Philosophy (Logic) - 9/4/25

Distribution of Terms in AEIO Propositions

• Four standard forms (propositions) in categorical logic:

  • $A:\ \text{All } S \text{ are } P$

  • $E:\ \text{No } S \text{ are } P$

  • $I:\ \text{Some } S \text{ are } P$

  • $O:\ \text{Some } S \text{ are not } P$

• Distribution values (how much a term is distributed within a proposition):

  • Each term has its own distribution value; they do not affect each other.

  • Subject distribution depends on the form being universal or particular:

  • If the proposition is universal (A or E), the subject term S is distributed.

  • If the proposition is particular (I or O), the subject term S is undistributed.

  • Predicate distribution depends on negation (whether the proposition is negative):

  • If the proposition is negative (E or O), the predicate term P is distributed.

  • If the proposition is affirmative (A or I), the predicate term P is undistributed.

• Quick summary of distribution for each form:

  • $A: \ S \text{ distributed; } P \text{ not distributed}$

  • $E: \ S \text{ distributed; } P \text{ distributed}$

  • $I: \ S \text{ not distributed; } P \text{ not distributed}$

  • $O: \ S \text{ not distributed; } P \text{ distributed}$

• How to determine distribution values (rule of thumb):

  • Look at the subject term: universal -> distributed, particular -> undistributed.

  • Look at the predicate term: negative proposition -> predicate distributed; affirmative proposition -> predicate undistributed.

  • Treat the two terms independently (their distribution values do not influence each other).

• Square of opposition and kinds of operations:

  • Contras (converse) switches the subject and predicate only; does not move on the square.

  • Obverse moves across the square (crossing) and negates the predicate term.

  • Contraposition (contrapositive) involves switching the terms and negating both terms (putting a "non-" in front of both subjects and predicates).

• Important note about equivalence and truth-values:

  • When a transformation is valid, the premise and conclusion are equivalent (mean the same thing, just different wording).

  • If one is true, the other is true; if one is false, the other is false, for valid transformations.

  • In the exam, you may be tempted to think obverse always works in a way that guarantees both true or both false; tests can try to trick you by giving an obverse that is true while the conclusion is false (i.e., not equivalent in that instance).

Converse

• Definition: interchange the subject and predicate terms.

• When is a converse valid?

  • Valid if the two terms have the same distribution value in the premise (i.e., the premise form is either E or I).

  • In practice:

  • Converse is valid for E (No S are P) and I (Some S are P).

  • Converse is not guaranteed valid for A or O (All S are P; Some S are not P).

• How to form the converse:

  • From $E: \text{No } S \text{ are } P$, the converse is $\text{No } P \text{ are } S.$ (same distribution on both terms)

  • From $I: \text{Some } S \text{ are } P$, the converse is $\text{Some } P \text{ are } S.$ (same distribution)

• Relationships to equivalence:

  • When a converse is valid, the premise and conclusion are equivalent statements (in form, they express the same content in different words).

  • If one is true, the other must be true; if one is false, the other must be false (within the valid cases).

• Common pitfall:

  • Do not assume converse always preserves truth-value for all forms; it only does so for E and I in standard practice.

Obverse

• Definition: two changes in one operation:
  1) Cross the square of opposition (move to the corresponding opposite form on the square).
  2) Place a negation in front of the predicate term (i.e., negate the predicate term).

• How to obtain an obverse:

  • From $A: \text{All } S \text{ are } P$, obverse gives $\text{No } S \text{ are not } P$ (i.e., No S are non-P).

  • From $E: \text{No } S \text{ are } P$, obverse gives $\text{All } S \text{ are not } P$.

  • From $I: \text{Some } S \text{ are } P$, obverse gives $\text{Some } S \text{ are not } P$.

  • From $O: \text{Some } S \text{ are not } P$, obverse gives \text{Some } S \ are } P.

• Obverse validity:

  • Obverse always works (is a valid immediate inference).

• Relationship to the square:

  • Obverse changes the form by moving across the square and negating the predicate; it does not swap the subject and predicate.

• Examples:

  • All dogs are mammals (A) ⇒ No dogs are non-mammals (E form obverse).

  • No dogs are mammals (E) ⇒ All dogs are not mammals (A form obverse).

  • Some dogs are mammals (I) ⇒ Some dogs are not non-mammals (O form obverse).

  • Some dogs are not mammals (O) ⇒ Some dogs are mammals (I form obverse).

• Note on distribution and obverse:

  • The obverse does not require changing the distribution values of the individual terms; it simply applies a global negation to the predicate and crosses the square.

Contraposition

• Definition: contrapositive involves switching the terms and negating both terms (i.e., applying non to both subject and predicate).

• How to form a contrapositive:

  • Start with $A: \text{All } S \text{ are } P$ → contrapositive: $\text{All non-}P \text{ are non-}S.$ (switch S and P; apply non to both)

  • Start with $E: \text{No } S \text{ are } P$ → contrapositive: $\text{No non-}P \text{ are non-}S.$

  • Start with $I: \text{Some } S \text{ are } P$ → contrapositive: $\text{Some non-}P \text{ are not } S.$ (Some non-P are not S)

  • Start with $O: \text{Some } S \text{ are not } P$ → contrapositive: $\text{Some non-}P \text{ are not } S.$ (Some non-P are not S)

• Validity of contrapositions (as discussed in the lecture):

  • Contraposition is not universally valid for all forms; its validity depends on the form and the steps used.

  • In the lecture examples, the following occurred:

  • The contrapositive of A (All S are P) was shown to be valid: All non-P are non-S.

  • The contrapositive of E (No S are P) was shown to be invalid in the given practice/example sequence.

  • The contrapositive of I (Some S are P) was explored and shown to be invalid in the given sequence.

  • The contrapositive of O (Some S are not P) was explored and shown to be invalid in the given sequence.

  • How they tested validity:

  • One method described: obtain contrapositive via obverse → convert (swap terms) → obverse, and check each step for validity; if any step is invalid, the overall contrapositive is invalid.

  • Alternate perspective: a contrapositive can be reached by the chain obverse → convert → obverse, and if each step is valid, then the contrapositive is valid.

• Practical takeaway from the lecture (based on the examples):

  • In this course's treatment, A forms tend to support contrapositive validity, while E, I, and O forms do not reliably yield valid contrapositives under the same rules used for converses/obverses.

  • Always verify contrapositive validity by following the stepwise method and checking distribution conditions; do not assume contrapositive is always valid.

• Examples from the lecture:

  • All dogs are mammals → contrapositive: All non-mammals are non-dogs (valid per the A contrapositive rule).

  • No dogs are mammals → contrapositive attempted as a sequence showed invalid steps in the given framework.

  • Some S are not P → contrapositive: Some non-P are not S (shown as invalid in the lecture’s worked example).

• Practical caution:

  • Although obverses are always valid, contrapositions can be tricky and are not universally valid for all forms in this framework; you should rely on the explicit validity tests shown in lectures and on practice sheets.

Quick identification tips for AEIO forms (memory aids)

• First word determines the form:

  • If the first word is All, it is form A.

  • If the first word is No, it is form E.

  • If the first word is Some, it is either I or O; determine which by looking at the predicate:

  • If the form says some S is P (no negation on P) → I form.

  • If the form says some S is not P → O form.

• A handy summary of how to read the forms on a test:

  • A: All S are P

  • E: No S are P

  • I: Some S are P

  • O: Some S are not P

• Common practical tips from the lecturer:

  • In practice, most problems on tests involve simple standard forms (All S are P, No S are P, Some S are P, Some S are not P) with straightforward subject and predicate terms.

  • If you get a clause with multiple negations (e.g., non-non S), you can reduce double negatives inside a term; do not apply negation across terms or symbols in a way that changes distribution types.

  • The difference between the word "non" (a built-in negation in the term) and the word "not" (explicit negation in predicate position) is important; non-non can cancel if within the same term, but you must respect the form and the distribution rules when manipulating terms.

• Practical study tip suggested in the lecture:

  • Draw the square of opposition and annotate each corner with its form (A, E, I, O) to develop a muscle memory for the relationships and the effects of obverse, converse, and contrapositive operations.

Worked examples (quick references)

• Example 1: Obverse of A

  • Start: $A: \text{All } S \text{ are } P$

  • Obverse gives: $\text{No } S \text{ are not } P$

• Example 2: Converse of E

  • Start: $E: \text{No } S \text{ are } P$

  • Converse: \text{No } P \ are } S (No P are S)

• Example 3: Contraposition of A (valid in this lecture)

  • Start: $A: \text{All } S \text{ are } P$

  • Contrapositive: $\text{All non-}P \text{ are non-}S$

• Example 4: Contraposition of I (invalid in this lecture’s sequence)

  • Start: $I: \text{Some } S \text{ are } P$

  • Contrapositive (per rule): $\text{Some non-}P \text{ are not } S$

  • The lecture shows a step where an intermediate transformation (e.g., a converse step) is invalid, thus the contrapositive is invalid in this case.

Quick reference formulas (LaTeX)

• Forms:

  • $A: \text{All } S \text{ are } P$

  • $E: \text{No } S \text{ are } P$

  • $I: \text{Some } S \text{ are } P$

  • $O: \text{Some } S \text{ are not } P$

• Distribution rules (summary):

  • Subject distributed if form is universal: $A, E \rightarrow D(S)=1; I, O \rightarrow D(S)=0$

  • Predicate distributed if form is negative: $E, O \rightarrow D(P)=1; A, I \rightarrow D(P)=0$

• Contraposition rule (formal):

  • From $A: \ All S \ are P$, contrapositive: $\text{All non-}P \text{ are non-}S$

  • From $E: \ No S \ are P$, contrapositive: \No\nonumber\text{non-}P \text{ are non-}S (as per the lecture’s notation)

  • From $I: \ Some S \ are P$, contrapositive: $\text{Some non-}P \text{ are not } S$

  • From $O: \ Some S \ are not P$, contrapositive: $\text{Some non-}P \text{ are not } S$

• Obverse rule (general): cross the square and negate the predicate:

  • $A \rightarrow E: \text{All } S \text{ are } P \;\Rightarrow\; \text{No } S \text{ are not } P$

  • $E \rightarrow A: \text{No } S \text{ are } P \;\Rightarrow\; \text{All } S \text{ are not } P$

  • $I \rightarrow O: \text{Some } S \text{ are } P \;\Rightarrow\; \text{Some } S \text{ are not } P$

  • O \rightarrow I: \text{Some } S \text{ are not } P \;\Rightarrow\; \text{Some } S \ are } P

Note: The above notes reflect the material and examples discussed in the transcript. When preparing for an exam, rely on the explicit validity tests and the step-by-step procedures demonstrated (converse, obverse, contrapositive) and practice with the given practice sheets to become fluent with these rules.