Assessing Argument Validity with Aristotelian Syllogisms

Introduction

Hello! In our last lesson, we took apart the Aristotelian categorical syllogism to understand its basic structure. We learned how every syllogism is built from three categorical propositions (A, E, I, O), is defined by three terms (Major, Minor, Middle), and can be fully classified by its mood and figure. This allowed us to describe the logical form of any such argument, like AAA-1 or EAE-1.

Today, we move from description to evaluation. Our goal is to apply Aristotelian syllogistic rules to determine the validity of simple arguments. An argument is valid if its conclusion necessarily follows from its premises. For syllogisms, validity depends entirely on the logical form we just learned to identify. We will learn a set of rules that can mechanically test any syllogistic form for validity, a process that should appeal to your mathematical background in its rigor and formality.

1. The Rules for Validity

Logicians have developed several equivalent sets of rules to test for syllogistic validity. We will focus on a common set of six rules. If a syllogism in standard form violates even one of these rules, it is invalid. If it passes all six, it is valid.

Before we list the rules, we need one crucial concept: distribution. A term is 'distributed' if the proposition says something about every member of the class designated by that term. Here's a quick review:

• A propositions (All S are P): S is distributed.
• E propositions (No S are P): Both S and P are distributed.
• I propositions (Some S are P): Neither term is distributed.
• O propositions (Some S are not P): P is distributed.

Now, let's dive into the rules themselves.

Here is a summary of the six rules for easy reference:

1. The middle term must be distributed in at least one premise.
   • Fallacy: Undistributed Middle.
2. If a term is distributed in the conclusion, it must also be distributed in the premise where it appears.
   • Fallacy: Illicit Major (if the major term fails) or Illicit Minor (if the minor term fails).
3. A syllogism cannot have two negative premises.
   • Fallacy: Exclusive Premises.
4. A negative premise must have a negative conclusion.
5. A negative conclusion must have a negative premise.
   (Rules 4 and 5 mean a syllogism has a negative conclusion if and only if it has exactly one negative premise).
6. Two universal premises cannot have a particular conclusion.
   (This is the modern, or 'Boolean', interpretation. Aristotle's original system, which assumed terms refer to existing things, allowed some exceptions. We will use the modern interpretation, as it is standard in formal logic.)

2. Applying the Rules: A Walkthrough

Let's test the validity of a syllogism we analyzed in the previous lesson: EAE-1.

• Mood/Figure: EAE-1
• Standard Form:
  • Major Premise: No M are P (E)
  • Minor Premise: All S are M (A)
  • Conclusion: No S are P (E)

Let's check the rules:

1. Undistributed Middle? The middle term is M. In the major premise (E-form), 'M' is the subject and is distributed. In the minor premise (A-form), 'M' is the predicate and is undistributed. Since M is distributed in the major premise, the rule is passed.
2. Illicit Process? The conclusion is "No S are P". Since it's an E proposition, both 'S' (minor term) and 'P' (major term) are distributed.
   • Is 'S' distributed in the minor premise ("All S are M")? Yes, as the subject of an A proposition, 'S' is distributed.
   • Is 'P' distributed in the major premise ("No M are P")? Yes, as the predicate of an E proposition, 'P' is distributed.
   • Both terms are fine. The rule is passed.
3. Two Negative Premises? We have one negative (E) and one affirmative (A) premise. The rule is passed.
4. Negative Premise, Affirmative Conclusion? We have a negative premise and a negative conclusion. The rule is passed.
5. Negative Conclusion, Affirmative Premises? We have a negative conclusion and one negative premise. The rule is passed.
6. Universal Premises, Particular Conclusion? The premises are universal (E, A) and the conclusion is universal (E). The rule is passed.

Since the form EAE-1 passes all six rules, any argument with this form is valid.

3. A Mathematical Perspective on Syllogisms

Your background in mathematics gives you access to a powerful way of thinking about syllogisms: set theory. We can translate the four categorical propositions into statements about sets, which can make the underlying logic very clear.

• All S are P translates to $S\subseteq P$ (The set S is a subset of the set P).
• No S are P translates to $S\cap P=\varnothing$ (The intersection of S and P is the empty set).
• Some S are P translates to $S\cap P\ne \varnothing$ (The intersection of S and P is not empty).
• Some S are not P translates to $S\setminus P\ne \varnothing$ (The set difference S minus P is not empty).

This perspective was explored by the mathematician and author Lewis Carroll (Charles Dodgson), who created many playful logic puzzles based on syllogistic reasoning. The following reading explains this connection and provides some of his puzzles as exercises.

Solving these puzzles requires you to do exactly what formal logic aims to do: abstract the logical structure from the distracting content of natural language, perform a valid inference, and then reinterpret the conclusion. This kind of active problem-solving is an excellent way to solidify your understanding of deductive validity.

Conclusion

In this lesson, we transitioned from describing the structure of syllogisms to evaluating their logical strength. We accomplished this by learning and applying a formal, rule-based method for determining validity.

Key Takeaways:

• The validity of a categorical syllogism depends solely on its logical form (mood and figure).
• A syllogism's validity can be tested by applying a set of six formal rules. Violating even one rule renders the argument invalid.
• The key rules involve the distribution of terms, the number and placement of negative propositions, and the relationship between universal premises and particular conclusions.
• Syllogistic logic can be elegantly represented using the language of set theory, providing a rigorous way to check inferences that aligns with a mathematical worldview.

Preview of the next lesson:
We have now explored Plato's Theory of Forms and Aristotle's empirical alternative, as well as the formal logic Aristotle invented to analyze reasoning. In the next lesson, we will synthesize these ideas to distinguish between Plato's rationalist and Aristotle's empiricist approaches to knowledge, concluding our first module on these two foundational figures of Western philosophy.