Boolean vs. Aristotelian
It is useful in logic to categorize topics into classes in order to relate things to one another and draw conclusions. This method of categorical propositions and the graphical representation of their relationships to one another, the square of opposition, were created by Aristotle and expanded upon over the generations. While the traditional views are used in application for certain ideas, the modern view, credited to George Boole, is different in one key notion that changes the way categorical propositions can be analyzed. The modern Boolean standpoint differs from the traditional Aristotelian standpoint in its treatment of existential import. These two views impact the square of opposition and what inferences can be made from categorical propositions.
A categorical proposition makes an assertion or a claim about a subject class or category in relation to another class. There are two terms within a proposition, a subject term and a predicate term. There are four standard forms of categorical propositions given the letter names A, E, I, and O. In the A form, all members of the subject class (S) are included in the predicate class (P): all S are P. In the E form, all member of S are excluded from P: no S are P. These forms A and E are universal propositions because they include all of the subject class. The forms I and O are particular propositions because they only include at least one of the subject class. In the I form, some members of S are included in P: some S are P. In the O form, some members of S are excluded from P: some S are not P.
Aristotle introduced “the doctrine of the square of opposition” in the 4th century BC (Luzeaux, Dominique, et al). The four different forms of propositions make up the corners of both the modern and traditional squares of opposition. The square of opposition illustrates the relation between categorical statements (Wei-Ming). It shows valid inferences that can be made and gives a concise way to check the validity of the immediate inferences. The difference between the modern and traditional squares of opposition lies in the interpretation of the universal propositions with respect to existential import. Existential import means that a claim of existence is made. The Aristotelian view is that all propositions, whether universal or particular, make a claim of existence if the subject terms refer to real things. The modern view is a bit more sophisticated. Boole asserted that particular propositions do have existential import, but universal propositions do not. They both agree that when a statement refers to things that are imaginary or do not exist, the claim has no existential import.
The question of whether universal statements make claims of existence has been widely debated. Particular propositions make claims about at least one member of a class, and since these statements are directly observable, both Aristotle and Boole agreed they have existential import. Aristotle, however, thought that observation of particulars over time supports making claims universally. For example, the universal A form proposition “All humans are things that require oxygen” has existential import in the traditional standpoint because no members of the human class have been found to not be things that require oxygen. Boole as a modern philosopher wondered if it is possible to really know anything with absolute verifiability without inspecting all things within the universe to make a claim of universal existence. For this reason, the modern Boolean standpoint does not give existential import for universal statements. This difference in view regarding universal categorical propositions makes the difference between the modern Boolean and traditional Aristotelian standpoints.
Both the modern and traditional squares of opposition are set up in the same way. The modern Boolean square of opposition addresses only the contradictory relationships between A and O, and E and I. Boolean logic is not concerned whether something exists or not, therefore certain inferences can’t be made. The horizontal and vertical relationships between the A and I, the I and O, the A and E, and E and O are logically unknown and therefore the modern square focuses on only the diagonal contradictory statements.
(Wei-Ming)

A contradiction occurs when two propositions declare that something that is and is not at the same time. For example, the statement “All cats are animals” is in the A form and asserts that there are no cats that are not animals. The O form statement is “Some cats are not animals.” This means there is at least one cat that is not an animal, and this directly opposes the A statement. If the A statement is true, then the O statement must be false. There cannot be one cat that is not an animal if all cats are animals. The contradiction is also valid if the O form is true. If the O form statement is “Some scrubber dubbers are not gas powered machines,” the A statement “All scrubber dubbers are gas powered machines” cannot be true. The E and I form statements exhibit the same contradictory nature. If the E statement “No cats are animals” is true, then the I statement “Some cats are animals” must be false. If the I statement “Some cats are animals” is true, it cannot be true that “No cats are animals.”
(Wei-Ming)

Upon looking at the Boolean square of opposition there are no assumptions about the existence of anything. Inferences validated from this standpoint are considered unconditionally valid because they don’t require existential import to be valid (Hurley, 212).
The traditional Aristotelian square of opposition allows more inferences to be made due to existential import. Aristotle held that there are relationships between the horizontally and vertically related forms in the square.
(Wei-Ming)

The traditional square of opposition identifies the four relations of opposition: “contradictory; which is opposite of truth value, contrary; which means at least one is false or (not both true), sub contrary; at least one is true (not both false), and sub alternate; truth flows downward and falsity flows upward” (Hurley, 228). The contradictory relationships in the traditional square are the same as those from the modern square, and also yield unconditionally valid results. The other relationships are conditionally valid because the validity of the inference is based on the existential import, which is only assumed until it is proven. At the point the existence of the class is proven, the statement then becomes unconditionally valid.
The contrary relationship exists between the universals forms A and E propositions. Since these statements make a claim about an entire category, they cannot both be true. If it is true that “All flowers are things that require water,” and flowers exist, then it cannot be true that “No flowers are things that require water.” The contrary relationship requires that at least one of the statements must be false, so it is possible for both of the statements to be false. Therefore if the proposition states that the truth value of an A or E statement is false, an inference cannot be made about the other. For example, if it is false that “All Toyota trucks are gold,” it is not valid to infer that “No Toyota trucks are gold.”
The subcontrary relationship exists between the particular forms I and O propositions. With subcontrary at least one of the statements is true but they are not both false. It is possible for the both of them to be true. If the I form statement “Some dogs are purple” is false, then the O statement “Some dogs are not purple” must be true. If the I form statement above is true, however, an inference regarding some dogs not being purple cannot be made. The subalternation relations are the vertical associations between A and I, I and A, or E and O, O and E. The different relationships of subalternation means that truth flows down from the universal to particular propositions and falsity flows upward from particular to universal on the traditional Aristotelian square of opposition. If the A claim is true, the downward I claim must be true, but if the A claim is false then no inferences can be made about the I claim. If the proposition “All cookies are delicious” is true, then it is true that “Some cookies are delicious.” Simultaneously with subalternation the falsity must flow upward. If the O claim is false then the E claim must be false, but if the O claim is true then the E claim would be unknown. If it is false that “Some rabbits are not animals,” then it must be false that “No rabbits are animals.”
Since the Aristotelian perspective recognizes existential import and gives relationships between the vertical and horizontal forms on the square of opposition, inferences that are valid due to these relationships would be invalid according to the Boolean standpoint due to the existential fallacy. For example, if “All rain is wet” is true, then traditionally “Some rain is wet” is a valid statement. In the modern view, however, it cannot be assumed that rain exists, so the inference regarding some rain is invalid by existential fallacy.
In conclusion the traditional Aristotelian standpoint believes that things exist within all propositions and recognizes the four relations of contradictory, contrary, subalternate and subcontrary within the square of opposition and the modern Boolean standpoint makes no assumptions about the existence of things and only recognizes the relation of contradictory on the square of opposition.

Works Cited
Hurley, P. J. H. A Concise Introduction to Logic (with Stand Alone Rules and Argument Forms Card): Patrick J. Hurley: 9780840034175: Cafesribe.com: Books. N.p., n.d. Web. 21 Sept. 2013.
Luzeaux, Dominique, et al. "Logical extensions of Aristotle's square." Publications du LIRMM. 2008. Management System Publications LIRMM. 11 November 2013. .
Wei-Ming, Wu. "Squares of Opposition." I Logic. 2013. Butte College. 3 November 2013. .