Logic: Categorical Propositions & Syllogisms Explained
Categorical Propositions and Syllogisms
What is a Categorical Proposition?
A Categorical Proposition contains a subject and a predicate, asserting that some or all of the members of the subject class are included in or excluded from the predicate class.
Example Premise Set:
- Whales are mammals.
- All mammals breathe by means of lungs.
- Therefore, whales breathe by means of lungs.
Categorical Syllogism
A Categorical Syllogism is a deductive argument consisting of three categorical propositions: two premises and one conclusion.
Taxonomic Example:
- Whales are a species of animal.
- Mammals are the genus to which that species belongs.
- Hierarchy: Animals (Genus) → Mammals (Genus) → Whales (Species)
Elements of a Categorical Proposition
- Subject (S): The term about which something is asserted (e.g., Whales).
- Predicate (P): The term that is asserted about the subject (e.g., Mammals).
- Quality (Copula): The verb ‘to be’ (is/are or is not/are not), indicating affirmation or negation.
- Quantity: Specifies how much of the subject class is included or excluded (Particular: some; Universal: all).
The Four Forms (A, E, I, O)
Categorical propositions have four standard forms based on quality (Affirmative/Negative) and quantity (Universal/Particular):
- A (Universal Affirmative): All S are P. (Affirmo)
- E (Universal Negative): No S is P. (Nego)
- I (Particular Affirmative): Some S are P. (Affirmo)
- O (Particular Negative): Some S are not P. (Nego)
Examples:
- (I) Some [movie stars] are [good actors]. (Quantity: Some, Subject: movie stars, Quality: are, Predicate: good actors)
- (A) All [phones] are [communication devices].
- (I) Some [phones] are [cordless devices].
- (A) All bread is nutritious.
- (E) No bread is nutritious.
- (I) Some bread is nutritious.
Note: The relationships between these forms can be visualized using the traditional Square of Opposition.
Relationships Between Propositions
Subcontraries (I and O): Both can be true, but both cannot be false.
Immediate Inferences
Immediate inferences are conclusions drawn from a single premise. Key types include:
Conversion
Conversion involves interchanging the subject and predicate terms.
- Valid for E and I propositions:
- E: No women have been US presidents. → No US presidents have been women.
- I: Some Englishmen are Scotch drinkers. → Some Scotch drinkers are Englishmen.
- Invalid for A and O propositions (generally):
- A: All pickpockets are criminals. → All criminals are pickpockets. (Invalid – Fallacy of Illicit Conversion)
- O: Some human beings are not Americans. → Some Americans are not human beings. (Invalid – Fallacy of Illicit Conversion)
Obversion
Obversion involves changing the quality of the proposition (affirmative to negative or vice versa) and replacing the predicate term (P) with its complement (non-P). The subject and quantity remain unchanged. Obversion is valid for all four forms (A, E, I, O).
Examples:
- A: All dogs are mammals. → E: No dogs are non-mammals. (Valid)
- A: All citizens are voters. → E: No citizens are non-voters.
- I: Some metals are conductors. → O: Some metals are not non-conductors.
Contraposition
Contraposition involves replacing the subject term with the complement of the predicate term, and replacing the predicate term with the complement of the subject term. Valid for A and O propositions.
Examples:
- A: All Ismailis are Muslims. → A: All non-Muslims are non-Ismailis. (Valid)
- A: All Texans are Americans. → A: All non-Americans are non-Texans. (Valid)
Distribution of Terms
A term is distributed if the proposition makes an assertion about every member of the class designated by that term. A term is undistributed if it does not.
- A (All S are P): S is distributed, P is undistributed.
- E (No S is P): Both S and P are distributed.
- I (Some S are P): Neither S nor P is distributed.
- O (Some S are not P): S is undistributed, P is distributed.
General Rules:
- The term following “All” (Subject of A) is distributed.
- Both terms following “No” (Subject and Predicate of E) are distributed.
- The term following “not” (Predicate of O) is distributed.
Rules for Valid Syllogisms
Rule 1: Middle Term Distribution
The middle term (the term appearing in both premises but not the conclusion) must be distributed in at least one premise.
Example of Fallacy (Undistributed Middle):
- Some terrorists support an independent Palestinian state. (Middle term ‘supporters of an independent Palestinian state’ is undistributed)
- Tom supports an independent Palestinian state. (Middle term is undistributed)
- Therefore, Tom is a terrorist. (Invalid Conclusion)
Rule 2: Term Distribution in Conclusion
If a term (Subject or Predicate) is distributed in the conclusion, it must also be distributed in the premise where it occurs.
Fallacy of Illicit Major/Minor: Occurs when the major term (predicate of conclusion) or minor term (subject of conclusion) is distributed in the conclusion but not in its respective premise.
Example Syllogism (Valid – AAA-1):
- Major Premise: All M are P (All mammals are animals that breathe by means of lungs).
- Minor Premise: All S are M (All whales are mammals). (Middle term ‘mammals’ is distributed in the major premise).
- Conclusion: All S are P (All whales are animals that breathe by means of lungs).
Example of Fallacy (Illicit Major):
- Major Premise: All dogs are mammals. (P: mammals – undistributed)
- Minor Premise: No cats are dogs.
- Conclusion: No cats are mammals. (P: mammals – distributed. Invalid because ‘mammals’ was not distributed in the major premise).
Example of Fallacy (Illicit Minor):
- Major Premise: All vertebrates reproduce sexually.
- Minor Premise: All vertebrates are animals. (S: animals – undistributed)
- Conclusion: All animals reproduce sexually. (S: animals – distributed. Invalid because ‘animals’ was not distributed in the minor premise).
Disjunctive and Hypothetical Propositions (Ch. 8)
Types of Propositions
- Disjunctive Propositions: Use the connective “or”. Example: Whales are mammals or they are very large fish.
- Hypothetical Propositions: Use the connective “if…then”. Example: If whales are mammals, then they cannot breathe underwater.
Disjunctive Syllogisms
A Disjunctive Syllogism is a deductive argument with a disjunctive premise, a second premise negating one of the disjuncts, and a conclusion affirming the remaining disjunct.
Example:
- The class is either in room 305 or room 307.
- The class is not in room 307.
- Therefore, the class is in room 305.
Hypothetical Syllogisms
A Hypothetical Syllogism (or Conditional Syllogism) involves compound propositions using the “if…then” connective.
- Structure: If Antecedent, then Consequent.
- Example: If it rains (antecedent), then the graduation will be held in the gym (consequent).
Pure Hypothetical Syllogism
A syllogism in which both premises and the conclusion are hypothetical propositions.
Example:
- If I do not wake up, then I cannot go to work.
- If I cannot go to work, then I will not get paid.
- Therefore, if I do not wake up, I will not get paid.
Another Example:
- If my wallet is not in my apartment, then I lost it.
- If I lost my wallet, then I will have to cancel my credit card.
- Therefore, if my wallet is not in my apartment, then I will have to cancel my credit card.
Mixed Hypothetical Syllogism
A syllogism with one hypothetical premise, one categorical premise, and a categorical conclusion. Two common valid forms are:
Modus Ponens (The Way that Affirms by Affirming)
Affirms the antecedent in the categorical premise.
Example:
- If you play with fire, then you will get burned. (Hypothetical Premise)
- You played with fire. (Categorical Premise – Affirms Antecedent)
- Therefore, you got burned. (Categorical Conclusion)
Modus Tollens (The Way that Denies by Denying)
Denies the consequent in the categorical premise.
Example:
- If God had wanted us to fly, He would have given us wings. (Hypothetical Premise)
- He has not given us wings. (Categorical Premise – Denies Consequent)
- Therefore, He did not want us to fly. (Categorical Conclusion)