46. Logic and Syllogistic Reasoning in Geometry and Theology

Summary

This lecture examines the practical application of syllogistic logic across disciplines, focusing on how syllogisms are used to construct proofs in geometry and establish theological truths. Berquist demonstrates that eight fundamental syllogistic forms (four categorical, two conditional, two disjunctive) recur throughout philosophical and theological reasoning, illustrating their use through Euclid’s geometric proofs and Thomas Aquinas’s theological arguments.

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Lecture Notes

Main Topics #

The Eight Fundamental Forms of Syllogism #

There are eight basic syllogistic forms that appear repeatedly in philosophical and theological reasoning:

Four categorical syllogisms: The primary forms used in demonstration
Two conditional syllogisms: If-then reasoning (affirming the antecedent; denying the consequent)
Two disjunctive syllogisms: Either-or reasoning (determining which alternative applies; eliminating false alternatives)

These eight forms constitute the core toolkit for all rigorous logical demonstration.

The Third Figure of Categorical Syllogism #

The middle term is the subject in both premises
The third figure has no universal conclusions—all conclusions are particular
Four cases with two universal statements: universal affirmative with universal negative; universal negative with universal affirmative; two universal negatives
Conversion is essential for working with second and third figure syllogisms, as the “set of all” or “set of none” does not apply directly
- Universal negative converts simply and remains universal negative
- Universal affirmative converts partially, yielding particular affirmative
- Particular affirmative converts to particular affirmative
- Particular negative does not convert

Negative Syllogisms in Theological Application #

Negative syllogisms are particularly important in theology because God is approached through the via negationis (negation). Examples include:

“Everything corruptible has parts. God has no parts. Therefore, God is not corruptible.”
“Whatever has parts is a composition of act and potentiality. God is not a composition of act and potentiality. Therefore, he has no parts.”
In Plato’s Phaedo, Socrates argues against the soul being the harmony of the body using this form: “The soul resists the body, but the harmony of the body does not resist the body. Therefore, the soul is not the harmony of the body.”

The Natural Road of Knowledge #

Berquist illustrates syllogistic structure through a self-constructed example about knowledge:

Conclusion to prove: The first road in our knowledge is the road from the senses into reason
Middle term needed: What unites “first” with “the road from the senses into reason”?
Answer: The natural road
Major premise: The first road is the natural road (because nature means what a thing is, and what is natural must be first)
Minor premise: The natural road is the road from the senses into reason (because human nature is to be an animal with reason; what is generic develops before what is particular; senses are generic, reason particular)

Key Arguments #

Euclid’s Proposition 1: Constructing an Equilateral Triangle #

Berquist analyzes Euclid’s construction of an equilateral triangle to demonstrate three syllogisms working in concert:

Syllogism 1 (Proving AC = AB):

All radii of the same circle are equal
AC and AB are radii of circle BCD
Therefore, AC = AB

Syllogism 2 (Proving BC = BA):

All radii of the same circle are equal
BC and BA are radii of circle ACE
Therefore, BC = BA

Syllogism 3 (Proving AC = CB):

All quantities equal to the same thing are equal to each other
AC and CB are both equal to AB
Therefore, AC = CB

The key insight: The first two syllogisms share the same form (radii of the same circle), but the third requires a different middle term (quantities equal to the same) because the two lines are not radii of the same circle.

The Use of Either-Or Reasoning in Natural Philosophy #

Berquist explains the disjunctive syllogism through Empedocles’s rejection of monism:

First alternative: If there is one first matter, either it has definite qualities OR it has no definite qualities
If it has definite qualities: Everything would share that quality (if water, everything would be wet; if fire, everything would be hot). But we observe a world of contraries, so this fails.
If it has no definite qualities: How do qualities emerge from a qualityless substrate? (This is equally problematic.)
Conclusion: Monism appears untenable; Empedocles must postulate multiple elements

Important Definitions #

Syllogism: A form of reasoning with two premises and a conclusion, where a middle term unites two extreme terms
Middle term (μέσον, medium): The term that appears in both premises but not in the conclusion; it is the means by which the mind sees the connection between the two extremes
Conversion (conversio): The logical operation of switching the subject and predicate of a proposition, subject to rules depending on the proposition’s quality and quantity
Via negationis (way of negation): The theological method of approaching God’s nature by denying what God is not, rather than affirming what He is
Figure: The position of the middle term in the premises (first figure: middle term is subject of major, predicate of minor; second figure: middle term is predicate of both; third figure: middle term is subject of both)

Examples & Illustrations #

Conditional Syllogisms in the Meno #

Socrates uses two forms in investigating virtue:

Affirming the antecedent: “If virtue is knowledge, then virtue can be taught. Virtue is knowledge. Therefore, virtue can be taught.”
Denying the consequent: “If virtue can be taught, then there are teachers of it. There are no teachers of it. Therefore, virtue cannot be taught.”

Negative Syllogisms in Theology and Philosophy #

Soul vs. harmony argument: “The soul resists the body, but harmony of the body does not resist the body. Therefore, the soul is not the harmony of the body.” (or: “The soul has a harmony, but a harmony does not have a harmony. Therefore, the harmony is not the soul.”)

Empedocles and the Problem of First Matter #

Either-or reasoning applied to natural philosophy: If all things come from one matter, that matter either has definite qualities or lacks them. Both alternatives face serious problems, suggesting monism is untenable.

Notable Quotes #

“There are really eight things you have to know in the sense that they’ll come up again and again, right? The four ones we have on the board here, right? And then the two ways of arguing from an if-then statement, right? And the two ways of reasoning from an either-or statement. And these will come up again and again, right?”

“Now, he has to prove that AC and AB and BC, that each of those lines is equal to the other two, right? Okay. Now, how many syllogisms does he need to prove that, right?”

“These are the four to remember. But then in addition to these, there’s the two forms of the if-then syllogism, right? One where you deny the consequent and then deny the antecedent, and the other where you affirm the antecedent and then affirm the consequent, right?”

Questions Addressed #

How do conversion rules affect second and third figure syllogisms? #

The second and third figures do not directly yield the “set of all” or “set of none” as the first figure does. Conversion—systematically switching subject and predicate according to logical rules—allows us to transform these syllogisms into recognizable forms and verify their validity.

Why are negative syllogisms central to theology? #

Because theology approaches God through negation (via negationis), the negative syllogism becomes a primary tool. We demonstrate truths about God by showing that certain properties cannot belong to Him (e.g., God is not corruptible, not composed, not a body).

How does Euclid employ multiple syllogisms in a single proof? #

Euclid uses three separate syllogisms to establish the three equalities required for an equilateral triangle. The first two share a form (radii of the same circle); the third uses a different middle term (quantities equal to the same thing) to relate quantities that are not radii of the same circle.

How does either-or reasoning address philosophical problems? #

Disjunctive syllogisms exhaust alternatives and eliminate impossible options. Empedocles uses this to show why monism (the theory that all things come from one first matter) faces insurmountable obstacles: whether that matter has definite qualities or lacks them, contradictions arise.

Structure and Pedagogy #

Berquist emphasizes that these eight forms recur “again and again” in philosophical and theological demonstration. Students are encouraged to memorize them because they form the backbone of rigorous reasoning across all disciplines. The lecture moves from abstract logical structure to concrete applications, showing that the same forms appear in Euclid’s geometry, Socratic dialogue, and Thomistic theology.