Mode maddening

Modus tollens ( Latin for "mode of lifting", literally: "canceling mode"), actually mode tollendo tollens (as opposed to modus ponendo tollens ), is a final figure that is used as a final rule in several calculations of classical logic . It says that from the prerequisites “ If , then . "And" Don't . "To" Not . “Can be closed. ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle B}$ ${\ displaystyle A}$

The Latin name modus tollendo tollens , "by canceling the ending", is explained by the fact that it is a final figure ( modus ) which, given the first premise , by the "canceling" ( tollendo ) of the sentence B, ie through the setting of its negation,, another sentence, namely , also “cancels” ( tollens ), thus to its negation,, leads. The tollendo tollens mode is thus a counterpart to the ponendo ponens mode . ${\ displaystyle A \ rightarrow B}$ ${\ displaystyle \ neg B}$ ${\ displaystyle A}$ ${\ displaystyle \ neg A}$

Forms and example

As a final form

Scheme

example

	${\ displaystyle A \ rightarrow B}$
	${\ displaystyle \ neg B}$
mode tollens	${\ displaystyle \ neg A}$

	If it has rained, the road is wet.
	The road is not wet.
mode tollens	It has not rained.

As a statement

Although the modus tollendo tollens is a final rule, i.e. a metalinguistic concept, the term modus tollens is occasionally used for object language expressions of the following form:

${\ displaystyle (\ neg B \ land (A \ rightarrow B)) \ rightarrow \ neg A}$

Statements of this form are tautologies in most propositional calculi; H. always true. But since inference rules and statements are different concepts, it is scientifically unsatisfactory to use the same name for the two terms. In general, mixing object and metalanguage is problematic.

proof

The logical equivalence of the statements and follows from the definitions of the subjunction and the negation . ${\ displaystyle A \ rightarrow B}$ ${\ displaystyle \ neg B \ rightarrow \ neg A}$

A.	→	B.	${\ displaystyle \ Leftrightarrow}$	¬B	→	¬A
f	w	f	w	w	w	w
f	w	w	w	f	w	w
w	f	f	w	w	f	f
w	w	w	w	f	w	f

Importance of the mode romping

According to Critical Rationalism , the mode (tollendo) tollens is the fundamental conclusion of scientific research. A is an abstract hypothetical theory , B an observation sentence that follows from the theory. The function of scientific experiments is to observe whether B is true or false. If B is wrong, then so is the theory on which it is based, which is thus falsified .