Understanding the Brilliance of Aristole…The Four Aristolelian Forms

Note:  A and E stand for universal and existential quantifiers.

Aristotle was a genius.  His beliefs and reasoning changed many disciplines, and logic was no exception.  He studied various kinds of reasoning that dealt with noun phrases such as “Every man,” “No man,” and “Some man.”  Today, using codified and established First Order Logic (FOL), one would translate these noun phrases using universal and existential quantifiers.  However, during Aristotle’s time, FOL and its quantification symbols were not available to him.  But, his conclusions about reasoning regarding these noun phrases laid the foundation for future work by logicians.

After studying what the above noun phrases entailed, Aristotle derived four main sentence forms:

1)     All P’s are Q’s.

2)     Some P’s are Q’s.

3)     No P’s are Q’s.

4)     Some P’s are not Q’s.

One should be able to recognize that essentially (1) and (3) are opposites, as are (2) and (4). [There is a mistake here. How can 2) and 4) be opposites if both can be true? For example: both Some men are honest and some men are not honest, are both true. In Aristotle's terminology, 1) and 3) are contraries and 2) and 4) are subcontraries. (This error was reported in an email message from a reader.0]

1)         Aristotle reasoned that objects existed with essential and inherent properties in that all objects embodied these properties.  Therefore, if red was an essential and inherent property of kites, then it followed that if x was a kite, then x was red.  All kites embodied the property of being red because having this property was essential to their being.  While Aristotle would have expressed “All kites are red” using the more general sentence form, “All P’s (kites) are Q’s (red),” today using FOL, that sentence would be translated as “‘Ax (P(x)à Q(x))’… For every object x, if x is a _____, then x is ______.”

2)         Aristotle reasoned that a second category of objects existed where some objects in this category embodied certain properties (or property) while others did not.  Therefore, these properties were not essential and necessary to an object’s being, since the object still existed even if it did not have the property.  For example, the property blue is not an essential, or a necessary, property of tables.  This is so because there are tables which exist which are not blue, but rather green or brown or some other color.  Only some tables are blue.    Aristotle would have expressed “Some tables are blue” using the more general form, “Some P’s (tables) are Q’s (blue).”  In FOL, that sentence would be translated as “ ‘Ex (P(x) ^ Q(x))’…For some object (but not all) x, x is a ____ and x is ____.”   

3)   Aristotle also reasoned that another category of objects existed where specific properties (or property) were never embodied in those objects.  There were certain properties that certain objects never embodied; these properties did not exist in these objects.  Obviously, these properties were not essential or necessary to an object’s being.  Rather, what was essential and necessary is that an object did not embody these properties.  For example, bunnies can not have the property of greenness- nothing that is a bunny is also green (has the specific property of greenness).  Aristotle would have expressed this by saying, “No P’s (bunnies) are Q’s (green).”  In FOL, this sentence is translated as “ ‘Ax (P(x)à ~Q(x))’… For any x, if x is _____, then x is not ______.”

4) The last category of objects Aristotle reasoned about were those objects that embodied a specific property, while other objects of the same category did not embody that same specific property.  Thus, some objects embodied a specific property, while others did not.  For example, some tables have the property of being blue, while some tables are not blue.  A table is still a table even if it is not blue.  Therefore, tables can exist even if they are not blue.  Aristotle would have expressed “Some tables are not blue” using the general sentence form, “Some P’s (tables) are not Q’s (blue).”  In FOL, this would be written as “ ‘Ex (P(x) ^ ~Q(x))’ … For some x, x is a ____ and x is not ____.”

What is essential to remember about the Aristotelian forms is that not only are they the very simplest and least complex sentences that can be constructed, but that all quantified FOL sentences, no matter how complex, have, depending on their left most quantifier, an overall Aristotelian structure. For example:

(5) Ax [Cube (x)à Ey (Tet (y) ^ LeftOf (x,y))]

has as its overall structure:

(1) Ax [P(x)à Q(x)]

which is identical to (1).  One trick to checking if your more complex sentences are written correctly is to compare, depending on the left most quantifier, your sentence with its Aristotelian form to determine whether their structures match.  For example, if one compares this sentence:

(6) Ax [Cube(x) ^  Ey (Tet(y) ^ LeftOf(x,y))]

To its Aristotelian form:

(1) Ax [P(x)à Q(x)]

One can clearly see that (6) is not written correctly as its structure does not match to (1).  The problem: the conjunction between Cube(x) and Ey needs to be a material conditional if (6) is to be written correctly.

To better acquaint yourself with this concept of Aristotelian Forms, refer to section 9.5, “The Four Aristotelian Forms” in Language, Proof and Logic, and the last paragraph on page 293 under section 11.2, “Mixed Quantifiers.”