39. Validity and Invalidity in First Figure Syllogisms

Subscribe in Podcast App | Download Transcript

Lecture Notes

Main Topics #

• The Two Fundamental Principles Applied to First Figure: The dictum de omni (set of all) and dictum de nullo (set of none) form the basis for evaluating all syllogistic validity. In the first figure, when these principles apply to the premises as they stand, the conclusion follows necessarily and obviously.
• The Sixteen Cases in First Figure: Berquist systematically examines all sixteen possible combinations of statement types (universal affirmative, universal negative, particular affirmative, particular negative) for the two premises.
• Valid versus Invalid Forms: Four forms with two universal statements are examined in detail; two are valid (following from set of all or set of none), two are invalid despite appearing deceptively convincing with true material content.
• Testing Invalidity Through Counterexample: Invalid forms are proven invalid not by finding single false instances, but by constructing paired examples showing that sometimes one conclusion holds true and sometimes its contradictory holds true, proving nothing is necessarily concluded.
• The Role of Accidents in Constructing Examples: Uses Porphyry’s doctrine of accidents (properties that can be present or absent) to efficiently construct sets of terms satisfying multiple conditions simultaneously.

Key Arguments #

Conditions for Immediate Validity in First Figure #

• The set of all or set of none must apply to the premises as they stand
• Set of all requires: (1) a universal statement and (2) an affirmative statement placing something under the subject of that universal
• Set of none requires: (1) a universal negative statement and (2) something coming under the subject of that universal negative
• When either applies, the conclusion is immediately evident

The Two-Condition Test for Invalidity #

To prove an invalid form is not a syllogism, one must find examples for the three terms A, B, and C such that:

1. When substituted into the premises, both premises are true
2. One example makes “every C is A” true (excluding both negatives from always being true)
3. Another example makes “no C is A” true (excluding both affirmatives from always being true)

If nothing is always true when the premises are true, nothing is necessarily true, and therefore it is not a syllogism.

Why Certain Forms Cannot Be Valid #

• Two particular premises: No universal statement exists, so neither set of all nor set of none can apply
• Two negative premises: Even the set of none requires an affirmative statement placing something under the universal negative; without an affirmative, neither principle applies

Reasoning from Necessity to Universality #

• If something is necessarily so, it must always be so
• If it is not always so (demonstrated by counterexamples), it is not necessarily so
• If it is not necessarily so, there is no syllogism

Important Definitions #

• Dictum de Omni (Set of All): If A is said of all B, then A is said of whatever B is said of. If every B is an A, then whatever is a B must also be an A.
• Dictum de Nullo (Set of None): If A is said of none of B, then A is denied of whatever B is said of. If no B is an A, then whatever is a B is not an A.
• Middle Term: The term appearing in both premises but not in the conclusion; its arrangement determines the figure of the syllogism. In the first figure, it is the subject of the major premise and predicate of the minor premise.
• Major Term: The predicate of the conclusion, appearing in the major premise
• Minor Term: The subject of the conclusion, appearing in the minor premise
• Universal Affirmative: A statement of the form “Every B is A”
• Universal Negative: A statement of the form “No B is A”
• Particular Affirmative: A statement of the form “Some B is A”
• Particular Negative: A statement of the form “Some B is not A”

Examples & Illustrations #

Valid First Figure Forms #

1. Every B is A; Every C is B; Therefore, Every C is A

   • Example: Every dog is an animal; Every cat is a dog; Therefore, every cat is an animal
   • Set of all applies: every dog is animal, and every cat is dog, so every cat must be animal

2. No B is A; Every C is B; Therefore, No C is A

   • Example: No plant is an animal; Every dog is a plant; Therefore, no dog is an animal
   • Set of none applies: no plant is animal, and every dog is plant, so no dog is animal

Invalid Forms Deceptively Convincing #

1. Every B is A; No C is B; Therefore [no valid conclusion]

   • Example: Every dog is an animal; No cat is a dog
   • Misleading conclusion “No cat is an animal” seems true (cats are indeed animals), but does not follow necessarily
   • Counterexamples:
     • Bush is not a dog but is a plant (if dog=animal, plant=A, dog=B, bush=C): Some C is A
     • Stone is not a dog but is not a plant: No C is A
   • Since both conclusion and its contradictory can be true when premises are true, nothing is necessarily concluded

2. Two Particular Premises

   • Using A=animal, B=white (an accident), C=dog/stone
   • Some white is animal; Some dog is white (premises true)
   • Yet sometimes “every dog is animal” (true) and sometimes “no stone is animal” (true)
   • All four possible conclusions sometimes hold; nothing necessarily follows

3. Two Negative Premises

   • Example: No plant is animal; Some cat is not plant
   • Counterexamples using A=animal, B=plant, C=cat/stone
   • Cat is not plant but is animal: Some C is A
   • Stone is not plant and not animal: No C is A
   • Neither principle applies; nothing necessarily follows

The Mother-Man Example #

• Premises: “Every mother is a woman; No man is a mother”
• Invalid form: Every B is A; No C is B
• Students tempted to conclude: “No man is a woman” (which happens to be true)
• This deceives students because all three statements are materially true, creating illusion of valid reasoning
• Yet this form is not a syllogism because the arrangement of terms does not guarantee the conclusion
• Contrast: “Every dog is an animal; No cat is a dog” does not tempt students to conclude “No cat is an animal” because material content is not misleading

Notable Quotes #

“If something is a B, and it’s not an A, then not every B is an A, right? Okay? So, this is one of the two beginnings, principles upon which the syllogism is based, right?”

“In the first figure, if it’s valid, we see the set of all, or the twin principle, the set of none in it, as it stands. And if either set of all or set of none applies to what it stands, you can guess it’s not going to be a syllogism.”

“You have to find examples for A, B, and C such that one, when I place them in this arrangement, the statements are true, right? And one of the examples where every C is A, which makes both negatives false ones, right? And one where no C is A is true, which makes both what? Affirmatives false ones.”

“If nothing is so always, then nothing is so what? Necessarily. And if nothing is so necessarily, you don’t have any syllogism, right?”

“The matter seems perfectly good, but it isn’t so. I know as a crafty magician, right? If I want the students to answer incorrectly, or show them that they don’t really, you know, more logic, I’ll give them this form of statements where they’re all true, right?”

Questions Addressed #

• How do we immediately recognize validity in the first figure? By checking whether the set of all or the set of none applies to the premises as they stand. If either principle applies, the conclusion is immediately evident.
• Why are two particular premises never valid? Because particular statements contain no universality, neither the set of all nor the set of none can apply.
• Why are two negative premises never valid? Because the set of all requires an affirmative statement, and even the set of none requires an affirmative statement placing something under the subject of the universal negative. Without affirmative premises, neither principle can apply.
• How do we prove an invalid form is truly invalid? By constructing concrete examples showing that when the premises are true, sometimes “every C is A” is true and sometimes “no C is A” is true, demonstrating that nothing is necessarily concluded.
• Why does “Every mother is a woman; No man is a mother” deceive students? Because all three statements happen to be materially true, creating an illusion that the invalid form produces a true conclusion. The form itself does not guarantee the conclusion despite the material truth of the statements involved.