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In propositional logic, modus tollens (/ ˈ m oʊ d ə s ˈ t ɒ l ɛ n z /) (MT), also known as modus tollendo tollens (Latin for "method of removing by taking away") and denying the consequent, is a deductive argument form and a rule of inference.
modus ponens and modus tollens, in propositional logic, two types of inference that can be drawn from a hypothetical proposition—i.e., from a proposition of the form “If A, then B” (symbolically A ⊃ B, in which ⊃ signifies “If . . . then”).
May 3, 2024 · Modus tollens is a valid form of argumentation in formal logic. It begins with a conditional (if–then) statement and proceeds to negate the consequent (the “then” statement). This structure is often expressed using the variables P and Q, as follows:
Jul 8, 2024 · Modus tollens is a valid argument form in propositional calculus in which p and q are propositions. If p implies q, and q is false, then p is false. Also known as an indirect proof or a proof by contrapositive.
Modus tollens is the rule of inference we get if we put modus ponens through the “contrapositive” wringer. \(\begin{array} & &¬B\\ &\underline{A \implies B} \\ ∴ &¬A \end{array}\) Modus tollens is related to the tautology \((¬B ∧ (A \implies B)) \implies ¬A\).
The Latin name here simply means reduced to absurdity. It is actually an application of modus tollens. It is a method to prove that a certain statement S is false: First assume that S is true. From the assumption that it is true, prove that it would lead to a contradiction or some other claim that is false or absurd. Conclude that S must be false.
The Latin name, modus tollens, translates to “mode that denies”. Notice that the second premise and the conclusion look like the contrapositive of the first premise, \(\sim q \rightarrow \sim p\), but they have been detached.
Modus tollendo tollens, usually simply called modus tollens or MT is a valid argument form in logic. It is also known as "denying the consequent". The form of modus tollens is: "If P, then Q.
Since modus tollens is a valid argument, using the substitution rule with the equivalences. \begin {equation*} r \land p \Leftrightarrow \neg (\neg r \lor \neg p) \Leftrightarrow \neg (r\rightarrow \neg p) \text {,} \end {equation*}
Feb 8, 2018 · The law of modus tollens is the inference rule which allows one to conclude ¬ P from P ⇒ Q and ¬ Q. The name “modus tollens” refers to the fact that this rule allows one to take away the conclusion of a conditional statement and conclude the negation of the condition.
