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Posts Tagged ‘Darii Syllogism’

Aristotle’s Formal Deductive Theory

Posted by allzermalmer on October 25, 2013

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.

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Logically Valid Arguments

Posted by allzermalmer on April 8, 2013

Categorically Valid Syllogisms

M stands for Middle Term; P stands for Major Term; S stands for Minor Term

Figure 1

(1) Barabara:If all M are P and all S are M, then all S are P
P. All M are P
P. All S are M
C. All S are P

(2) Celarent: If no M are P and all S are M, then no S are P
P. No M are P
P. All S are M
C. No S are P

(3) Darii: If all M are P and some S are M, then some S are P
P. All M are P
P. Some S are M
C. Some S are P

(4) Ferio: If no M are P and some S are M, then some S are not P
P. No M are P
P. Some S are M
C. Some S are not P

Figure 2

(1) Camestres: If all P are M and no S are M, then no S are P
P. All P are M
P. No S are M
C. No S are P

(2) Cesare: If no P are M and all S are M then no S are P
P. No P are M
P. All S are M
C. No S are P

(3) Baroko: If all P are M and some S are not M, then some S are not P
P. All P are M
P. Some S are not M
C. Some S are not P

(4) Festino: If no P are M and some S are M, then some S are not P
P. No P are M
P. Some S are M
C. Some S are not P

Figure 3

(1) Datisi: If all M are P and some M are S, then some S are P
P. All M are P
P. Some M are S
C. Some S are P

(2) Disamis: If some M are P and all M are S, then some S are P
P. Some M are P
P. Some M are S
C. Some S are P

(3) Ferison: if no M are P and some M are S, then some S are not P
P. No M are P
P. Some M are S
C. Some S are not P

(4) Bokardo: If some M are not P and all M are S, then some S are not P
P. Some M are not P
P. All M are S
C. Some S are not P

Figure 4

(1) Camenes: If all P are M and no M are S, then no S are P
P. All P are M
P. No M are S
C. No S are P

(2) Dimaris: If some P are M and all M are S, then Some S are P
P. Some P are M
P. All M are S
C. Some S are P

(3) Fresison: If no P are M and some M are S, then some S are not P
P. No P are M
P. Some M are S
C. Some S are not P

Propositional Logic

Modus Ponens: Given the conditional claim that the consequent is true if the antecedent is true, and given that the antecedent is true, we can infer the consequent.
P. If S then P
P. S
C. Q

Modus Tollens: Given the conditional claim that the consequent is true if the antecedent is true, and given that the consequent is false, we can infer that the antecedent is also false.
P. If S then P
P. Not P
C. Not S

Hypothetical Syllogism: Given two conditional such that the antecedent of the second is the consequent of the first, we can infer a conditional such that its antecedent of the first premise and its consequent is the same as the consequent of the second premise.
P. If S then M
P. If M then P
C. If S then P

Constructive Dilemma: Given two conditionals, and given the disjunction of their antecedents, we can infer the disjunction of their consequents.
P. If S then P                 P. If S then P
P. If M then N               P. If M then P
P. S or M                        P. S or M
C. P or N                        C. P or P

Destructive Dilemma: Given two conditionals, and given the disjunction of the negation of their consequents, we can infer the disjunction of the negation of their antecedents.
P. If S then P                P. If S then P
P. If M then N              P. If S then N
P. Not P or Not N        P. Not P or Not N
C. No S or Not M        C. Not S or Not S

Biconditional Argument: Given a biconditional and given the truth value of one side is known, we can infer that the other side has exactly the same truth value.
P. S<–>P    P. S<–>P   P. S<–>P   P. S<–>P
P. S               P. P              P. Not S      P. Not P
C. P              C. S               C. Not P     C. Not S

Disjunctive Addition: Given that a statement is true, we can infer that a disjunct comprising it and any other statement is true, because only one disjunct needs to be true for the disjunctive compound to be true.
P. S
C. S or P

Disjunctive Syllogism: Because at least one disjunct must be true, by knowing one is false we can infer tat the other is true.
P. S or P   P. S or P
P. Not P   P. Not S
C. S          C. P

Simplification: Because both components of a conjunctive argument are true, it is permissible to infer that either of its conjuncts is true.
P. S & P   P. S & P
C. S          C. P

Adjunction: Because both premises are presumed true, we can infer their conjunction.
P. S
P. P
C. S & P

Conjunctive Argument: Because the first premise says that at least one of the conjuncts is false and the second premise identifies a true conjunct, we can infer that the other conjunct is false.
P. ~(S & P)   P. ~(S & P)
P. S                P. P
C. Not P        C. Not S

 

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