Monotony (logic)

Monotony is a property of a deducibility relation or an inference operation and means that the addition of further premises (assumptions) always preserves the previous conclusions.

Formal definition

For derivative relations : Let be a derivability relation and let formula sets. is monotonic if and only if : ${\ displaystyle \ vdash}$ ${\ displaystyle \ Gamma, \ Delta}$ ${\ displaystyle \ vdash}$

If then ${\ displaystyle \ Gamma \ vdash \ mathrm {A}}$ ${\ displaystyle \ Gamma \ cup \ Delta \ vdash \ mathrm {A}}$

For inference operations: Let be an inference operation and let formula sets. is monotonic if and only if: ${\ displaystyle Cn}$ ${\ displaystyle \ Gamma, \ Delta}$ ${\ displaystyle Cn}$

If then ${\ displaystyle \ Gamma \ subseteq \ Delta}$ ${\ displaystyle Cn (\ Gamma) \ subseteq Cn (\ Delta)}$

Explanation

The property of monotony says that if a certain statement follows from a set of assumptions , that statement still follows when further assumptions are added.

First case For example, in propositional logic, it follows from the set of the two statements

Peter's grief.
When Peter is troubled, he drinks.

the statement

Peter drinks.

Second case This statement still follows even if we also include the statement "Peter is Austrian." assume: From the set of the three statements

Peter's grief.
When Peter is troubled, he drinks.
Peter is Austrian

also follows

Peter drinks.

Third case The conclusion remains correct even if we add the negation of the conclusion to the assumptions: From the set of the three statements

Peter's grief.
When Peter is troubled, he drinks.
Peter doesn't drink.

follows

Peter drinks,

although the conclusion contradicts one of the assumptions. This can be explained by the fact that the set of assumptions is inconsistent ; This means that the assumptions contradict each other and everything follows from a contradiction (see Ex contradictione sequitur quodlibet ).

discussion

Monotony applies to the classical propositional and predicate logic derivability concept, as well as to many other types of logic , such as modal logic . Nevertheless, monotony was also criticized as a general property of inferential relationships. The critics can be divided into two groups.

Non-monotonous inferential relationships

Consequences in everyday life often have a non-monotonous character. For example, if we learn that Tux is a bird, we would conclude that Tux can fly. If, however, we learn that Tux is a penguin, we would no longer conclude that Tux can fly, since we know that penguins cannot fly. Out

Tux is a bird.

seems so

Tux can fly.

to follow while out

Tux is a bird.
Tux is a penguin.

certainly not

Tux can fly.

follows, but rather

Tux can't fly.

The controversy can be resolved by distinguishing between conclusions that are 100% certain to hold and those that make the conclusion likely. From the assumptions that Peter is grieved and that Peter drinks when he's grieved, it follows with 100% certainty that Peter drinks. But it is not with the same certainty that if it is true that Tux is a bird, then he can also fly. Most birds can fly, but penguins or those with broken wings cannot.

In summary it can be said that one should not assume monotony for inferential relationships that prove themselves in everyday life, but which often do not apply with absolute certainty. In order to do justice to this intuition, so-called non-monotonic logics were developed (see for example induction ). However, the criticism does not concern conclusions, for example in mathematics , since there is always conclusions with absolute certainty ( deductive ).

The relevance point of view

This criticism, too, is based on everyday inferential actions. Let us look again at the above example with Peter and assume that someone would say to us: “We know three things: Peter is grief, if Peter is grief, he drinks and Peter is Austrian. So we also know that Peter drinks. ”Then we would rightly ask:“ What does that he drinks have to do with the fact that he is Austrian? ”. The problem becomes even more obvious when we replace the irrelevant assumption “Peter is Austrian” with “Grass is green”. We would be irritated if someone explained to us that from the fact that Peter has sorrow, from the fact that he drinks when he is sore, and from the fact that grass is green, it follows that Peter drinks.

So the criticism says that the addition of irrelevant assumptions is not allowed, and that therefore monotony is not a universal property of inferences. Logics that want to do justice to this intuition are called relevance logics .

A defender of the monotony property could object to the criticism: The fact that inferences with irrelevant assumptions are so rare in everyday life has more practical reasons. You save time if you avoid irrelevant assumptions and still achieve your (argument) goal. From this point of view, arguments with irrelevant assumptions are rare and unusual, but therefore not wrong.