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Archive for October, 2013

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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The Aim is the Search for Universal Laws

Posted by allzermalmer on October 22, 2013

I would rather discover one true cause than gain the kingdom of Persia.
Read more at http://www.brainyquote.com/quotes/authors/d/democritus.html#Ti0otIFKyo1RksRu.99“I would rather discover one true cause than gain the kingdom of Persia1. There exists at least one (empirical) principle that is true at describing an invariable relationship of the empirical world. (expressed) 2. There exists at least one possible (empirical) principle that is true at describing an invariable relationship  of the empirical world.  (expressed or not express3. There exists at least one invariable relationship of the empirical world.

“I would rather discover one true cause than gain the kingdom of Persia.” Democritus

Question: “Does there exist one true cause?”

The answer to this question, is answered in the affirmative by some and answered in the negative by some.

Affirmative: There does exist at least one causal relationship between events.
Negative: There doesn’t exist at least one causal relationship between events.

Both answers to the question are logically possible. There is no self-contradiction in either proposition. So it is possible that there does exist at least one causal relationship between events & it is possible that there doesn’t exist at least one causal relationship between events.

Suppose that we have someone similar to Democritus, they would be searching for at least one true causal relationship. It is possible that they don’t find what they are searching for since it is possible that there doesn’t exist at least one causal relationship between events.

So we can have the Aim of Searching for Universal Laws, but that doesn’t mean that what we search for is true or that we will find what we search for. In both cases we would neither be able to find what we search for (since it doesn’t exist) nor search for what is true (since it isn’t true).

We can have a methodological principle that there does exist at least one causal relationship between events. All our actions would be consistent with this methodological principle, but none of this is asserting that there actually does exist at least one causal relationship between events because it is possible that there doesn’t actually exist at least one causal relationship. This implies that our assertion was false, which implies that we didn’t find at least one true causal relationship.

The whole endeavor would be predicated on a fiction, and everything produced within the endeavor would be fictional as well. Even the single processes, or plural processes, used in this endeavor would only have input of fictions and output of fictions. It would be similar to comic books, having fictional characters as input in their process and fictions as the output in their process.

Suppose that there actually does exist a causal relationship between events. It is logically possible that both there exists at least one causal relationship between events & we don’t know this one causal relationship between events. However, it is logically impossible that we know both there exists at least one causal relationship between events & we don’t know this one causal relationship between events.

Suppose we know that both there exists at least one causal relationship between events & we don’t know this one causal relationship between events. It follows that, we know there exists at least one causal relationship between events and we know we don’t know this one causal relationship between events. We know there exists at least one causal relationship between events & we don’t know this one causal relationship between events. This is a contradiction, so it is necessary we don’t know both there exists at least one causal relationship between events & we don’t know this one causal relationship between events. From all this it follows we don’t know there exists at least one causal relationship. Since we don’t know there exists at least one causal relationship, it means that we can’t know that there exists at least one causal relationship.

The very Aim of the Search for Universal Laws would be of something that you can’t possibly know, even if you did obtain what you were searching for.

 

 

 

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Other Minds on Other Planets

Posted by allzermalmer on October 16, 2013

“That’s the perspective of a new book by science journalist titled Five Billion Years of Solitude, a nod to the Gabriel García Márquez masterpiece One Hundred Years of Solitude. Billings constructs a moving tale of our collective yearning to find companionship in the vastness of space, using interviews with a few of the key players in the search for intelligent life..If we are alone, or if life is rare, we must be the protectors of life and take charge to preserve it at all costs, possibly spreading it to other planetary platforms.”

Suppose there is some strong desire that there exists at least one other mind & it is possible that there exists at least one other mind.

This would ignore that there is a strong desire that there does exist at least one other mind & it isn’t possible that there does exist at least one other mind. It would also ignore that if there is at least another mind then there aren’t a finite amount of other mind. So it would assume that there are a finite amount of other minds.

An individual mind has a strong desire that another mind exists as well. This other mind would have at least one, or more, similarities to the individual mind. There tends to be an immediacy of other minds, when we accept at least one. The individual mind will have another mind, and this leads to a relationship between them. Similar to how we say that one person is closer to me relative to another person. This is the most primary relationship of immediacy of other minds.

Given that there are only two individuals minds that exist, then it doesn’t matter about a higher, or more complex, immediacy of other minds. Should another mind be added, now we obtain a more complex immediacy of other minds. A>B>C, or A>C>B, or B>A>C, or B>C>A, or C>A>B, or C>B>A. But there is always at least one other mind that has principle immediacy of other minds.

This type of addition of other minds can also of be of different types of minds. There different types would be those specific differences that would at least at least to between the principle of immediacy of other minds. This is because no two minds are alike. So there would be at least one difference. This in turn implies that there exists at least one difference between each mind that exists.

Take the example of a human being who is married to another human being. John and Jane are married, and John holds that only one other mind, which is Jane, exists & vice versa. The immediacy would be of John to Jane. This would also hold with different types of minds, like those of John and the mind of a cat like Tibbles. This would mean it would also hold with what are called aliens species.

Human beings like to believe that other individual human beings exist. They further project the existence of different types of minds that are non-human beings. Once it has been projected to the immediate environment, it is also projected out into outer space.

From the presumed number of stars, planets, galaxies, and necessary conditions for life, that at least one other mind not of terrestrial origin exists. This is even supposing that goes beyond our solar system or local neighborhood in the galaxy. Just from the abundance of things it is immediately thought that there exist other minds from this abundance.

 

 

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