Exercises

Exercises for "Example"

Go to "Example" (use the link at the bottom line of the main pages Part 1 - 3 or the link to "Example" at the top of this page).

Exercise 1

Task: Mark "All men are mammals" and press "Execute".

Result:The first column of the result reproduces the input; the second column shows the consequences of the input proposition: I(man,mammal) (Some men are mammals) by the so called rule of subalternation, and I(mammal,man) (Some mammals are men) by conversion. The third column shows which rules have been applied to which proposition:

Caution: Do not interprete I(man,mammal) as an "existential" proposition! It just says that the concept "man" and the concept "mammal" are connected by the "Aristotelian quantor" I. For the syntax of Aristotelian logic, it is not necessary to make any assumption about the existence of individuals within classes. This allows to formalize propositions about concepts which are not known to be predicated of existing individuals. See the next exercise!

Exercise 2

Task: Mark "All men are mammals" and "All mammals are living beings" and press "Execute".

Result: Now the only interesting proposition in the result is proposition 2, because it has been generated by the application of the syllogism "Barbara" to the two input propositions 1 and 2. The following six propositions 4 - 9 do not give new insight into the structure of the input propositions. I- and O- propositions are often a kind of uninteresting "noise" produced by subalternation and partial conversion. (I think of inserting a switch which will turn off the display of these propositions - however I shall wait for reactions to this idea).

Reference: Go to "Short introduction" or to "Part 2" in order to see a list of all rules of the Aristotelian system!

Exercise 3

Task: Like in Exercise 2, mark "All men are mammals" and "All mammals are living beings". In addition, write A(dog,mammal) into the input box - either below or instead of the commentary. Then, press "Execute". - Interprete the result!

Caution: Be careful to write "mammal", not "mammals", because this is the way the concept has been denoted within the framework of our example. Experiment with denoting it the wrong way, too, and try to interprete the result.

Exercise 4

Task: Like in Exercise 3, mark "All men are mammals" and "All mammals are living beings". In addition, write A(bird,mammal) into the input box - either below or instead of the commentary. Then, press "Execute". - Interprete the result!

Question: Are you surprised that the program does not complain about the proposition "All birds are mammals", which is, as WE know, not true?

Remark:The program does not "know" what the concepts "bird", "mammal" etc. denote in the English language. By the way, do you know, which concept the word "bird" denotes in any of the languages of the world? Maybe there is an indianic language where "bird" denotes the English concept "penguin"? If we do not know, how could the program on the server? - All this is to explain why you cannot hope that the program is able to detect errors of "semantic" nature, i.e. errors which stem from an - in our language -incorrect USE of concepts.

Exercise 5

Task: Mark "All men are mammals" and "All mammals are living beings". In addition, write E(man,mammal) (No men are mammals) into the input box. Then, press "Execute". - Interprete the result!

Question: Are you surprised that the program does not complain about having to process a contradiction like A(man,mammal) together with E(man,mammal)?

Remark:This problem is at a quite different and a bit more sophisticated level than the one before. Here we have a syntactical problem which consists in the fact that apparently the rules of the system do not care about "contrary" propositions occurring together within the premise set. This is an important difference to standard propositional and predicate logic, where a contradiction within the set of premises always has the consequence that ALL propositions follow logically by the premises. This is not true in Aristotelian logic! In our example, the proposition E(livingbeing,man), f.i., does not belong to the set of conclusions. Thus, Aristotelian logic is said to be "paraconsistent" - a contradiction normally does not ruin everything like in standard prdedicate logic!

Additional exercise: Show that, in this example, there are 24 possible different propositions, (of which only 16 are consequences of the premises ). If you count propositions, remember or be informed now that Aristotle's original system did not allow the same term as subject as well as predicate of a proposition).

Exercise 6

Task: Mark "All men are mammals", "All mammals are living beings" and "Some mammals are quadrupedals". Click "Term graph". Try to identify the input propositions within the "term graph" which is produced by the program.

Question: What does "w1" signify?

Answer: One of the input propositions is "Some mammals are quadrupedals". "w1" is the name - provided automatically by the program - of those creatures who are as well mammals as also quadrupedals.

Exercise 7

Task: Mark "All men are mammals", "All mammals are living beings","All horses are mammals". In addition, write "I(horse,man)" into the input box.

Question: Why does the program write "w1" instead of "centaur"?

Answer: "w1" is the name - provided automatically by the program. You never told the program that creatures which are both men and horses are usually called "centaurs". If you want the program to know this, goto the next example.

Exercise 8

Task: Mark "All men are mammals", "All mammals are living beings","All horses are mammals". In addition, write "A(centaur,man) A(centaur,horse)" into the input box.

No Question.

WILL BE CONTINUED!