English 1 - Critical Reasoning: Square of Opposition Analysis

This essay provides an in-depth analysis of the Square of Opposition, a fundamental concept in categorical logic. It begins with an introduction to the Square of Opposition, explaining its purpose as a chart that visually represents logical relationships between categorical propositions. The essay then details the importance and features of the Square of Opposition, including the four types of propositions (A, E, I, and O) and their relationships (contradictory, contrary, subcontrary, and subalternation). The essay uses a diagram to illustrate these relationships and explain how they function within Aristotelian categorical logic. Furthermore, it clarifies the differences between Boolean and Aristotelian definitions of categorical logic. The essay concludes by summarizing the various relationships within the Square of Opposition, emphasizing their significance in critical reasoning and logical analysis. The content is supported by several academic references.

Running head: ENGLISH 1
Critical Reasoning
Name
Institution

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Introduction
Square of opposition is a chart that is used within a categorical logic to symbolize a
logical relationship that holds between certain propositions by a virtue of their structure.
Traditionally, it is represented in the form of a diagram that exhibits four forms of the categorical
propositions, represented as q.v or statements (Choudhury & Chakraborty, 2016). However,
these statements have the same subjects and predicate together with their corresponding pairs.
Conversely, categorical logic is the branch of mathematics, in which tools and concepts from a
category assumption is applied to study the mathematical logic. In other words, categorical
statements are those statements that asserts to a whole or partly true to the subject and predicate
terms of the statements (Béziau & Basti, 2017).
Importance and features of Square Opposition
A square of opposition assists in inferring the truth value of the proposition grounded on
the truth features of the other propositions that have the same terminologies as per the below
figure. As a result, one should be able to ascertain the differences between the Boolean and the
Aristotelian definitions of the categorical logics depending on the use of the square of opposition
(Peterson, 2018). To confirm the features of the square of opposition, the following diagram will
be used to inclusively describe the purpose of the square opposition in the categorical logics;

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Figure 1.0: Square of Opposition
From the diagram above, it can be observed that the diagram has four corner charts that represent
four forms of propositions in the square opposition. These include;
a. “A” proposition which is also known as the universal affirmative. It takes the form of ‘all
S are P”.
b. “E” proposition which is the universal negation and takes the form of “no S are P”
c. “I” proposition, also known as the particular affirmative and it takes the form of “some S
are P”
d. “O” proposition, also known as the particular negation that takes the form of “some S are
not P”

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From the above four features of the square opposition, using Aristotelian categorical logic that
contains at least one member of propositions, the following relationships are bound to hold in the
square of opposition;
Firstly, it can be argued that A and O propositions are contradictory, so as E and I
propositions. In general, propositions are said to be contradictory when the fact of one proves
falseness of the other, and when the falseness of the other proves the truth of the other (Arenhart
& Krause, 2014). Taking the proposition of the form “all S are P” shows that there is a false of
the other argument that “some S are not P”. For instance, if we argue that “all industrialists are
capitalists” then it means that A is true showing that the argument that “some industrialists are
capitalists” must be false, for O. Similarly, if we argue that “no mammals are aquatic” then it
shows that E is a false statement by agreeing that “some mammals are aquatic”
Secondly, one can argue that proposition A and E are contrary when both of A and E are
true. For instance, if we argue that “all hippopotamuses have short tails” then it cannot be true at
the same time when arguing the corresponding proposition E: “no hippopotamuses have short
tail”. It is therefore clear that while proposition A and E cannot be true at the same time, they
cannot still be false at the same time. Lastly, proposition I and O are said to be subcontrary when
it is only impossible for both I and O to be regarded to be false at the same time (Demey &
Smessaert, 2018). For instance, if we say that “some birth day party are alcohol-free” is false,
then it is true that “some birth day party are not alcohol-free”. However, it is also possible for
both proposition I and O to be true if we say that it is true that “some presidents are dictators”,
and “some dictators are presidents”. It therefore shows that proposition I and O can be
subcontrary, but neither a contractor nor contrary.

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Lastly, if proposition A is in the subalternation relationship with proposition I, then the
truth of A, for example “all cars are German made” infers the truth that “ some cars are German
made”. However, if we argue that “some cars are German made” does not imply that “no cars are
German made”. In traditional logic, facts in the proposition A or in E provide truth in either
proposition I or O correspondingly. However, false in proposition I or O provides corresponding
falseness in A or E respectively (Vidal-Rosset, 2017).
In summary, contrary relationship is a partial opposition where at least one contrary must
not be true while subcontrary is one partial opposition with at least one of the subcontrary is true
or both. In addition, two propositions are regarded to be subalternate if the truth for the first,
known as the superaltern is the truth for the second proposition, known as the subaltern.

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References
Arenhart, J. R., & Krause, D. (2014). Contradiction, quantum mechanics, and the square of
opposition. arXiv preprint arXiv:1406.1836.
Béziau, J. Y., & Basti, G. (2017). The square of opposition: A cornerstone of thought. In The
Square of Opposition: A Cornerstone of Thought (pp. 3-12). Birkhäuser, Cham.
Choudhury, L., & Chakraborty, M. K. (2016). Singular propositions, negation and the square of
opposition. Logica Universalis, 10(2-3), 215-231.
Demey, L., & Smessaert, H. (2018, June). Aristotelian and duality relations beyond the square of
opposition. In International Conference on Theory and Application of Diagrams (pp. 640-
656). Springer, Cham.
Peterson, P. L. (2018). Intermediate Quantities: Logic, Linguistics and Aristotelian Semantics:
Logic, Linguistics and Aristotelian Semantics. Routledge.
Vidal-Rosset, J. (2017). The exact intuitionistic meaning of the square of opposition. In The
Square of Opposition: A Cornerstone of Thought (pp. 291-303). Birkhäuser, Cham.