A diagrammatic calculus of syllogisms
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1 A diagrammatic calculus of syllogisms Ruggero Pagnan DII, University of Genova
2 Categorical Propositions A P : All is P (universal affirmative) E P : No is P (universal negative) I P : ome is P (particular affirmative) O P : ome is not P (particular negative)
3 yllogisms A syllogism is a rule of inference P 1,P 2 C where P 1, P 2 and C are categorical propositions.
4 yllogisms A syllogism is a rule of inference P 1,P 2 C where P 1, P 2 and C are categorical propositions. More precisely, in a syllogism appear exactly three terms, P and M as follows: - M must appear in both P 1 and P 2 but is not allowed to appear in C. - must appear in both P 2 and C, as the subject of the latter. - P must appear in both P 1 and C, as the predicate of the latter.
5 yllogisms A syllogism is a rule of inference P 1,P 2 C where P 1, P 2 and C are categorical propositions. More precisely, in a syllogism appear exactly three terms, P and M as follows: - M must appear in both P 1 and P 2 but is not allowed to appear in C. - must appear in both P 2 and C, as the subject of the latter. - P must appear in both P 1 and C, as the predicate of the latter. Example: O PM,E M I P
6 Valid yllogisms A MP,A M A P E PM,A M E P I MP,A M I P A PM,E M E P E MP,A M E P A PM,E M E P A MP,I M I P I PM,A M I P A MP,I M I P E PM,I M O P O MP,A M O P E PM,I M O P E MP,I M O P A PM,O M O P E MP, I M O P A MP,A M I P A PM,E M O P A MP,A M I P A PM,E M O P E MP,A M O P E PM,A M O P E MP, A M O P E PM,A M O P A PM,A M I P
7 The calculus A P P E P P I P P O P P
8 Example A MP,A M A P
9 Example A MP,A M A P A M M M A MP P
10 Example A MP,A M A P A M M M A MP P A M M A MP P
11 Example A MP,A M A P A M M M A MP P A M M A MP P A P P
12 Example E PM,I M O P
13 Example E PM,I M O P I M M M E PM P
14 Example E PM,I M O P I M M M E PM P I M M E PM P
15 Example E PM,I M O P I M M M E PM P I M M E PM P O P P
16 Example A PM,O M O P
17 Example A PM,O M O P A PM O M M M P
18 Example A PM,O M O P A PM O M M M P O M M A PM P
19 Example A PM,O M O P A PM O M M M P O M M A PM P O P P
20 Example A MP,A M I P
21 Example A MP,A M I P A M M M A MP P
22 Example A MP,A M I P A M M M A MP P A M M I MM M A MP P
23 Example A MP,A M I P A M M M A MP P A M M I MM M A MP P I P P
24 Terminology and notation A syllogistic inference is any instance of the previous computation process.
25 Terminology and notation A syllogistic inference is any instance of the previous computation process. yllogistic inferences will be henceforth represented by planar diagrams like A M M A MP P A P P A M M I MM M A MP P P I P
26 Recovering the valid syllogisms Theorem A syllogism is valid if and only if there is a (necessarily unique) syllogistic inference from its premisses to its conclusion.
27 Recovering the valid syllogisms Theorem A syllogism is valid if and only if there is a (necessarily unique) syllogistic inference from its premisses to its conclusion. ketch of proof: A syllogistic inference yields a diagram between A P, E P, I P, O P in exactly the following cases: M P M P M P M P M P M P M P M P M M P M P P M P M P M M P M P
28 ome features of the calculus - the calculus is algorithmic.
29 ome features of the calculus - the calculus is algorithmic. - the calculus allows the representation of the premisses of a syllogism in any order. Example: E MP,I M O P P M
30 ome features of the calculus - the calculus is algorithmic. - the calculus allows the representation of the premisses of a syllogism in any order. Example: E MP,I M O P P M - the calculus easily extends to n-term syllogisms.
31 Non-valid syllogisms yllogistic inferences do not delete or create the bullet symbols.
32 Non-valid syllogisms yllogistic inferences do not delete or create the bullet symbols. Example: The syllogism O MP,E M E P is not valid.
33 Non-valid syllogisms yllogistic inferences do not delete or create the bullet symbols. Example: The syllogism O MP,E M E P is not valid. E M M P O MP P E P
34 Non-valid syllogisms yllogistic inferences do not delete or create the bullet symbols. Example: The syllogism O MP,E M E P is not valid. E M M P O MP P E P Example: The syllogism E MP,E M A P is not valid.
35 Non-valid syllogisms yllogistic inferences do not delete or create the bullet symbols. Example: The syllogism O MP,E M E P is not valid. E M M P O MP P E P Example: The syllogism E MP,E M A P is not valid. E M M E MP P A P P
36 The rules of syllogism - From two negative premisses nothing can be inferred. - From two particular premisses nothing can be inferred. - From a particular first premise and a negative second premise nothing can be inferred. - If one premise is particular, then the conclusion is such. - The conclusion of a syllogism is negative if and only if so is one of its premisses.
37 Rules of syllogism From two negative premisses nothing can be inferred.
38 Rules of syllogism From two negative premisses nothing can be inferred. E M M M E MP P
39 Rules of syllogism From two negative premisses nothing can be inferred. E M M M E MP P E M M M O MP P
40 Rules of syllogism From two negative premisses nothing can be inferred. E M M M E MP P E M M M O MP P O M M M E MP P
41 Rules of syllogism From two negative premisses nothing can be inferred. E M M M E MP P E M M M O MP P O M M M E MP P O M M M O MP P
42 The quare of Opposition A P contradiction subalternation contrariety E P subalternation I P subcontrariety O P
43 The laws of the quare of Opposition A P,I I P E P,I O P ubalternation A P,O P O E P,I P O Contradiction E PP,A P E P Contrariety E PP,I P O P ubcontrariety
44 The laws of the quare of Opposition Contradiction: O P P ubcontrary: A P O I P P O P E PP P P
45 Extending the calculus: n-term syllogisms An n-term syllogism is a rule of inference P 1,...,P n 1 C in which P 1,...,P n 1,C are categorical propositions any two contiguous of which have exactly one term in common. Notation: the appearing terms will be henceforth denoted by T 1,T 2,...,T n.
46 Recovering the valid n-term syllogisms Theorem For every positive natural number n, an n-term syllogism is valid if and only if there is a (not necessarily unique) pasting of syllogistic inferences from its premisses to its conclusion.
47 Recovering the valid n-term syllogisms Theorem For every positive natural number n, an n-term syllogism is valid if and only if there is a (not necessarily unique) pasting of syllogistic inferences from its premisses to its conclusion. ketch of proof: For every positive natural number n, a syllogistic inference yields a diagram between A P, E P, I P, O P in exactly the following cases: - T 1 T 2 T i T i+1 T n 1 T n. - T 1 T 2 T i T i+1 T n 1 T n, with 1 i n 1. - T 1 T 2 T i T i+1 T n 1 T n, with 1 i n 1. - T 1 T 2 T i T i T n 1 T n, with 1 i n. - T 1 T 2 T i T i+1 T n 1 T n, with 1 i n 1. - T 1 T i T i+1 T j 1 T j T n, with 1 i < j n. - T 1 T i T i T j 1 T j T n, with 1 i < j n.
48 Example A T5 T 4,E T3 T 4,I T3 T 2,A T2 T 1 O T1 T 5 T 1 A T2 T 1 I T3 T 2 E T 2 T3 T 4 A T 3 T5 T 4 T 4 T 5 T 1 T 3 T 4 T 5 I T1 T 3 E T3 T 4 A T5 T 4 T 1 A T5 T 4 T O 4 T 5 T1 T 4 T 1 T O 5 T1 T 5
49 Example A T5 T 4,E T3 T 4,I T3 T 2,A T2 T 1 O T1 T 5 A T2 T 1 I T3 T 2 A T5 T 4 E T 1 T 2 T3 T 4 T 3 T 4 T 5 A T2 T 1 T 1 T 2 A T5 T 4 T O 4 T 5 T2 T 4 T 1 A T2 T 1 T 2 T 5 O T2 T 5 T 1 T O 5 T1 T 5
50 Valid n-term syllogisms The valid n-term syllogisms are 3n 2 n: syllogism quantity A Tn 1 T n,...,a T1 T 2 A T1 T n 1 A TnT n 1,...,E Ti T i+1,...,a T1 T 2 E T1 T n n-1 A TnT n 1,...,E Ti+1 T i,...,a T1 T 2 E T1 T n n-1 A Tn 1 T n,...,i Ti T i+1,...,a T2 T 1 I T1 T n n-1 A Tn 1 T n,...,i Ti+1 T i,...,a T2 T 1 I T1 T n n-1 A Tn 1 T n,...,i Ti T i,...,a T2 T 1 I T1 T n n A TnT n 1,...,O Ti T i+1,...,a T2 T 1 O T1 T n n-1 A TnT n 1,...,E Tj 1 T j,...,i Ti T i+1,...,a T2 T 1 O T1 T n A TnT n 1,...,E Tj 1 T j,...,i Ti+1 T i,...,a T2 T 1 O T1 T n A TnT n 1,...,E Tj T j 1,...,I Ti T i+1,...,a T2 T 1 O T1 T n A TnT n 1,...,E Tj T j 1,...,I Ti+1 T i,...,a T2 T 1 O T1 T n A TnT n 1,...,E Tj 1 T j,...,i Ti T i,...,a T2 T 1 O T1 T n A TnT n 1,...,E Tj T j 1,...,I Ti T i,...,a T2 T 1 O T1 T n (n 1)(n 2) 2 (n 1)(n 2) 2 (n 1)(n 2) 2 (n 1)(n 2) 2 n(n 1) 2 n(n 1) 2
51 Particular cases: n = 1, 2 n = 1: two valid syllogisms, that is the laws of identity A T1 T 1 A T1 T 1 I T1 T 1 I T1 T 1 n = 2: ten valid syllogisms, - A T1 T 2 A T1 T 2, E T1 T 2 E T1 T 2, I T1 T 2 I T1 T 2, O T1 T 2 O T1 T 2, plus the laws of subalternation A T1 T 2,I T1 T 1 I T1 T 2, E T1 T 2,I T1 T 1 O T1 T 2. - E T2 T 1 E T1 T 2, I T2 T 1 I T1 T 2 which are the laws of simple conversion, and I T2 T 2,A T2 T 1 I T1 T 2, E T2 T 1,I T1 T 1, O T1 T 2 which are the laws of conversion per accidens.
52 n-graphs An n-graph is a diagram of sets and functions s 0 G 0 G 1 t 0 s 1 t 1 G 2 G n 1 s n 1 t n 1 G n such that s k s k+1 = s k t k+1 and t k s k+1 = t k t k+1.
53 n-graphs An n-graph is a diagram of sets and functions s 0 G 0 G 1 t 0 s 1 t 1 G 2 G n 1 s n 1 t n 1 G n such that s k s k+1 = s k t k+1 and t k s k+1 = t k t k+1. - A 0-graph is just a set.
54 n-graphs An n-graph is a diagram of sets and functions s 0 G 0 G 1 t 0 s 1 t 1 G 2 G n 1 s n 1 t n 1 G n such that s k s k+1 = s k t k+1 and t k s k+1 = t k t k+1. - A 0-graph is just a set. - A 1-graph is an ordinary multilabelled oriented graph.
55 Categories A category is a multilabelled oriented graph G 0 G 1 equipped t 0 with a composition operation on arcs, required to be associative and with neutral element. s 0
56 Categories A category is a multilabelled oriented graph G 0 G 1 equipped t 0 with a composition operation on arcs, required to be associative and with neutral element. Example: (h g) f =h (g f ) s 0 e A A f B g f g e B h f h C e C h g
57 Categories A category is a multilabelled oriented graph G 0 G 1 equipped t 0 with a composition operation on arcs, required to be associative and with neutral element. Example: (h g) f =h (g f ) e A A f B g f g e B h f s 0 h C e C h g Examples: sets and functions, each monoid, well formed formulas and logical consequence...
58 The free category over a graph s 0 Let G = (G 0 G 1 ) be a graph. A word in G is a sequence t 0 (f 1,f 2,,f k ) where f i G 1 and t 0 (f i ) = s 0 (f i+1 ): s 0 (f 1 ) f 1 t 0 (f 1 ) = s 0 (f 2 ) f 2 t 0 (f 2 ) s 0 (f k ) f k t 0 (f k ) For k = 0, one gets the empty word ().
59 The free category over a graph s 0 Let G = (G 0 G 1 ) be a graph. A word in G is a sequence t 0 (f 1,f 2,,f k ) where f i G 1 and t 0 (f i ) = s 0 (f i+1 ): s 0 (f 1 ) f 1 t 0 (f 1 ) = s 0 (f 2 ) f 2 t 0 (f 2 ) s 0 (f k ) f k t 0 (f k ) For k = 0, one gets the empty word (). The free category over G is the the graph G = (G 0 G1 ) t 0 where G1 is the set of words in G equipped with concatenation as composition, with neutral element given by the empty word: (f 1,f 2,,f k ) (g 1,g 2,,g r ) = (f 1,f 2,,f k,g 1,g 2,,g r ) () (f 1,f 2,,f k ) = (f 1,f 2,,f k ) = (f 1,f 2,,f k ) () s 0
60 2-polygraphs A 2-polygraph is a diagram of sets and functions G 1 G 2 s 0 t0 s 1 t1 t 0 G 0 G1 s 0 in which G = (G 0 G 1 ) t 0 s 0 is the free category over s 0 G = (G 0 G 1 ) t 0 2-graph. t 0 and where G 0 G 1 s 0 t 1 s 1 G 2 is a
61 Rewriting systems A rewriting system is a 2-polygraph G 1 G 2 s 0 t0 s 1 t1 t 0 G 0 G1 s 0 in which the set G 0 consists of exactly one element.
62 Rewriting systems A rewriting system is a 2-polygraph G 1 G 2 s 0 t0 s 1 t1 t 0 G 0 G1 s 0 in which the set G 0 consists of exactly one element. - G 2 identifies a subset of G 1 G 1, namely the set of rewriting rules (f 1,f 2,,f k ) (g 1,g 2,,g r )
63 Rewriting systems A rewriting system is a 2-polygraph G 1 G 2 s 0 t0 s 1 t1 t 0 G 0 G1 s 0 in which the set G 0 consists of exactly one element. - G 2 identifies a subset of G1 G 1, namely the set of rewriting rules (f 1,f 2,,f k ) (g 1,g 2,,g r ) Remark: the emphasis is on the directed nature of.
64 The 2-polygraph of n-term syllogisms For every positive natural number n: - G 0 = {T 1,..., T n}. - G 1 = {A Ti T j, E Ti T j, I Ti T j,o Ti T j 1 i j n} - G 2: (A Ti T i ) (A Ti T i ) (I Ti T i ) (I Ti T i ) 1 i n (A Ti T j ) (A Ti T j ) (I Ti T j ) (I Ti T j ) 1 i < j n (E Ti T j ) (E Ti T j ) (O Ti T j ) (O Ti T j ) 1 i < j n (A Ti T j ) (I Ti T i ) I Ti T j (E Ti T j ) (I Ti T i ) O Ti T j 1 i < j n (E Tj T i ) (E Ti T j ) (I Tj T i ) (I Ti T j ) 1 i < j n (I Tj T j ) (A Tj T i ) (I Ti T j ) (E Tj T i ) (I Ti T i ) (O Ti T j ) 1 i < j n (A Tj T k ) (A Ti T j ) (A Ti T k ) (E Tj T k ) (A Ti T j ) (E Ti T k ) 1 i < j < k n (A Tj T k ) (I Ti T j ) (I Ti T k ) (E Tj T k ) (I Ti T j ) (O Ti T k ) 1 i < j < k n (E Tk T j ) (A Ti T j ) (E Ti T k ) (A Tk T j ) (E Ti T j ) (E Ti T k ) 1 i < j < k n (E Tk T j ) (I Ti T j ) (O Ti T k ) (A Tk T j ) (O Ti T j ) (O Ti T k ) 1 i < j < k n (I Tj T k ) (A Tj T i ) (I Ti T k ) (A Tj T k ) (I Tj T i ) (I Ti T k ) 1 i < j < k n (O Tj T k ) (A Tj T i ) (O Ti T k ) (E Tj T k ) (I Tj T i ) (O Ti T k ) 1 i < j < k n (A Tk T j ) (E Tj T i ) (E Ti T k ) (I Tk T j ) (A Tj T i ) (I Ti T k ) 1 i < j < k n (E Tk T j ) (I Tj T i ) (O Ti T k ) 1 i < j < k n
65 Theorem For every positive natural number n, the rewriting system for the calculus of n-term syllogisms is complete.
66 Theorem For every positive natural number n, the rewriting system for the calculus of n-term syllogisms is complete. ketch of proof: It must be observed that the length of the words that undergo rewriting strictly decreases. Then, the proof proceeds by cases depending on n. - n = 2: the test is on the two possible rewritings of the word (E Aj A i ) (I Ai A i ). - n = 3: the test is on the possible rewritings of the words (E Ak A j ) (A Ai A j ) (A Ak A j ) (E Aj A i ) (E Ak A j ) (I Ai A j ) (E Aj A k ) (I Aj A i ) (A Aj A k ) (I Aj A i ) (E Aj A k ) (I Aj A i ) (I Ak A j ) (A Aj A i ) (E Aj A i ) (I Ai A i ). (A Aj A k ) (A Ai A j ) (I Ai A i ) (E Aj A k ) (A Ai A j ) (I Ai A i ) (A Ak A j ) (E Ai A j ) (I Ai A i ) (E Ak A j ) (A Ai A j ) (I Ai A i ) (E Aj A k ) (I Aj A j ) (A Aj A i ) (A Ak A j ) (E Aj A i ) (I Ai A i ) (E Ak A j ) (I Aj A j ) (A Aj A i ) (A Aj A k ) (I Aj A j ) (A Aj A i )
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