Monotonicity of entailment

Monotonicity is a property of a Ableitbarkeitsrelation or an inference, stating that the addition of further premises always preserves the previous conclusions.

Formal definition

Be a Ableitbarkeitsrelation. iff is monotone. applies

If so, then

Alternative: Let be an inference. iff is monotone. applies

If so, then

Explanation

The property of monotonicity states that if a certain statement follows from a set of assumptions, this statement still follows if additional assumptions are added. For example, follows from the assumption quantity

{ "Peter has sorrow. ", " If Peter Kummer has he drinks ."}

The statement

" Peter is drinking ".

This statement should then still follow if we as a further premise " Peter is from Austria. " to accept. from

{ "Peter has sorrow. ", " If Peter Kummer has he drinks. ", " Peter is from Austria "}

So also follows

" Peter is drinking. "

Discussion

Monotony is true for the classical propositional logic and predicate logic Ableitbarkeitsbegriff, as well as for many other types of logic such as modal logic. Nevertheless, criticism of the monotony as a general property of inferential relations has become noisy. The critics can be divided into two groups.

Non- monotonic inference relations

The criticism sparked by the observation that inferences in everyday life often have non-monotonic character. Do we, for example, that Tux is a bird, so we could conclude that Tux can fly. Learn we then, however, that Tux is a penguin, so we would not conclude that Tux can fly, because we know that penguins can not fly. from

{" Tux is a bird ."}

So it seems to follow

" Tux can fly "

While from

{" Tux is a bird. ", " Tux is a penguin ."}

Certainly does not follow

" Tux can fly ," but rather

" Tux can not fly ."

The controversy can be resolved to some extent by that one between conclusions that apply with 100 % certainty and those that are likely only to a certain extent, different. From { "Peter has sorrow. ", " If Peter Kummer has he drinks ."} Follows with 100 % certainty " Peter drinks ". ( This is of course not to say that it is certain that Peter is drinking it is only said that if these two premises are true, then the conclusion must be true -. ., Or one of the premises is in fact not true) It but does not apply with the same certainty that if it is true that Tux is a bird, then he can also fly needs. Tux could be a penguin, Tux could have broken a wing, it can be many situations construct in which the premise is true but the conclusion false.

In summary, it can be said, therefore, that we should expect for inferential relationships in everyday life, which often do not apply with absolute certainty, no monotony. To meet this intuition justice, so called were designed "non- monotonic logics ". But the criticism does not affect inferences in mathematics and in other formal sciences, because there is always concluded with absolute certainty.

The relevance aspect

Also, this criticism is based partly on everyday inference actions. Let us consider again the above example with Peter, the troubled drinkers from Austria. Suppose someone were to say to us: "We know three things: Peter has grief when Peter Kummer, he drinks and Peter is from Austria, so we also know that Peter is drinking. . ." Then we would rightly ask: "What does that mean that he drinks to do with the fact that he is Austrian? ". Even more obvious is the problem if we the irrelevant premise " Peter is an Austrian " replaced by " grass is green ". We would certainly highly irritated when someone explained to us that the fact that Peter Kummer, from the fact that he drinks with grief, and the fact that grass is green, it follows that Peter is drinking.

The criticism therefore means in a nutshell, that the addition of irrelevant premises is not allowed, and that therefore monotony is not a universal property of conclusions. Logics that want to meet this intuition, hot Relevance logics.

A defender of the monotony property may be objected against criticism following: The fact that inferences occur so rarely with irrelevant premises in everyday life, has more practical reasons. People can save breath, and time when they avoid extraneous assumptions, and are still at it ( Reasoning ) target. So why perform such premises. Arguments with irrelevant premises are thus from this perspective, rare and unusual, but they are not so wrong.

• Logic