Square of Opposition in Categorical Logic - Free Essay Example

The square of opposition is a chart that is widely applied under the categorical logic to give a representation of the relationships of logic that hold between given propositions. Categorical logic, in essence, is the proposition that can either deny or insinuate that all of the members that belong to a specific category are also included in another category. The former category is the subjective term while the latter category is the predicate term. The square of opposition gained prominent application in the field of deductive reasoning through the Ancient Greek philosophers, notably Aristotle (Beziau & Read, 2014). The square of opposition comprises of four corners within the chart that are representative of the four fundamental forms of propositions in categorical logic. The four corners are thus labeled as A, E, I, & O. Given these propositions, it is easy to make inferences about a given form based on its relationship to the other forms.

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The corner A is a proposition that represents all the universal affirmatives that take the form that all the subject terms are predicate terms. The E corner constitutes of all the propositions that represent the universal negations which take the form that no subject terms are predicate terms. At the I corner there are the propositions that assert particular affirmatives that mainly take the form that some of the subject terms are predicate terms. The O corner of the square opposition entails the propositions that represent particular negations that take the form that some of the subject terms are not predicate terms. All the four corners are important as they show the prevailing relationships between the different forms. Under the Aristotelian categorical logic, it is assumed that each of the four categories contains at least a member as indicated on the opposition square (Pfeifer & Sanfilippo, 2017).

The relationships that hold true on the square opposition begin with the A and O propositions which happen to be contradictory in nature just like are the propositions found in corners E and I. The contradictory element comes out in the fact that the truth of one proposition proves to be the falsity of the other proposition. The truth of the proposition that all subject terms are predicate terms automatically implies falsity in the proposition that some subject terms are not predicate terms. Then, there are the propositions A and E that are contrary. Contrary propositions are the ones that cannot be true at the same time. It is important to note that as much as A and E propositions are contrary they cannot be construed to be contradictory. Contrary means that they both cannot be correct although the two propositions can be false (Beziau & Read, 2014).

The I and O propositions as depicted in the square opposition happen to be subcontrary. Subcontrary propositions are the ones that are impossible to express a falsity at the same time. However, it is worth noting that the I and O propositions that correspond to each other can possibly be both true. The subcontrary nature of the I and O propositions means that they can neither be contrary nor contradictory. Lastly, when the truth of the first proposition is an implication of the truth in the second proposition, then the two propositions are said to be in subalternation. The truth contained in propositions A or E are implicit of the truth in the corresponding I or O propositions. This is similar to the falsity found in the propositions I or O which is implicit of the falsity of the corresponding propositions A or E.

An original example that can used to illustrate how to translate statements into standard-form claims and corresponding standard-form claims including description of the relationships that are depicted within the square entails the use of mangoes and oranges. Translating the square of opposition is done through the help of operations like contraposition, obversion, and conversion (Pfeifer & Sanfilippo, 2017). Given the two items namely mangos and oranges, it is advisable to commence with term complements in which a term is made into an opposite term. Examples of terms and their complements in this case are mangoes and non-mangoes as well as non-oranges and oranges respectively. The addition of the prefix non- before the term will be enough to transform the term into its complement. Adhering to these basics beforehand will help in simplifying the translation process of the square of opposition through any of the three operations.

With the conversion operation, there is switching of the subject and predicate terms and this works with the E and I propositions. The new proposition will retain the same truth or falsity as the original statement. An example would be 'no mangoes are oranges' would be converted to 'no oranges are mangoes' thus retaining the same truth. Obversion entails all the A, E, I, and O propositions with this operation changing the quality of the affirmative or negative proposition and then replacing the predicate term with its complement. An example would be the statement 'no mangoes are oranges' would be translated to 'all mangoes are non-oranges'. Contraposition is another simple translating operation that involves the A and O propositions where the value of truth remains intact. It involves switching the subject and predicate terms like in the conversion operation, with the difference being the replacement of both the subject and predicate terms with their complements. The proposition 'some mangoes are not yellow fruits' would be translated into 'some non-yellow fruits are not non-mangoes'.

References

Beziau, J. & Read, S. (2014). Square of opposition: A diagram and a theory in historical perspective. History and Philosophy of Logic, Volume 35, Issue 4, Pages 315-316. doi: 10.1080/01445340.2014.917836.

Pfeifer, N. & Sanfilippo, G. (2017). Square of opposition under coherence. Soft Methods for Data Science, Vol. 456, Pages 407-414. doi: 10.1007/978-3-319-42972-4_49.