48. Demonstration, Dialectic, and the Matter of the Syllogism

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Lecture Notes

Main Topics #

Demonstration (Ἀπόδειξις/Apodeixis) #

Demonstration is reasoning from premises that are necessarily true to conclusions that are necessarily true. The Greek word apodeixis means “to show” in the sense of making something clear or evident. Berquist emphasizes that the rigor of the syllogistic form does not guarantee the truth of the conclusion—only that if the premises are true, the conclusion follows necessarily. A demonstration proposes to show something, and when completed, indicates Q.E.D. (quod erat demonstrandum—what was to be demonstrated).

The Rigor Paradox: Correct logical operation (like correct addition) does not guarantee correct results if the premises themselves are false or incorrect. A syllogism can validly derive a false conclusion from false premises.

Dialectical Syllogism #

Dialectical reasoning uses the same eight forms of syllogisms as demonstration but proceeds from probable premises rather than necessarily true ones. The Greek work Topika (Topics) addresses this: it teaches reasoning from probable opinions. Unlike demonstration, dialectical reasoning can proceed to contradictory conclusions because both sides of a contradiction may have some probability.

The Nature of Probable Opinion #

According to Aristotle (Topics), an opinion is probable (ἔνδοξος) if it is:

• The opinion of all or most people, OR
• The opinion of all or most people in a given art or science regarding that art or science, OR
• The opinion of the most famous and illustrious thinkers in that art or science

Berquist notes these reduce to two sources of probability: the quantity of people holding an opinion or the quality (eminence) of those holding it.

Demonstration vs. Dialectic: Key Difference #

In demonstration, if one side of a contradiction is necessarily true, the other is necessarily false—reasoning never proceeds to contradictory conclusions. In dialectic, since probable opinions can exist on both sides of a contradiction, one can legitimately reason from probable premises to contradictory conclusions. However, if we reason dialectically to opposite conclusions, it reveals that at least one side has weakness in its probable grounds.

Key Arguments #

The Problem of False Premises #

Berquist illustrates that a syllogism can be formally valid yet materially false:

• Major: Every man is a stone
• Minor: Every stone is an animal
• Conclusion: Therefore, every man is an animal (necessarily follows, but from false premises)

This shows the distinction between the form of the syllogism (which ensures logical necessity) and the matter of the syllogism (the truth of the premises).

Virtue and Knowledge in the Meno #

In Plato’s Meno, Socrates reasons dialectically to opposite conclusions about virtue:

First side (virtue can be taught):

• If-then: If virtue is knowledge, then virtue can be taught
• Minor: Virtue is knowledge (supported by: virtue directs us to the good; what directs us to the good is knowledge)
• Conclusion: Virtue can be taught

Second side (virtue cannot be taught):

• Observation: If virtue could be taught, there would be teachers of virtue
• Fact: There appear to be no real teachers of virtue (only sophists of questionable credibility)
• Conclusion: Virtue cannot be taught

Berquist notes that the weakness in the first argument is the major premise that only knowledge (not right opinion) directs us to the good. He gives the example of a man at a fork in the road: he can reach Boston either by knowing the right fork leads there or by having the correct opinion about it. Similarly, great leaders of Athens may have succeeded through right opinion rather than knowledge.

Protagoras’ Response #

In the Protagoras, the sophist counters with a dialectical argument about the diffuse teaching of virtue: we don’t point to a single teacher of English or Greek because many people (parents, relatives, society) teach these implicitly. Yet we don’t conclude they’re unteachable. Similarly, virtue may be taught diffusely rather than formally.

Important Definitions #

Demonstration (Latin) / Ἀπόδειξις (Greek): A syllogism whose premises are necessarily true and seen as necessarily true, yielding a necessarily true conclusion.

Dialectical Syllogism: A syllogism whose premises are probable (held by all/most people or by experts in a field) but not necessarily true, capable of yielding only probable conclusions or even contradictory conclusions.

Probable (Ἔνδοξος): An opinion having probability due to the number of people holding it or the eminence of those holding it.

Q.E.D. (Quod erat demonstrandum): “What was to be demonstrated”—the concluding formula in demonstrations (especially in Euclid) indicating that what was proposed to be shown has been shown.

The Matter of the Syllogism: The truth-status and nature of the premises, distinct from the form (the logical structure that ensures necessity of conclusion).

Examples & Illustrations #

Aristotle’s Critique of Early Greek Physics #

Aristotle examines the early Greek natural philosophers (melesis) and finds their conclusions don’t follow and their premises are neither true nor probable. Berquist notes Aristotle kindly attributes this to their lack of logic as a tool.

Einstein on Euclid #

Berquist quotes Einstein’s biographical sketch: “If Euclid did not arouse your youthful enthusiasm, you were not born to be a scientist.” Einstein speaks of Euclid’s effect upon him as a child and emphasizes the rigor of reasoning where one thing is shown from another thing.

Probable Authorities #

• Shakespeare: “To hold the mirror up to nature” (on the purpose of drama)—probable because Shakespeare is an illustrious grammist
• Mozart: In his letters on representing anger in music, stating music must be pleasing to the ear or it ceases to be music—probable because Mozart is the most illustrious composer
• Einstein: Scientific hypotheses are freely imagined, not reasoned out—probable because Einstein is such a famous scientist

MacArthur and the Inchon Landing #

Berquist uses MacArthur’s decision to execute the Inchon landing as an example of probable reasoning with risk. MacArthur was convinced of success but still had fear that it could fail. He reasoned through the entire plan aloud, then read his Bible and waited—illustrating how even strong probable reasoning maintains some uncertainty about its contradictory.

Questions Addressed #

Why does the rigor of logical form not guarantee correct conclusions? Because the rigor of the syllogistic form only ensures that the conclusion follows from the premises; it says nothing about whether the premises themselves are true. One can correctly add two incorrect numbers and still reach an incorrect sum.

Can we legitimately reason to contradictory conclusions? Yes, in dialectic we can reason to contradictory conclusions because both sides of a contradiction may have some probability. However, in demonstration (reasoning from necessarily true premises), we never reason to contradictory conclusions—one side must be necessarily true, making its opposite necessarily false.

How do we evaluate competing dialectical arguments? We examine which side has greater probability or fewer weaknesses in its probable grounds. Often, further analysis reveals which side has stronger support from authorities or from the opinions of all/most people in a relevant field.

What is the difference between demonstration and adding numbers? Both have rigorous form: mathematical operations are rigorous, and syllogistic form is rigorous. But just as correct addition of incorrect numbers yields an incorrect result, correct syllogistic form applied to false or merely probable premises yields conclusions that are not necessarily true.

Referenced Material #

Note: The lecture is quite fragmented with background discussion, student questions, and administrative matters interspersed. The primary philosophical content references are noted below.