allzermalmer

Truth suffers from too much analysis

Posts Tagged ‘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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Proof of Disjunctive Syllogism

Posted by allzermalmer on July 28, 2013

anguage

(I) Symbols: Ø = contradiction, → = conditional, and [] = Modal Operator
(II) Variables: p, q, r, p’, q’, r’. (Variables lower case)

Well Formed Formula for Language

(i) Ø and any variable is a modal sentence.
(ii) If A is a modal sentence, then []A is a modal sentence.
(iii) If A is a modal sentence and B is a modal sentence, then A implies B (A→B) is a modal sentence.

* A, B, and C are modal sentences, i.e. upper case letters are modal sentences. These upper case letters are “variables as well”. They represent the lower case variables in conjunction with contradiction, conditional, or modal operator.

So A may possibly stand for p, or q, or r. It may also possibly stand for a compound of variables and symbols. So A may stand for q, or A may stand for p→Ø, and etc.

Negation (~) = A→Ø
Conjunction (&) = ~(A→B)
Disjunction (v) = ~A→B
Biconditional (↔) = (A→B) & (B→A)

Because Ø indicates contradiction, Ø is always false. But by the truth table of material implication, A → Ø is true if and only if either A is false or Ø is true. But Ø can’t be true. So A → Ø is true if and only if A is false.

This symbol ∞ will stand for something being proved.

(1) Hypothesis (HY) : A new hypothesis may be added to a proof anytime, but the hypothesis begins a new sub-proof.

(2) Modus Ponens (MP) : If A implies B and A, then B must lie in exactly the same sub-proof.

(3) Conditional Proof (CP): When proof of B is derived from the hypothesis A, it follows that A implies B, where A implies B lies outside hypothesis A.

(4) Double Negation (DN): Removal of double negation ~~A & A lie in the same same sub-proof.

(5) Reiteration (R): Sentence A may be copied into a new sub-proof.

Proof of Disjunctive Syllogism: Because at least one disjunct must be true, by knowing one is false we can infer tat the other is true.

If either p or q and not p, then necessarily true q.

Premise (1) p v q (Hypothesis)
Premise (2) ~p (Hypothesis)
(3) ~p implies q ((1) and Definition v)
Conclusion (4) q (Modus Ponens by (2) and (3))

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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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Categorical Logic

Posted by allzermalmer on May 16, 2011

So what is Logic? Logic is “the study of methods for evaluating whether the premises of an argument adequately support its conclusion.” (The Power of Logic 4th edition). Now there are three words in such a definition that would need some answers as to what they themselves mean. Those words are argument, premises, and conclusion.
An argument  is a “set of statements where some of the statements, called the premises, are intended to support another, called the conclusion.” (The Power of Logic 4th edition).  So we can take away something from this. We have a set of statements that lead to another statement. Now we could wonder, what is a statement? A statement is “declarative statement that is either true or false” (The Power of Logic 4th edition. In some cases you can interchange between the words statement and proposition. In some texts they use the word statement, and in some other texts they use the word proposition.

Now that we know what statements are, and premises and conclusions are statements, we might wonder what is a premise and conclusion. A premise is “a statement we know, or strongly believe, to be true”. A conclusion is “a statement derived, or supported, by the premises”

Now sometimes it works that we come up with a statement, this would be our conclusion. Now we want to support this conclusions, this statement, and so we support this conclusion with a set of premises. We say, from these premises, we are lead to this conclusion.

Let me give an example: I say, “Socrates is mortal”. Now we want to know how did I arrive at this statement, which is the conclusion. So I would support this statement (conclusion), with another set of statements (premises). These premises would be, “All men are mortal” & “Socrates is a man”. Thus, our argument would look like this: (Premise 1) All men are mortal. (Premise 2) Socrates is a man. (Conclusion) Socrates is mortal. This forms an argument, and we find we have a conclusion that necessarily follows form its premises. Now this argument is valid.

Now the question would be? What does it mean for an argument to be valid? A valid argument is where “the premises succeed in guaranteeing the conclusion”. The other way of seeing this is as, the conclusion necessarily follows from the premises. Now the opposite of a valid argument is an invalid argument. So what is an invalid argument? An invalid argument is in which “the premises fail to guarantee the conclusion”. The other way of seeing this is as, the conclusion does not necessarily follow from the premises.

Valid and invalid are terms that are only used in deductive logic. These terms are used for deductive arguments. So the question becomes, what is a deductive argument? A Deductive argument is “an argument in which the premises are intended to guarantees the conclusion”. Deductive arguments are valid arguments, because the conclusion necessarily follows from the premises. The conclusion is guaranteed to follow from the premises.

Now one main point we can take is this: Logic is concerned about the form of the argument, which includes that the conclusion necessarily follows from the premises.

Now there are three axioms of logic. These are the law of identity, the law of excluded middle, and the law of non-contradiction.

The Law of Identity is “every individual thing is identical to itself”. It carries this symbolic form: A–>A or A=A. (the symbol of “–>” of “If…then” statements in propositional logic). An example would be like this: “If the sun is yellow in color, then the sun is yellow in color” or “the sun is yellow in color is the sun is yellow in color”

The Law of Excluded Middle is “every statement is either true or false”.  It carries this symbolic form: Av~A ( the symbol “v” is an “or” operator that we will come to in propositional logic). An example would be like this: “The sun is yellow in color or the sun is not yellow in color”. (The law of excluded middle is used when we get to truth-tables in propositional logic)

The Law of Non-Contradiction is “given any statement and its opposite, one is true and the other false”. It carries this symbolic form: ~(A&~A) (the symbol of “~” stands for “not“, and the symbol “&” stands for “and“). An example would be like this: “Not both the sun is yellow in color and the sun is not yellow in color”.

Now that we set up some of the basics, we can get into categorical syllogisms and propositional logic.

Categorical Syllogisms

Now categorical syllogisms deal with statements that tell us something about the relationship between categories. So we can give a definition to categorical statement. A Categorical Statement is “an assertion or denial that all or some members of the subject class are included in the predicate class”
( Logic: An Introduction 2nd edition) Now before we continue on with Categorical Syllogisms, now that we defined a Categorical statement, we should define what a syllogism is.

A Syllogism is “any deductive argument in which the conclusion is inferred from two premises” (Logic: An Introduction 2nd edition)

We have three types of statements in syllogisms. (1.) Universal statements, (2.) Particular statements, & (3.) Singular statements.

A Universal Statement is “a proposition that refers to all the members of a class”. A Particular Statement is “a proposition that refers to some but not all the members of a class. A Singular Statement is “a proposition that asserts that a particular individual has (or has not) some specified attribute”. (Introduction to Logic 13th edition)

Now I will try to break these three statements down some. All Universal and Particular statements involve general terms (categories). A General Term is “any word or phrase that describes or designates a general class of individuals” ( Logic: An Introduction 2nd edition). General terms are indefinite descriptions that follow the form of “a so-and-so”. They would be something like these:  “a cute baby”, “charming”, “drives a Buick”. General terms are symbolized with capital letters; So “a cute baby” can be symbolized with “C”. A cute baby=C.

All Singular statements have at least one singular term. A Singular Term is “a word or expression that truly can designate only one individual (e.g., a proper name or a definite description)”. (Logic: An Introduction 2nd edition). Singular terms are definite descriptions that follow the form of “the so-and-so”. They would be something like this: “the world’s cutest baby”, “this child”, “David”. Singular terms are symbolized with small letters; So “the world’s cutest baby” can be symbolized with “c”. The world’s cutest baby=c.

We have 8 forms of statements in syllogisms:
Four Categorical statements:
1. (A) Universal Affirmative: All A are B
2. (E) Universal Negative: No A are B
3. (I) Particular Affirmative: Some A are B
4. (O) Particular Negative: Some A are not B

The other four are singular statements, and two have a general term involved, and two have no general term:
5. Singular with one general term: a is B
6. Singular with one general term: a is not B
7. Singular with no general term: a is b
8. Singular with no general term: a is not b

Now I shall give an example of all 8 of these forms with actual english sentences.

All A are B: ‘All men are a cute person’
No A are B: ‘No men are a cute person’
Some A are B: ‘Some men are a cute person”
Some A are not B: ‘Some men are not a cute person’
a is B: ‘Gondoliere is a cute person’
a is not B: ‘Gondoliere is not a cute person’
a is b: ‘Gonoliere is the cutest person’
a is not b: ‘Gondoliere is not the cutest person’

Now with our four categorical statements of A, E, I, and O, they each have their own logical equivalents. Being Logically equivalent  means that the statements “have the same meaning, and may therefore replace one another whenever they occur”. For example, in english Bachelor means unmarried male. So any time we say bachelor, we are saying they are an unmarried male. And when we say someone is an unmarried male, we are also saying that they are a bachelor. They are equivalent to each other. So we can make an immediate inference. An Immediate inference is “an inference that is drawn directly from one premise without the mediation of any other premise”. (Introduction to Logic 13th edition)

We have the immediate inferences of conversion, obversion, and contraposition.

Conversion is “the result of interchanging the subject and predicate terms in a categorical statement”. (The Power of Logic 4th edition). The statement that we immediately derive in conversion is known as the converse. The Converse is “the result of interchanging the subject and predicate terms in a categorical statement”. We can do conversion with E and I categorical statements to obtain their converse. So this is how it would look:

(E) No A is B= No B is A
No plants are animals= No animals are plants
(I) Some A is B= Some B is A
Some plants are trees= Some trees are plants

Now before I talk about obversion, I have to set something up so we can better understand obversion. Each class/category, has its complement. A Complement is “the complement of class/category X is the class containing all things that are not a member of class/category X”. (The Power of Logic 4th edition). For example, the complement of the class/category of tree is the class/category containing all nontrees, which is, everything that is not a tree (horse, hawks, humans, hamburgers, and son).

Now each term has a term-complement. A Term-Complement is “a word or phrase that denotes a class/category complement” (The Power of Logic 4th edition). So the term-complement of “dogs” is “nondogs”, which denotes the class of everything that is not a dog. Now a term-complement should not be confused with a contrary term. So, the term “winner” should not be confused with “loser”, but is “nonwinner”, which includes nonwinners like players who tie, nonplayers, and losers.

Now when we get to term-complement with more than one word, like “wild dogs”, the term complement would not be “non wild-dogs”, since it contains a class that includes everything outside of the class connotated by the term wild dogs. So the term-complement of “wild dogs” is “things that are not wild dogs”, which includes a class that includes both tame dogs and nondogs. So this would include, in nondogs, as wild geese, since they are nondogs.

So Obversion is “a process of immediate inference by which a logically equivalent statement is formed from a categorical statement by changing its quality and negating the predicate term” (Logic: An Introduction 2nd edition). Now we change the statement to its obverse, and the Obverse is ” the result of changing the quality of a categorical statement and replacing the predicate terms with its term-complement”. (The Power of Logic 4th edition)

So here is how obversion works with all four categorical terms:
(A) All A are B= No A are not-P
All trees are plants= No trees are nonplants
(E)
No A are B= All A are not-B
No cats are trees= All cats are nontrees
(I)
Some A are B= Some A are not not-B
Some trees are oaks= Some trees are not nonOaks (this involves double negation)
(O) Some A are not B= Some A are non-B
Some trees are not oaks= Some trees are nonoaks

Contraposition is “a process of immediate inference by which the subject and predicate of a categorical statement trade places, and each is negated. (Logic: An Introduction 2nd edition).

(A) All A are B= All non-B are non-A
All cats are mammals= All nonmammals are noncats
(O) Some A are not B= Some non-B are non-A
Some plants are not weeds= Some nonweeds are not nonplants

Now the four categorical statement forms form what we call the square of opposition. The Square of opposition is “A diagram displaying the logical relations between categorical statements having the same subject and predicate terms”. (Logic: An Introduction 2nd edition)

There are two diagrams of the square of opposition, and here is the Traditional Square of Opposition:

Now here is the Modern Square of Opposition:


Now, there is a vital difference between the traditional square of opposition, and the modern square of opposition. But before we go over those differences, we should start out with how the Traditional Square of Opposition works.

The traditional square of opposition has the terms contradictories, contraries, subcontraries, and subalternation. Now we should go over what some of these terms mean, and how they work.

Contradictories are “two propositions so related that one is the denial or negation of the other”. (Introduction to Logic 13th edition). This means that contradictory statements cannot both be true, or false, at the same time. This means that both statements, which are contradictory, have the same subject and predicate terms, but they are different to each other in both quantity and quality. So the categorical statements of (A) and (O) are contradictories, as are the categorical statements of (E) and (I). So All A are B is contradicted by Some A is not B, and No A is B is contradicted by Some A are B.

Contradictories:
(A) All A is B :: (O)  Some A is not B
All cats are animals :: Some cats are not animals
(E) No A is B :: (I) Some A is B
No cats are animals :: Some cats are animals

Contraries  are “two propositions so related that they cannot both be true, although both may be false” (Introduction to Logic 13th edition). They cannot both be true, but they can both be false. Take this example of two statements: ‘The Ravens will win the game against the Steelers”‘ or “The Steelers will win against the Ravens”. If these two propositions refer to the same game, then one has to be true and the other false. Now they are not contradictory since the game could be a draw, and so both would be false. So they cannot both be true, but they could both be false. So (A) propositions are contrary to (E) propositions.

Contraries:
(A) All A are B :: (O) No A are B
All poets are dreamers :: No poets are dreamers
* Now they cannot both be true, but they could both be false since there might not be any poets at all.

Subcontrariers are “two propositions so related that they cannot both both be false, although they may both be true”. (Introduction to Logic 13th edition).  Both particular statements have the same subject and predicate terms, but they are differing in quality (one affirming the other denying). So “some diamonds are precious stone” and “some diamonds are not precious stone” could both be true, but they could not both be false. (I) and (O) statements are subcontrariers.

Subcontrariers:

(I) Some A are B :: (O) Some A are not B
Some diamonds are precious stones :: Some diamonds are not precious stones

Subalternation are “are the relation on the square of opposition between a universal proposition and its corresponding particular proposition.” (Introduction to Logic 13th edition). Two statements have the same subject and predicate terms, and agree on quality (affirming/denying), but differ in quantity (one universal and one particular). So the (I) statement is the subalternation of (A) statement, and the (O) statement is the subalternation of the (E) statement.

Subalternation:
(A) All A are B :: (I) Some A are B
All spiders are 8 legged animals :: Some spiders are 8 legged animals
(E) No A are B :: (O) Some A are not B
No whales are fish :: Some whales are not fish

Now that we have gone over traditional square of opposition, it is time to speak of the problems with the traditional square of opposition. It is because of these problems that a modern square of opposition was developed. Once we go over the problems of the traditional square of opposition, we will go over the modern square of opposition and see why we use the modern square of opposition now.

Problems of traditional square of opposition:

This problem revolves around existential import. Existential import is “an attribute of those propositions which normally assert the existence of objects of some specified kind.”(Introduction to Logic 13th edition). (I) and (O) statements carry existential import, and traditional interpretation of categorical statements say that (A) and (E) statements carry existential import. However, the modern interpretation differs from traditional interpretation on these issues.

So the (I) statement of “Some soldiers are heroes” says that there exists at least one soldier who is a hero. The (O) statement of “Some dogs are not companions” says that there exists at least one dog that is not a companion. So (I) and (O) statements do assert that the class designated by their subject terms are not empty, and each has at least one member in it. So by subalteration we find that (I) and (O) statements validly follow from (A) and (E) statements. So (A) and (E) statements must also have existential import, because a statement with existential import cannot be derived validly from another that does not have such import.

We know that (A) and (O) statements are contradictories on the traditional square of oppositions. So “All Danes speak English” is contradicted by “Some Danes do not speak English”. Contradictories cannot both be true, and both cannot both be false, since one must be true. Now if both (A) (Universal statement) and (O)  (particular statement) have existential import, then both contradictories could both be false. Let us take another example: “All inhabitants of Mars are blond” and “Some inhabitants of Mars are not Blond”. They are contradictories and would, under traditional assumption, have existential import. So we assume that both statements are asserting that there are inhabitants of Mars, then both propositions are false if Mars has no inhabitants. So if Mars has no inhabitants, then the class of its inhabitants are empty. Thus, both of the propositions would be false. Now if they are both false, then they cannot be contradictories.

Now if the traditional square of opposition is correct about existential import, and it states that (A) is the contradictory of (O) and (E) is the contradictory of (I), then the square is not correct that they are contradictory based on the idea of existential import. We are clearly left with a contradiction, which violates the Law of Non-contradiction ~(A&~A). Not both the traditional square of opposition has contradictories and the traditional square of opposition does not have contradictories.

Modern solution to Traditional Square of Opposition:

We can solve this problem to the traditional square of opposition, but the price can seem high in doing so. In order to solve this problem, we presuppose that the categories are not empty. For example, take the question “Have you stopped beating your wive?”. This question presupposes that you have beat your wife, so it presuppose that the category is not empty. So if we can answer this question “yes” or “no” that I have stopped, or not, of beating my wife. Now if we we don’t presuppose that I have beat my wife, then I don’t have to answer “yes” or “no”. The category is empty when we don’t make that presupposition, and the category is not empty when we do make the presupposition.

So to rescue the square of opposition, then we insist that all statements, like (A), (E), (I), and (O), that the categories that they refer do have members  (so they are not empty). So the truth and falsehood of the statements, and logical relation among them, can be answered (under this interpretation) if we never presuppose that they are empty. Under this interpretation of presupposition; (A) and (E) will remain contraries; (I) and (O) will remain contraries; (A) and (O) will remain contradictories, as well (E) and (I).

The first problem with these presuppositions is this: If we presuppose that the class designated has members, we can never be able to formulate a statement that denies that it has members, since we presuppose that they do have members. The second problem with these presuppositions is this: Sometimes what we say doesn’t presuppose that there are any members of the category that we are talking about.

Now the modern square of opposition does not make any presupposition. Instead, it works as follows:

(1.) (I) and (O) statements continue to have existential import. So the statement “Some A are B” is false if the category of A is empty, and same for “Some A are not B”.

(2.) Universal statements, (A) and (E), are contradictories of particular statements, (O) and (I).

(3.) Universal statements are interpreted as having no existential import. So if category A is empty, the statement “All A are B” can be true, and so can the statement “No A are B”. Take this example: “All unicorns have horns” and “no unicorns have wings” may both be true, even if there are no unicorns. However, if there are no unicorns, the (I) statement “Some unicorns have horns” is false, and so is the (O) statement “Some unicorns do not have wings”.

(4.) We can utter universal statements which we intend to assert existence. However, in doing so requires two statements, one existential in force but particular, and the other universal but not existential in force.

(5.) Corresponding (A) and (E) statements can both be true and are thus not contraries. They would carry this propositional logic form, which we get to later on. (A) All unicorns have wings= If there is a unicorn, then it has wings; (E) No unicorns have wings= If there is a unicorn, then it does not have wings. And both of these “if…then” statements can be true even if there are no unicorns.

(6.) Corresponding (I) and (O) statements are not subcontraries. They do have existential import, and can both be false if the subject class is empty.

(7.) Subalternation, inferring an (I) statement from an (A) statement or an (O) statement from an (E) statement, is generally not valid. This is because one may not validly infer a statement that has existential import from one that does not.

(8.) Now for this point, I have given conversion, contraposition, and obversion, all based on the modern interpretation. So this point is not necessary to explain what it was. However, with the traditional square of opposition, there were more conversions and contrapositions, but they are not really allowed with the modern interpretation.

(9.) Relations along the sides of the square are undone, but the diagonal, contradictory relations remain in force.

The existential presupposition is rejected by modern logicians. So it is a mistake to assume that a class has members if it is not asserted explicitly that it does. So if an argument relies on assumption of existential import, without explicitly stating it, relies on the existential fallacy.
Finding out if a syllogism is valid:

Here is a syllogism, and one that is valid:

Premise 1: No heroes are cowards (No A are B) (E)
Premise 2: Some soldiers are cowards (Some A are B) (I)
Conclusion: Some soldiers are not heroes (Some A are not B) (O)

The proposition above has three terms in it. It carries the terms heroes, cowards, and soldiers. Our conclusion is important in figuring out how to create a valid syllogism. Remember how I said that we usually come up with the conclusion first, and then support it with some premises? Well, this is where that comes back into play. In the conclusion, we find that the last sentence has a subject and predicate.The subject is soldiers and the predicate is heroes.

There are three terms involved in a valid syllogisms. They are major term, middle term, and minor term. The Major Term is “the term of the categorical syllogism that appears as the predicate term of the conclusion”, which would be “heroes” in the foregoing argument. (Logic: An Introduction 2nd edition). The Minor Term is “the term in the categorical syllogism that occurs as the subject term of the conclusion”, which would be “soldiers” in the foregoing argument. (Logic: An Introduction 2nd edition). The Middle Term is “the term in a categorical syllogism that appears in both premises but not in the conclusion”, which would be “cowards” in the foregoing argument. (Logic: An Introduction 2nd edition)

The premises of a syllogism also have terms to go along with them, besides the terms that show up in them. There are three premises in a syllogism. They are major premise, minor premise, and conclusion. We have already gone over conclusion, so now it is time to deal with major premise and minor premise.

A Major premise is “the premise of a categorical syllogism containing the major term”, which would be the first premise of the foregoing argument (Logic: An Introduction 2nd edition). A Minor premise is “the premise of a categorical syllogism containing the minor term”, which would be the second premise of the foregoing argument. (Logic: An Introduction 2nd edition).

So let us go back over the syllogism that I started out with, but list what they are and what is going on.

(E) No heroes are cowards:: It is the major premise, and has “heroes” as the major term and “cowards” as the middle term.
(I) Some soldiers are cowards:: It is the minor premise, and has “soldiers” as the minor term and “cowards” as the middle term.
(O) Some soldiers are not heroes:: It is the conclusion, and has “soldiers” as the minor term (subject) and “heroes” as the major term (predicate)

Each categorical syllogism has a mood. A Mood is “the specification of the categorical forms of a syllogism arranged in standard order (major premise, minor premise, and conclusion)”. (Logic: An Introduction 2nd edition). For example, each premise of a categorical syllogism is made up (A), (E), (I), and (O) premises. These types of statements make up the mood of an argument. For example, the argument that I presented carries the mood of (E), (I), and (O).

Now categorical syllogisms also have a figure. A figure is “the arrangement of the middle term in the premise of a categorical syllogism”. (Logic: An Introduction 2nd edition). There are four figures to a categorical syllogism. Here are the four figures of a categorical syllogism.

That is the four figures of a categorical syllogism, and here are the rules to it.

(1.) Middle term may be the subject term of the major premise and predicate term of the minor premise.
(2.) Middle term may be the predicate term of both premises.
(3.)  Middle term may be the subject term of both premises.
(4) Middle term may be the predicate term of the major premise and the subject term of the minor premise.

Now there are 256 possible syllogisms that one can come up with (4×64=256). However, there are only 15 syllogisms that are valid. Let us go over the 15 valid forms of a categorical syllogism.

There are four valid syllogisms following figure 1. They all carry a certain mood along with them, as they follow a certain figure. So these syllogisms all follow figure 1.
1. AAA-1:: barabara
All M are P
All S are M
All S are P
2. EAE-1:: celarent
No M are P
All S are M
No S are P
3. AII-1:: darii
All M are P
Some S are M
Some S are P
4. EIO-1:: ferio
No M are P
Some S are M
Some S are not P

There are four valid syllogisms following figure 2. They all carry a certain mood along with them, as they follow a certain figure. So these syllogisms all follow figure 2.
1. AEE-2:: camestres
All P are M
No S are M
No S are P
2. EAE-2:: cesare
No P are M
All S are M
No S are P
3. AOO-2:: baroko
All P are M
Some S are not M
Some S are not P
4. EIO-2:: festino
No P are M
Some S are M
Some S are not M

There are four valid syllogisms following figure 3. They all carry a certain mood along with them, as they follow a certain figure. So these syllogisms all follow figure 3.
1. AII-3:: datisi
All M are P
Some M are S
Some S are P
2. IAI-3:: disamis
Some M are P
All M are S
Some S are P
3. EIO-3:: ferison
No M are P
Some M are S
Some S are not P
4. OAO-3:: bokardo
Some M are not P
All M are S
Some S are not P

There are three valid syllogisms following figure 4. They all carry a certain mood along with them, as they follow a certain figure. So these syllogisms all follow figure 4.
1. AEE-4:: camenes
All P are M
No M are S
No S are P
2. IAI-4:: dimaris
Some P are M
All M are S
Some S are P
3. EIO-4:: fresison
No P are M
Some M are S
Some S are not P

There are six rules to make a valid syllogism. I shall list those 6 rules, but it should be known that if you follow the figure and mood that I gave above, then you have a valid syllogism.

Rule 1. Standard-form syllogism must contain exactly three terms, each of which is used in the same sense throughout the argument. Violation: Fallacy of four terms

Rule 2. In a valid standard-form categorical syllogism, the middle term must be distributed in at least one premise. Violation: Fallacy of Undistributed Middle

Rule 3. In a valid standard-form categorical syllogism, if either term is distributed in the conclusion, then it must be distributed in the premises. Violation: Fallacy of illicit major, or fallacy of illicit minor

Rule 4. No standard-form categorical syllogism having two negative premises is valid. Violation: Fallacy of exclusive premises

Rule 5. If either premise of a valid standard-form categorical syllogism is negative, the conclusion must be negative. Violation: Fallacy of drawing an affirmative conclusion from a negative premise

Rule 6. No valid standard-form categorical syllogism with a particular conclusion can have two universal premises. Violation: Existential fallacy

However, Harry Gensler in Introduction to Logic, gives us a very simple way to find out if a syllogism is valid or not. It is called the Star Test.

So let us return to the 8 statements of syllogisms:
1. All A are B
2. No A are B
3. Some A are B
4. Some A are not B
5. a are B
6. a are not B
7. a are b
8. a are not b

Pay attention to the ones that are in bold and italicized, since that is very important to the Star Test.

A letter is distributed in a valid syllogism if it occurs just after “All” or anywhere after “No” or “not”.

So by definition, (1.) The first letter after “All” is distributed, but not the second. (2.) Both letters after “No” are distributed. (3.) The Letter after “not” is distributed.

So the Start Test for a syllogism goes as follow:  Star the distributed letters in the premises and undistributed letters in the conclusion. Then the syllogism is VALID if and only if every capital letter is starred exactly once and there is exactly one star on the right-handed side.

So a valid argument must satisfy two conditions: (1.) Each capital letter is starred in one and only one occurrence. (Small letters can be starred any number of times.) (2.) Exactly one right-handed letter (letter after “are” or “are not”) is starred.

Let me give you an example of a valid argument with the Star Test.

Example 1:
All A* are B
Some C are A
Some C* are B*

Now we notice that the argument is in figure 1, and has the mood of AII. So it is AII-1, which is a Darii syllogism.

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