9.3 Sorites - My Notes on "The Reasonable Person" by Mark Grannis

Sorites
"a syllogism that links a chain of propositions together as premises without any intermediate conclusions." 

It comes from the Greek word for "heap". The word is used for both singular and plural. 

Aristotelian Sorites 

"a categorical sorites that ends in a conclusion that relates the subject of the first premise with the predicate of the last premise." 

This is a working "inside out," from the more specific into larger categories. "Aristotelian sorites starts with A and moves alphabetically." 

Example: 

All kittens are cats;                                            All A is B;

All cats are mammals;                                        All B is C;

All mammals are vertebrates;                             All C is D;

All vertebrates are animals;                               All D is E; 

Therefore All kittens are animals.              Therefore All A is E. 

"Each new term has greater extension than the term that precedes it." 

To assess this formally we need to interpolate the missing premises and conclusions. 

Interpolating:

All B is C;

All A is B;

[Therefore All A is C]

All C is D;

[All A is C;]

[Therefore All A is D.]

All D is E;

[All A is D;]

Therefore All A is E.

All of these were universal and affirmative propositions. Could there be particular or negative propositions? Only in certain cases. 

For Particular Propositions - "the rule is that only the premise that contains the minor term may be particular." (minor term being the subject of the conclusion)

For Negative Propositions - "Only the premise that contains the major term may be negative -- and in an Aristotelian sorites, only the premise just before the conclusion contains the major premise." (The major term being the predicate of the conclusion)

Goclenian Sorites

"a categorical sorites that ends in a conclusion that relates the predicate of the first premise with the subject of the last premise." 

This is a working "outside-in" from larger categories into smaller ones. "Golcenian sorites starts with the most general terms and goes clean backwards." 

Example: 

All D is E; 

All C is D;

All B is C;

All A is B; 

Therefore All A is E. 

Interpolating:

All D is E;

All C is D;

[Therefore All C is E.]

All B is C;

[All C is E;] 

[Therefore All B is E.]

[All B is E;]

All A is B;

Therefore All A is E;

Particular Premises - "only a premise with the minor term may be particular..."

Negative Premise - "and only a premise with a major term may be negative."

Pure Conditional Sorites

"a sorites that consists entirely of conditional propositions, linked together without intermediate conclusions." 

If A, then B;

If B, then C;

If C, then D;

If D, then E; 

Therefore If A, then E.

Interpolating:

If B, then C;

If A, then B; 

[Therefore If A, then C.]

If C, then D; 

[If A, then C;]

[Therefore If A, then D.]

If D, then E; 

[If A, then D;]

Therefore If A , then E.

"Once we have interpolated, the rules of validity for the pure conditional sorites are exactly the same as for pure conditional syllogisms..."

Mixed Conditional Sorites

"is a sorites that consists of two or more conditional propositions, linked together without intermediate conclusions, followed by a categorical proposition affirming or denying one element of the conditionals and a categorical conclusion." 

Examples:

If A, then B;                        If A, then B; 

If B, then C;                        If B, then C;

If C, then D;                        If C, then D;

A;                                         Not D;

Therefore D.                        Therefore Not A

Interpolating:

If A, then B;                        If A, then B; 

If B, then C;                        If B, then C;

If C, then D;                        If C, then D;

[If A, then D;]                      [If A, then D;]

A;                                         Not D;

Therefore D.                        Therefore Not A

modus ponens                        modus tollens