45. Syllogistic Form, Matter, and Demonstration versus Dialectic

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Lecture Notes

Main Topics #

The Form versus Matter of a Syllogism #

• Form: Whether a conclusion follows necessarily from given premises (the logical structure)
• Matter: Whether the premises themselves are true or false, necessary or probable
• A syllogism can have correct form but false matter, or correct form and true matter
• The distinction is crucial for understanding both logic and theology

The Arithmetic Analogy #

Berquist illustrates the form/matter distinction through the example of multiplying numbers:

• Form: Performing multiplication correctly (the mathematical operation)
• Matter: Having the correct numbers to multiply (actual quantities)
• One can multiply correctly with wrong numbers (correct form, wrong matter)
• One can have correct numbers but multiply incorrectly (correct matter, wrong form)
• Both components are necessary for a correct result

Demonstration (Scientia) versus Dialectical Syllogism #

Demonstration #

• Premises are seen as necessarily true
• Either evident in themselves (axioms, postulates) or derived through prior demonstrations
• Example: Euclid’s geometry—axioms like “all right angles are equal” and “the whole is greater than the part”
• The conclusion is necessarily true when it follows from necessarily true premises
• Demonstrates not only that something is so, but why it must be so (knowledge of the cause)

Dialectical Syllogism #

• Premises are seen as probable opinions (endoxa)
• Conclusion follows necessarily in form, but is only probable in matter
• The conclusion is necessarily probable (follows necessarily from probable premises)
• Does not prove that something is necessarily true, only that it is probably so

Definition of Probable Opinion (endoxa) #

According to Aristotle, an opinion is probable because of:

1. Quantity of men holding it: All men or most men think it
2. Quality of men holding it: All or most men famous in a particular art or science, when speaking of matters pertaining to that art or science
   • Example: Doctors’ opinions about smoking and health
   • Example: Euclid’s opinions about geometry

The Conversion of Universal Negative Propositions #

Berquist establishes that “No A is B” necessarily entails “No B is A” through reductio ad absurdum:

• Assume “No A is B” but “Some B is A”
• Call the B that is A by the name X
• Then X is both B and A
• Therefore “Some A is B”
• This contradicts “No A is B”
• Therefore, we must accept “No B is A”
• This principle holds even for singular propositions (e.g., “Socrates is not a woman” entails “No woman is Socrates”)

Syllogistic Figures and Validity #

Berquist discusses converting premises to achieve valid forms:

• Not all figures equally make conclusions evident
• First figure is clearest: premises can be directly applied
• Other figures may require conversion to reduce to first figure form
• Example: “No B is C” and “Every A is B” requires conversion to establish “No C is A” clearly

Science as Reasoned-Out Knowledge #

Berquist defines scientia (science) as:

• Reasoned-out knowledge of things
• Not merely facts, but understanding through reasoning
• Includes thinking out, making distinctions, divisions, definitions, picking out statements, and forming conclusions
• Distinguished from modern experimental science, though both involve reasoning

Modern Science and Hypothesis #

Berquist discusses Einstein’s characterization of scientific hypotheses:

• Einstein said hypotheses in science are “freely imagined”
• Great discoveries often made by young scientists (20s-40s) with free imagination
• The testing of hypotheses occurs through prediction: if hypothesis is true, then certain phenomena should follow
• This is not identical to the syllogistic form of demonstration
• Example: Newton (light as particles) versus Huygens (light as waves) and the later photoelectric effect showing light behaves as both
• Young physicists like Heisenberg and Bohr made breakthrough discoveries through insight combined with rigorous reasoning

Key Arguments #

The Conversion of Negatives (Reductio ad Absurdum) #

1. Assume “No A is B” but concede “Some B is A”
2. Let X be a B that is A
3. Then X is both A and B
4. So there exists an A that is B
5. This contradicts our premise
6. Therefore, “No B is A” must be true

The Equality of Vertical Angles (Geometric Demonstration) #

1. When two straight lines intersect, they form angles
2. Each angle plus its adjacent angle equals two right angles (from the definition of a right angle)
3. So: a + x = 2 right angles AND b + x = 2 right angles
4. Therefore: a + x = b + x
5. Subtracting equal things from equal things yields equal results
6. Therefore: a = b (vertical angles are equal)

Form versus Matter in Validity #

1. A syllogism with correct form makes its conclusion follow necessarily from its premises
2. If the premises are true, the conclusion must be true
3. If the premises are false or merely probable, the conclusion is still only as certain as the premises allow
4. Therefore, correct form is necessary but not sufficient for knowledge of truth

Important Definitions #

Demonstratio #

A syllogism with necessarily true premises that produces a necessarily true conclusion, providing understanding of the cause of what is demonstrated.

Dialectical Syllogism #

A syllogism with probable premises (endoxa) that produces a conclusion that is necessarily probable but not necessarily true.

Endoxa #

Probable opinions—statements held by all or most people, or by all or most experts in a given field, regarding matters pertaining to that field.

Prior Analytics #

Aristotle’s logical treatise concerning the form of the syllogism (using letters rather than concrete examples), which asks whether something follows necessarily given certain premises.

Scientia #

Reasoned-out knowledge of things through demonstrations; knowledge of both the fact and the cause.

Examples & Illustrations #

The Ice Cream Cone Calculation #

• Berquist was at the beach with 19 grandchildren and adults needing ice cream
• Had to multiply: (number of people) × (cost per cone) = total cost
• Correct form: performing multiplication correctly
• Correct matter: knowing actual cost and actual number of people
• Either could be wrong independently

Geometric Examples #

1. Vertical Angles: When two straight lines intersect, opposite angles are equal
   • Demonstrated through the definition of right angles and the axiom that equal things subtracted from equal things are equal
2. Material Universals: No animal is a stone, No cat is a stone, No tree is a stone
   • These premises are all true, but they don’t necessarily entail “Every cat is an animal”
   • Shows that true premises don’t guarantee a conclusion follows necessarily if the form is invalid

Famous Figures and Probable Opinions #

1. Einstein on scientific hypotheses: His statement that hypotheses are freely imagined is probable because Einstein is famous in physics
2. Mozart on opera: Mozart is famous for operas and says words must be secondary to music—his opinion on opera composition is probable
3. Euclid’s Elements: Euclid’s geometric theorems carry weight because he is famous in geometry

The Nature of Light #

• Newton hypothesized light is particles (corpuscles)
• Huygens hypothesized light is waves
• Experiments showed light behaves as waves
• Einstein showed photoelectric effect only explained by particle hypothesis
• Light seems to be both waves and particles—a genuine puzzle about nature, not about reasoning

Philosophical Application #

• Pythagoras claimed not to be wise (sophos) but rather a lover of wisdom (philosophos)
• God is wise (He is wisdom itself) in the most absolute sense, yet can be called a “lover of wisdom” analogously
• The word “philosopher” itself indicates one who recognizes he lacks the complete wisdom of God

Questions Addressed #

How does the form of a syllogism differ from its matter? #

• Form: The logical structure—whether a conclusion necessarily follows from premises
• Matter: The truth or probability of the premises themselves
• Both are necessary: correct form ensures logical validity; correct matter ensures true conclusions

Can one validly convert “No A is B” to “No B is A”? #

• Yes, necessarily
• Denying this leads to contradiction: if some B were A when no A is B, then we’d have both A being B and A not being B
• This principle applies even to singular statements (e.g., “Socrates is not a woman” ↔ “No woman is Socrates”)

What is the difference between demonstration and dialectical syllogism? #

• Difference in form: None—both follow the rules of syllogistic logic
• Difference in matter: Demonstration uses necessarily true premises; dialectic uses probably true premises
• Result: Demonstration yields necessary knowledge; dialectic yields probable knowledge

Is modern science (as exemplified by Einstein) reasoned-out knowledge? #

• Yes, though it involves more than pure demonstration in the Aristotelian sense
• Scientific hypotheses involve free imagination, but they are tested through reasoning
• When predictions drawn from hypotheses are confirmed by experiment, this provides probable support
• But the confirmation is not quite syllogistic—it’s the affirmation of the consequent (if P then Q; Q is true; therefore P)

How can different scientific theories (particle vs. wave) both be supported by evidence? #

• This suggests that reality itself may be more complex than classical logic accommodates
• It indicates the need for deeper understanding, as both properties seem to inhere in the same subject
• Modern quantum mechanics eventually developed the concept of complementarity to address this

Notable Quotes #

“I’m from Missouri. How do you know you can turn that around, huh?” — Berquist, challenging a student’s assumption about converting negatives

“You’re forced by the truth itself, right?” — Berquist, explaining how reductio ad absurdum compels assent

“The difference between demonstration and dialectic right is in the form or in the matter?” — Berquist’s core question distinguishing the two types of syllogisms

“The words must be all together to obedience [to music].” — Mozart, cited by Berquist on opera composition (probable opinion from an expert)

“Don’t call me wise. God alone is wise.” — Pythagoras, cited to illustrate how even the word “philosopher” (lover of wisdom) properly indicates one aware of lacking God’s wisdom