This is an axiomatic presentation of Aristotle’s Formal Deductive System. This was formalized by polish logician Jan Lukasiewicz in his book Elements of Mathematical Knowledge.

Aristotelian Formal Deductive System has the condition that “empty names may not be the values of our name variables, i.e., such as “square circle”.” This is another way of stating that the subject must actually exist. This is known as Existential Assumption, which modern deductive systems in logic don’t hold to. So some of the theorems in this system aren’t acceptable in modern logic.

I have used polish notation, which was used by Lukasiewicz.

C= Conditional Implication (–>)

K= Conjunction (&)

N= Negation (~)

A= Universal Affirmative

E= Universal Negative

I= Particular Affirmative

O= Particular Negative

Axioms

S1: Aaa

All a are a

S2 Iaa

Some a are a

S3: CKAmbAamAab

If all m are b & all a are m then all a are b

S4: CKAmbImaIab

If all m are b & some m are a then some a are b

Definitions

D1: Oab=NAab

Some a are not b= Not all a are b

D2: Eab= NIab

No a are b= Not some a are b.

Postulates

T1: Cpp

p implies p

T2: CCpqCCqrCpr

If p implies q then if q implies r then p implies r

T3: CCpqCNqNp

If p implies q then not q implies not p

T4: CCpNqCqNp

If p implies not q then q implies not p

T5: CCNpqCNqp

If not p implies q then not q implies p

T6: CCKpqrCpCqr

If p & q implies r then if p then q implies r

T7: CCKpqrCqCpr

If p & q implies r then if q then p implies r

T8: CCKpqrCKpNrNq

If p & q implies r then p & not r implies not q

T9: CCKpqrCKNrqNp

If p & q implies r then not r & q implies not p

T10: CCKpqrCCspCKsqr

If p & q implies r then if s implies p then s & q implies r

T11: CCKpqrCCsqCKpsr

If p & q implies r then if s implies q then p & s implies r

T12: CCKpqrCCrsCKqps

If p & q implies r then if r implies s then q & p implies s

Theorems

Law of the Square of Opposition

S5 COabNAab

1. Cpp [sub p/NAab in T1]

2. CNAabNAab

3. COabNAab [by D1 of antecedent in 2]

Q.E.D.

S6 CNAabOab

1.Cpp [sub p/NAab in T1]

2. CNAabNAab

3. CNAabOab [D1 of consequent in 2]

Q.E.D.

S7 CAabNOab

1. CCpNqCqNp [sub. p/Oab, q/Aab in T4]

2. CCOabNAabCAabNOab

3. CAabNoab [by (2)/(S5) & MP]

Q.E.D.

S8 CNOabAab

1. CCNpqCNqp [sub. p/Aab, q/Oab in T5]

2. CCNAabOabCNOabAab

3. CNOabAab [by (2)/(S6) & MP]

Q.E.D.

S9 CEabNIab

1. Cpp [sub. p/NIab in T1]

2. CNIabNIab

3. CEabNIab [by D2 of antecedent in 2]

Q.E.D.

S10 CNIabEab

1. Cpp [sub. p/NIab in T1]

2. CNIabNIab

3. CNIabEab [by D2 of consequent in 2]

Q.E.D.

S11 CIabNEab

1. CCpNqCqNp [sub. p/Eab, q/Iab, in T4]

2. CCEabNIabCIabNEab

3. CIabNEab [by (2)/(S9) & MP]

Q.E.D.

S12 CNEabIab

1. CCNpqCNqp [sub. p/Iab, q/Eab in T5]

2. CCNIabEabCNEabIab

3. CNEabIab [by (2)/(S10) & MP]

Q.E.D.

Laws of Subalternation

S13 CAabIab

1. CCKpqrCqCpr [sub. p/Aab, q/Iaa, r/Iab in T7]

2. CCKAabIaaIabCIaaCAabIab

3. CKAmbImaIab [sub. m/a in S4]

4. CKAabIaaIab

5. CIaaCAabIab [by (2)/4) & MP]

6. CAabIab [by (5)/(S2) & MP]

Q.E.D.

S14 CNIabNAab

1. CCpqCNqNp [sub. p/Aab, q/Iab in T3]

2. CCAabIabCNIabNab

3. CNIabNAab [by (2)/(S13) & MP]

Q.E.D.

S15 CEabOab

1. CNIabNAab [reiteration of S14]

2. CEabNAab [by D2 of antecedent of 1]

3. CEabOab [by D1 of consequent of 2]

Q.E.D.

Laws of Contrariety

S16 CNOabNEab

1. CCpqCNqNp [sub. p/Eab, q/Oab, in T3]

2. CCEabOabCNOabNEab

3. CNOabNEab [by (2)/(S15) & MP]

Q.E.D.

S17 CEabNAab

1. CNIabNAab [reiteration S14]

2. CEabNAab [D2 of antecedent (1)]

Q.E.D.

S18 CAabNEab

1. CCpNqCqNp [sub. p/Eab, q/Aab in Th 4]

2. CCEabNAabCAabNEab

3. CAabNEab [by (2)/(S17) & MP]

Q.E.D.

Laws of Subcontrariety

S19 CNIabOab

1. CNIabNAab [reiteration S14]

2. CNIabOab [by (1) & D1 consequent]

Q.E.D.

S20 CNOabIab

1. CCNpqCNqp [sub. p/Iab, q/Oab, in Th 5]

2. CCNIabOabCNOabIab

3. CNOabIab [by (S19)/(2) & MP]

Q.E.D.

Laws of Conversion

S21 CIabIba

1. CCKpqrCpCqr [sub p/Aaa, q/Iab, r/Iba in Th 6]

2. CCKAaaIabIbaCAaaCIabIba

3. CKAmbImaIab [sub m/a, b/a, a/b in (S4)]

4. CKAaaIabIba

5. CAaaCIabIba [(2)/(4) & MP]

6. CIabIba [(S1)/(5) & MP]

Q.E.D.

S22 CAabIba

1. CCpqCqrCpr [sub p/Aab, q/Iab, r/Iba in Th 2]

2. CCAabIabCIabIbaCAabIba

3. CIabIbaCAabIba [by (2)/(S3) & MP]

4. CAabIba [by (3)/(S21) & MP]

Q.E.D.

S23 CNIabNIba

1. CCpqCNqNp [sub p/Iba, q/Iab, r/Iab in Th 3]

2. CCIbaIabCNIabNIba

3. CIabIba [sub a/b, b/a in (S21)]

4. CIbaIab

5. CNIabNIba [by (4)/(2) & MP]

Q.E.D.

S24 CEabNIba

1. CNIabNIba [reiteration (S23)]

2. CEabNIba [D2 of antecedent (1)]

Q.E.D.

S25 CEabEba

1. CEabNIba [reiteration (S24)]

2. Eba=NIba [sub a/b, b/a in D2]

3. CEabEba [by (1)/(2) D2 consequent (1)]

Q.E.D.

Syllogisms Figure 1

S26 CKAmbAamIab (Barbari)

1. CCKpqrCCsqCKpsr [sub p/Amb, q/Ima, r/Iab, s/Aam in Th 11]

2. CCKambImaIabCCAamImaCKAmbAamIab

3. CCAamImaCKAmbAamIab [by (2)/(S4) & MP]

4. CAamIma [sub b/m in (S22)]

5. CKAmbAamIab [by (3)/(4) & MP]

Q.E.D.

S27 CKAmbNIabNIma

1. CCKpqrCKpNrNq [sub p/Amb, q/Ima, r/Iab in Th 8]

2. CCKAmbImaIabCKAmbNIabNIma

3. CKAmbNIabNIma [by (2)/(S4) & MP]

Q.E.D.

S28 CKAmbEbaNIma

1. CCKpqrCCsqCKpsr [sub p/Amb, q/NIab, r/NIab, s/Eba in Th 11]

2. CCKAambNIabNImaCCEbaNIabCKAmbEbaNIma

3. CCEbaNIabCKAmbEbaNIam [by (2)/(S27) & MP]

4. CEabNIba [sub a/b, b/a in (S24)]

5. CEbaNIab

6. CKAmbEbaNIam [by (3)/(5) & MP]

Q.E.D

S29 CKEmbAamEab (Celarent)

1. CCKpqrCCrsCKqps [sub p/Aam, q/Emb, r/NIab, s/Eab in Th 12]

2. CCKAamEmbNIabCCNIabEabCKEmbAamEab

3. CKAmbEbaNIma [sub m/a, b/m, a/b in (S28)]

4. CKamEmbNIab

5. CCNIabEabCKEmbAamEab [by (3)/(4) & MP]

6. CKEmbAamEab [by (5)/(S10) & MP]

Q.E.D.

S30 CKEmbAamOab (Celaront)

1. CCpqCCqrCpr [sub p/KEmbAam, q/Eab, r/Oab in Th 2]

2. CCKEmbAamEabCCEabOabCKEmbAamOab

3. CCEabOabCKEmbAamOab [by (2)/(S29) & MP]

4. CKEmbAamOab [by (3)/(S15) & MP]

Q.E.D.

S31 CKEmbIamIab (Darii)

1. CCKpqrCCsqCKpsr [sub p/Amb, q/Ima, r/Iab, s/Iam in Th 11]

2. CCKAmbImaIabCCIamImaCKAmbIamIab

3. CCIamIMaCKAmbIamIab [by (2)/(S4) & MP]

4. CIabIba [reiteration (S21) & sub m/b]

5. CIamIma

6. CKAmbIamIab [by (3)/(5) & MP]

Q.E.D.

S32 CKNIabImaNAmb

1. CCKpqrCKNrqNp [sub p/Amb, q/Ima, r/Iab, in Th 9]

2. CCKAmbImaIabCKNIabImaNAmb

3. CKNIabImaNAmb pby (2)/(S4) & MP]

Q.E.D.

S33 CKEmbIamOab (Ferio)

1. CKNIabImaNAmb [sub a/m, m/a in (S32)]

2. CKNImbIamNAab

3. Emb=NImb [D2 sub a/m]

4. CKEmbIamNAab

5. CKEmbIam Oab [by (4) & D1 on consequent of (4)]

Q.E.D.

Syllogism Figure 2

S34 CKEbmAamEab (Cesare)

1. CCKpqrCCspCKsqr [sub p/Emb, q/Aam, r/Eab, s/Ebm in Th 10]

2. CCKEmbAamEabCCEbmEmbCKEbmAamEab

3. CCEbmEmbCKEbmAamEab [(2)/(S29) & MP]

4. CEabEba [sub a/b, b/m in (S24)]

5. CEbmEmb

6. CKEbmAamEab [by (5)/(3) & MP]

Q.E.D.

S35 CKEbmAamOab

1. CCpqCCqrCpr [sub p/KEbmAam, q/Eab, r/Oab in Th 2]

2. CCKEbmAamEabCCEabOabCKEbmAamOab

3. CCEabOabCKEbmAamOab [(2)/(S34) & MP]

4. CKEbmAamOab [by (3)/(S15) & MP]

Q.E.D.

S36 CKAbmEamEab (Camestres)

1. CCKpqrCCrsCKqps [sub p/Eam, q/Abm, r/Eba, S/Eab in Th 12]

2. CCKEamAbmEbaCCEbaEabCKAbmEamEab

3. CKEbmAamEab [sub b/a, a/b in (S34)]

4. CKEamAbmEba

5. CCEbaEabCKAbmEamEab [by (2)/(4) & MP]

6. CEabEba [sub a/b, b/a in (S25)]

7. CEbaEab

8. CKAbmEamEab [by (5)/(7) & MP]

Q.E.D.

S37 CKAbmEamOab (Camestrop)

1. CCpqCCqrCpr [sub p/KAbmEam, q/Eab, r/Oab in Th 2]

2. CCKAbmEamEabCCEabOabCKAbmEamOab

3. CCEabOabCKAbmEamOab [by (2)/(S36) & MP]

4. CKAbmEamOab [by (3)/(S15) & MP]

Q.E.D.

S38 CKEbmIamOab (Festino)

1. CCKpqrCCspCKsqr [sub p/Emb, q/Iam, r/Oab, s/Ebm in Th 10]

2. CCKEmbIamOabCCEbmEmbCKEbmIamOab

3. CCEbmEmbCKEbmIamOab [by (2)/(S33) & MP]

4. CEabEba [sub a/b, b/m by (S25)]

5. CEbmEmb

6. CKEbmIamOab [by (3)/(5) & MP]

Q.E.D.

S39 CKAmbNAabNAam

1. CKpqrCKpNrNq [sub p/Amb, q/Aam, r/Aab in Th 8]

2. CCKAmbAamAabCKAmbNAabNAam

3. CKAmbNAabNAam [by (2)/(S3) & MP]

Q.E.D.

S40 CKAbmOamOab (Baroco)

1. CKAmbNAabNAam [sub m/b, b/m in (S39)]

2. CKAbmNAamNAab

3. Oab=NAab [sub b/m in D1]

4. Oam=NAam

5. CKAbmOamOab [D1 of consequent (2)]

Q.E.D.

Syllogism Figure 3

S41 CKAmbAmaIab (Darapti)

1. CCKpqrCCsqCKpsr [sub p/Amb, q/Ima, r/Iab, s/Ama in Th 11]

2. CCKAmbImaIabCCAmaImaCKAmbAmaIab

3. CCAmaImaCKAmbAmaIab [by (2)/(S4) & MP]

4. CAabIab [sub a/m, b/a in (S13)]

5. CAmaIma

6. CKAmbAmaIab [by (5)/(3) & MP]

Q.E.D.

S42 CKEmbAmaOab (Felapton)

1. CCKpqrCCsqCKpsr [sub p/Emb, q/Iam, r/Oab, s/Ama in Th 11]

2. CCKEmbIamOabCCAmaIamCKEmbAmaOab

3. CCAmaIamCKEmbAmaOab [by (2)/(S33) & MP]

4. CAabIba [sub a/m, b/a in (S22)]

5. CAmaIam

6. CKEmbAmaOab [by (5)/(3) & MP]

Q.E.D.

S43 CKImbAmaIab (Disamis)

1. CCKpqrCCrsCKqps [sub p/Ama, q/Imb, r/Iba, s/Iab in Th 12]

2. CCKAmaImbIbaCCIbaIabCKImbAmaIab

3. CKAmbImaIab [sub b/a, a/b in (S4)]

4. CKAmaImbIba

5. CCIbaIabCKImbAmaIab [by (4)/(2) & MP]

6. CIabIba [sub a/b, b/a in (S21)]

7. CIbaIab

8. CKImbAmaIab [by (5)/(7) & MP]

Q.E.D.

S44 CKNAabAamNAmb

1. CCKpqrCKNrqNp [sub p/Amb, q/Aam, r/Aab in Th 9]

2. CCKAmbAamAabCKNAabAamNAmb

3. CKNAabAamNAmb [by (2)/(S3) & MP]

Q.E.D.

S45 CKOmbAmaOab (Bocardo)

1. CKNabAamNAmb [sub a/m, m/a in (S44)]

2. CKNAmbAmaNAab

3. Oab=NAab [sub a/m in D1]

4. Omb=NAmb

5. CKOmbAmaNAab

6. CKOmbAmaOab [by D1 of consequent in (5)]

Q.E.D.

S46 CKEmbImaOab (Fersion)

1. CCKpqrCCsqCKpsr [sub p/Emb, q/Iam, r/Oab, s/Ima in Th 11]

2. CCKEmbIamOabCCImaIamCKEmbImaOab

3. CCImaIamCKEmbImaOab [by (2)/(S33) & MP]

4. CIabIba [sub a/m, b/a in (S21)]

5. CImaIam

6. CKEmbImaOab [by (5)/(3) & MP]

Q.E.D.

Syllogism Figure 4

S47 CKAbmAmaIab (Bamalip)

1. CCKpqrCCspCKsqr [sub p/Imb, q/Ama, r/Iab, s/Abm in Th 10]

2. CCKImbAmaIabCCAbmImbCKAbmAmaIab

3. CCAbmImbCKAbmAmaIab [by (2)/(S43) & MP]

4.CAabIba [sub a/b, b/a in (S22)]

5. CAbmImb

6. CKAbmAmaIab [by (5)/(3) & MP]

Q.E.D.

S48 CKAbmEmaEab (Calemes)

1. CCKpqrCCsqCKpsr [sub p/Abm, q/Eam, r/Eab, s/Ema in Th 11]

2. CCKAbmEamEabCCEmaEamCKAbmEmaEab

3. CCEmaEamCKAbmEmaEab [by (2)/(S36) & MP]

4. CEabEba [sub a/m, b/a in (S25)]

5. CEmaEam

6. CKAbmEmaEab [by (5)/(3) & MP]

Q.E.D.

S49 CKAbmEmaOab (Calemop)

1. CCpqCCqrCpr [sub p/KAbmEma, q/Eab, r/Oab in Th 2]

2. CCKAbmEmaEabCCEabOabCKAbmEmaOab

3. CCEabOabCKAbmEmaOab [by (2)/(S48) & MP]

4. CKAbmEmaOab [by (3)/(S15) & MP]

Q.E.D.

S50 CKIbmAmaIab (Diamtis)

1. CCKpqrCCspCKsqr [sub p/Imb, q/Ama, r/Iab, s/Ibm in Th 10]

2. CCKImbAmaIabCCIbmImbCKIbmAmaIab

3. CCIbmImbCKIbmAmaIab [by (2)/(S43) & MP]

4. CIabIba [sub a/b, b/m in (S21)]

5. CIbmImb

6. CKIbmAmaIab [by (3)/(5) & MP]

Q.E.D.

S51 CKEbmAmaOab

1. CCKpqrCCspCKsqr [sub p/Emb,q/Ama, r/Oab, s/Ebm]

2. CCKEmbAmaOabCCEbmEmbCKEbmAmaOab

3. CCEbmEmaCKEbmAmaOab [by (2)/(S46) & MP]

4. CEabEba [sub a/b, b/m in (S25)]

5. CEbmEmb

6. CKEbmAmaOab [by (5)/(3) & MP]

Q.E.D.

S52 CKEbmImaOab (Fression)

1. CCKpqrCCspCKsqr [sub p/Emb, q/Ima, r/Oab, s/Ebm in Th 10]

2. CCKEmbImaOabCCEbmEmbCKEbmImaOab

3. CCEbmEmbCKEbmImaOab [by (2)/(S46) & MP]

4. CEabEba [sub a/b, b/m in (S25)]

5. CEbmEmb

6. CKEbmImaOab [by (5)/(3) & MP]

Q.E.D.