argument is a Set of one or more Declarative sentences (or "propositions") known as the Premises along Affirming the consequent, sometimes also called Converse error, is a Formal fallacy, committed by reasoning in the form: If P Denying the antecedent, sometimes also called Inverse error, is a Formal fallacy, committed by reasoning in the form: If P, then It is closely related to another valid form of argument, modus ponens or "affirming the antecedent". In Classical logic, modus ponendo ponens ( Latin: mode that affirms by affirming; often abbreviated to MP or modus ponens) is a In Classical logic, modus ponendo ponens ( Latin: mode that affirms by affirming; often abbreviated to MP or modus ponens) is a

Modus tollens has the following argument form:

If P, then Q. In Logic, the argument form or test form of an Argument results from replacing the different words or sentences that make up the argument with letters

¬Q

Therefore, ¬P. ^[5]

1 Formal notation
2 Explanation
3 Relation to modus ponens
4 Justification via truth table
5 See also
6 Notes
7 External links

Formal notation

The modus tollens rule may be written in logical operator notation:

$P\to Q, \neg Q \vdash \neg P$

where $\vdash$ represents the logical assertion. Table of logic symbolsIn Logic, two sentences (either in a formal language or a natural language may be joined by means of a logical connective to form a compound sentence In Proof theory, a sequent is a formalized statement of provability that is frequently used when specifying calculi for deduction.

Or in set-theoretic form:

$P\subseteq Q$

$x\notin Q$

$\therefore x\notin P$

("P is a subset of Q. x is not in Q. Therefore, x is not in P. ")

It can also be written as:

$\frac{P\to Q ~,~~ \neg Q}{\neg P}$

Explanation

The argument has two premises. The first premise is the conditional "if-then" statement, namely that P implies Q. The second premise is that Q is false. From these two premises, it can be logically concluded that P must be false.

Consider an example:

If there is fire here, then there is oxygen here.

There is no oxygen here.

Therefore, there is no fire here.

Supposing that the premises are both true, if there is a fire here, then there must be oxygen. It is a fact that there is no oxygen here. It follows, then, that there cannot be a fire here. An argument is valid if it is not possible for the premises to be true and the conclusion false. The term validity (also called logical truth, analytic truth, or necessary truth) as it occurs in Logic refers generally to a property of (A counter-example demonstrates that Hydrogen gas burns efficiently with Halogen gases like Chlorine and Fluorine and will combust with Iodine, with no Oxygen present. Hydrogen (ˈhaɪdrədʒən is the Chemical element with Atomic number 1 Abundance Owing to their high Reactivity, the halogens are found in the environment only in compounds or as Ions Halide ions and oxoanions Chlorine (ˈklɔriːn from the Greek word 'χλωρóς' ( khlôros, meaning 'pale green' is the Chemical element with Atomic number 17 and Fluorine, fluorum meaning "to flow" is the Chemical element with the symbol F and Atomic number 9 Iodine (ˈaɪədaɪn ˈaɪədɪn or /ˈaɪədiːn/ from ιώδης iodes "violet" is a Chemical element that has the symbol I and Atomic Oxygen (from the Greek roots ὀξύς (oxys (acid literally "sharp" from the taste of acids and -γενής (-genēs (producer literally begetteris the )

Another example:

If Lizzie were the murderer, then she owns an axe.

Lizzie does not own an axe.

Therefore, Lizzie was not the murderer.

Modus tollens became well known when it was used by Karl Popper in his proposed response to the problem of induction, falsificationism. Sir Karl Raimund Popper ( July 28 1902 &ndash September 17 1994) was an Austrian and British Philosopher and a professor The problem of induction is the philosophical question of whether inductive reasoning is valid Falsifiability (or "refutability" is the logical possibility that an assertion can be shown false by an observation or a physical experiment However, here the use of modus tollens is much more controversial, as "truth" or "falsity" are inappropriate concepts to apply to theories (which are generally approximations to reality) and experimental findings (whose interpretation is often contingent on other theories). Thus (to take a historical example)

If Special Relativity is true, then the mass of the electron has a specific dependence on velocity

Experimentally, the mass of the electron does not have this dependence (Kauffmann (1906))

Therefore, Special Relativity is false

Einstein rejected this argument on the grounds that the alternative theories that appeared to be validated by the experiment were inherently less plausible than his own.

Relation to modus ponens

Every use of modus tollens can be converted to a use of modus ponens and one use of transposition to the premise which is a material implication. In Classical logic, modus ponendo ponens ( Latin: mode that affirms by affirming; often abbreviated to MP or modus ponens) is a Transposition (mathematics -->In the methods of Deductive reasoning in Classical logic, " The material conditional, also known as the material implication or truth functional conditional, expresses a property of certain Conditionals in Logic For example:

If P, then Q. (premise -- material implication)

If Q is false, then P is false. (derived by transposition)

Q is false. (premise)

Therefore, P is false. (derived by modus ponens)

Likewise, every use of modus ponens can be converted to a use of modus tollens and transposition.

Justification via truth table

The validity of modus tollens can be clearly demonstrated through a truth table. A truth table is a Mathematical table used in Logic — specifically in connection with Boolean algebra, Boolean functions and Propositional

p	q	p → q
T	T	T
T	F	F
F	T	T
F	F	T

In instances of modus tollens we assume as premises that p → q is true and q is false. There is only one line of the truth table - the fourth line - which satisfies these two conditions. In this line, p is false. Therefore, in every instance in which p → q is true and q is false, p must also be false.

Notes

^ Sanford, David Hawley. In Classical logic, modus ponendo ponens ( Latin: mode that affirms by affirming; often abbreviated to MP or modus ponens) is a Modus tollendo ponens (literally mode which by denying affirms) or MTP, is a valid, simple Argument form that is today known as Modus ponendo tollens (Latin mode that denies by affirming) is a valid Rule of inference, sometimes abbreviated MPT. Affirming the consequent, sometimes also called Converse error, is a Formal fallacy, committed by reasoning in the form: If P Denying the antecedent, sometimes also called Inverse error, is a Formal fallacy, committed by reasoning in the form: If P, then Falsifiability (or "refutability" is the logical possibility that an assertion can be shown false by an observation or a physical experiment Non sequitur ( Latin for "it does not follow" in formal logic is an argument where its conclusion does not follow from its premises 2003. If P, Then Q: Conditionals and the Foundations of Reasoning. London, UK: Routledge: 39 "[Modus] tollens is always an abbreviation for modus tollendo tollens, the mood that by denying denies. "
^ Stone, Jon R. 1996. Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language. London, UK: Routledge: 60.
^ Politzer, Guy & Carles, Laure. 2001. 'Belief Revision and Uncertain Reasoning'. Thinking and Reasoning, 7:217-234.
^ Suppes, Patrick & Hill, Shirley A. 1964. First Course in Mathematical Logic. Dover Publications:54-55.
^ University of North Carolina, Philosophy Department, Logic Glossary. Accessdate on 31 October 2007. Events 445 BC – Ezra reads the Book of the Law to the Israelites in Jerusalem (see Nehemiah 91 NLTse Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century.

External links

Modus Tollens at Wolfram MathWorld

Dictionary

-noun

(philosophy, logic) A valid form of argument in which the consequent of a conditional proposition is denied, thus implying the denial of the antecedent. Modus tollens has this form:

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