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Posts Tagged ‘Theorem’

Impossibility Paradox of Computers by Curry Paradox

Posted by allzermalmer on July 29, 2013

There was a paper called “Computer Implication and the Curry Paradox”. It was authored by both Wayne Aitken and Jeffery A. Barrett, which appears in Journal of Philosophical Logic vol. 33 in 2004.

Suppose an Implication Program has two input statements that are about the behavior of the program, then it tries to deduce the second statement from the first statement by some specified rules in it’s library.

If program finds a deduction of the second statement from the first statement, then the program halts and has output of 1 to signal proof has been found.

The Implication Program can prove statements involving the Implication Program itself.

It is assumed throughout the paper that programs are written in a fixed language for a computer with unlimited memory.

The Impossibility Theorem basically states that “no sufficiently powerful implication program can incorporate an unrestricted form of modus ponens.”

One of the consequences of this Impossibility Theorem is that “modus ponens is an example of a valid rule of inference that can be defined algorithmically, but cannot be used by the implication program.”

Assume (1) that property C(X) is defined to hold if and only if X having property X implies Goldbach’s Conjecture. Furthermore, suppose (2) that C (C). Thus, by the definition of C (X) it implies that C (C) implies Goldbach’s Conjecture. Since C (C) is true by assumption, then it follows by Modus Ponens that Goldbach’s Conjecture is true.

This doesn’t prove Goldbach’s Conjecture yet. However, it does prove that C (C) implies Goldbach’s Conjecture. So by the definition of C (X), it follows that C (C) is true. And by the use of Modus Ponens, Goldbach’s Conjecture is true.

A statement is a list, i.e.  [prog, in, and out]. Prog is a program considered as data, and in is an input for prog, and out is an anticipated output.

A statement is called true if the program prog halts with input in and output out. A statement that isn’t true is called false.

“There is a program to check if but not to test whether [prog, in, out] is a true statement. Given [prog, in, out] as an input, it first runs prog as a sub-process with input in. If and when prog halts, it compares the actual output with out. If they match then the program outputs 1; if they do not match, the program does something else (say, outputs 0). This program will output 1 if and only if [prog, in, out] is true, but it might not halt if [prog, in, out] is false. Due to the halting problem, no program can check for falsity.”

So there is a program that check whether [prog, in, out] is a true statement, but the program can’t test whether [prog, in, out] is a true statement. From the Halting problem, no program can check for falsity. So the program can’t check for it’s own falsity, but can check for it’s truth.

It will use 1 as a signal of positive result, and 0 to signal a negative results. However, failure to halt also indicates, but is not a signal to a real use,  a negative result. So failure of the program to halt doesn’t signal a negative result, but it does indicate a negative result.

“A rule is a program that takes as input a list of statements and outputs a list of statements…A valid rule is a rule with the property that whenever the input list consists of only true statements, the output list also consists of only true statements.”

The output list will include the input list as a sublist, and that the rule halt for all input lists.

Take the program AND. This specific program expects as an input a list of two statements. These two statements are [A,B]. The AND Program first checks the truth of A in manner indicated above. If the program determines A is true, then it checks the truth of B. If B is true, then it outputs 1. Now if either A or B is false, the AND program fails to halt.

“A library is a list of rules used by an implication program in its proofs. We assume here that the library is finite at any given time. A valid library is one that contains only valid rules.”

“Consider the implication program ⇒ defined as follows. The program ⇒ expects as input a list of two statements [A, B]. Then it sets up and manipulates a list of sentences called the consequence list. The consequence list begins as a singleton list consisting only of A. The program ⇒ then goes to the library and chooses a rule. It applies the rule to the consequence list, and the result becomes the new consequence list. Since rules are required to include the input list as a sublist of the output list, once a statement appears on any consequence list it will appear on all subsequent consequence lists. After applying a rule, the program ⇒checks whether the consequent B is on the new consequence list. If so, it outputs 1; otherwise it chooses another rule, applies it to update the consequence list, and checks for B on the new consequence list. It continues to apply the rules in an exhaustive way until B is found, in which case ⇒outputs 1. If the consequent B is never found, the implication program ⇒does not halt.”

Take the Modus Ponens Program. This program expects an input list of statements, and from this it starts by forming empty result list. It searches the input list for any statement of the form [–>,[A,B],1] where A and B are statements. From all the statements, it searches to check if A is a statement on the input list. If A is found, then Modus Ponens Program adds B to the result list. The Program outputs a list that shows all the statements in the input list that are followed by all the statements of the result list (if any statements).

“The Modus Ponens program is a rule. A rule is valid if, for an input list of true statements, it only adds true statements. From the definition of –>, if [–>, [A,B],1] and A are on the input list and if they are both true and if the library is valid, then B will be true. So, MP is a valid rule if the library used by –> is valid. “

The EQ program expects an input list that contains [m,n], which are two natural numbers. Supposing m=n, then the EQ outputs 1, or outputs O. This is an example that some statements are clearly false. So let false be the false statement [EQ,[0,1],1]. If 0=1, then the EQ outputs 1, which is truth. This shows some statements are clearly false.

“Consider the program CURRY defined as follows. It expects a program X as input. Then it runs ⇒ as a subprocess with input [[X, X, 1], false]. The output of the subprocess (if any) is then used as the output of CURRY. If X checks for a particular property of programs, then the statement [X, X, 1] asserts that the program X has the very property for which it checks. The program CURRY when applied to program X can be thought of as trying to find a proof by contradiction that the statement [X, X, 1] does not hold.”

There is only way that CURRY can output 1 with input X. This is done by if –> outputs 1 with input [[X,X,1}, false].  This is what lies behind the Ad Hoc Rule (AH).

The AH expects a list of statements as input. From there it begins producing an empty result list. It than checks its input for statements that take the form of [CURRY, X, 1] where X is a program. For all such statements on input list, AH adds the statement [–>,[[X,X,1], false] to the result list. The AH will than construct a result list, which contains statements in the input list followed by the statements of the result list (if any).

AH is a valid rule because the statements on the input list are true and AH only adds true statements to form the output list. AH is ad hoc because it is specifically designed for the CURRY program.

“We now describe an algorithmic version of the Curry paradox. We assume that the library is valid and contains MP and AH. Consider what happens when we run ⇒ with input [[ CURRY, CURRY, 1], false]. First a consequence list containing the statement [ CURRY, CURRY, 1] is set-up. Next rules from the library are applied to the consequence list. At some point the Ad Hoc Rule AH is applied and, since [ CURRY, CURRY, 1] is on the consequence list, [⇒, [[CURRY, CURRY, 1], false], 1] is added to the consequence list. Because of this, when MP is next applied to the consequence list, false will be added to the list. Since the initial input had the statement false as the second item on the input list, ⇒will halt with output 1 when false appears on the consequence list.”

So the Implication Program outputs 1 with input of [[CURRY, CURRY, 1], false]. Based on the definition of CURRY Program, it implies that CURRY outputs 1 as CURRY is given as an input. Basically, the statement [CURRY, CURRY, 1] is true. A false statement is true.

Suppose that –> is applied to [[CURRY, CURRY, 1], false]. Because the antecedent [CURRY, CURRY, 1] is true, all statements added to the consequence list will also be true. But the statement false is added to the consequences list, which means that false is true, which is a contradiction.

The Curry Paradox has occurred in a concrete setting of a perfectly well-defined program and careful reasoning about the expected behavior.

The Curry Paradox proves that any library containing the Modus Ponens program and Ad Hoc Rule are not valid. AH is unconditionally valid, so we can conclude that MP is not valid in the case where all the other rules in the library are valid.

“We conclude from this that there are valid inference rules (including MP) that are valid only so long as they are not included in the library of rules to be used. Informally, we can say that there are valid rules that one is not allowed to use (in an unrestricted manner) in one’s proofs. It is the very usage of the rule in inference that invalidates it.”

In order to maintain a valid open library, one must check that the rule is valid itself and that it remains valid when added to the library. A rule is independently valid if it is valid regardless of which library is used by the implication library. The Ad Hoc Rule is an example of an independently valid rule. Any library consistent of only independently valid rules is valid.

The Mods Ponens rule isn’t independently valid. The Modus Ponens rule is contingent on the nature of the library. The Curry Paradox itself provides an example of libraries which MP is not valid.

It is thought that the source of the paradox can be considered to be the misuse of MP. It is suggested that modus ponens is the source of the classical Curry paradox

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Logical Analysis of Consciously Held Beliefs

Posted by allzermalmer on June 10, 2013

Axioms of Consciously Believing

Axiom: “I believe that p” if and only if “I believe that I believe that p”
EBpBBp = Bp <-> BBp

Axiom: “I don’t believe that p” if and only if “I believe that I don’t believe that p”
ENBpBNBp = ~Bp <-> B~Bp

Axiom: If “I believe that not p” then “I don’t believe that p”
CBNpNBp = B~p -> ~Bp

Axiom: If “I believe that p implies y” then “I believe that p implies I believe that y”
CBCpyCBpBy = B(p->y) -> (Bp->By)

Theorems of Consciously Believing

T1: “I don’t believe that both p and not p”
NBKpNp = ~B(p&~P)

T2: “I believe that p is equivalent to y” implies “I believe that p is equivalent to I believe that y”.
CBEpyEBpBy = B(p<->y) -> (Bp<->By)

T3: “I believe that p or I believe that y” implies “I believe that both p or y”
CABpByBApy = (BpvBy) -> B(pvy)

T4: “I believe that both p & y” if and only if “I believe that p & I believe that y”
EBKpyKBpBy = B(p&y) <-> (Bp&By)

T5: “I believe that p” implies “I don’t believe that not p”
CBpNBNp = Bp->~B~p

T6: “I believe that either p or y & I don’t believe that p” implies “I don’t believe not p”
CKBApyNBpNBNp = B(pvy)&~Bp -> ~B~p

T7: “I believe that either p or y & I believe that not p” implies “I believe that y”
CKBApyBNpBy = B(pvy)&B~p -> By

T8: If “I believe that p implies y” then “I believe that not y implies I believe that not p”
CBCpyCB~yB~p = B(p->y) -> (B~y->B~p)

T9: If “I don’t believe that p implies y” then “I don’t believe that not p & I don’t believe that y”.
CNBCpyKNBNpNBy = ~B(p->y) -> (~B~p&~By)

T10: If “I believe that I believe that p or I believe that y” then “I believe that either p or y”
CBABpByBApy = B(BpvBy) -> B(pvy)

T11: If “I believe that p implies y” then “I believe that I believe that p implies y”
CBCpyBCBpy = B(p->y) -> B(Bp->y)

T12: “I believe that I believe that p implies p”
BCBpp = B(Bp->p)

http://video.msnbc.msn.com/mitchell-reports/34510812#52158575

http://bcove.me/bqpor8gd

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Max Tegmark and Multiverse Hypothesis

Posted by allzermalmer on May 26, 2013

Max Tegmark, a theoretical physicist that teaches at the Massachusetts Institute of Technology, has proposed that hypothesis that “all logically acceptable worlds exist“. Not only has Max Tegmark proposed this hypothesis itself, he believes that it is an empirical hypothesis or scientific hypothesis.

Possibly and Necessarily: Modal Logic

Before I go into some of the ideas proposed by Tegmark, I will first go into a rough sketch of a form of logic known as Modal logic. More specifically, this form of modal logic is known as the S-5 system of modal logic and was formally created by Clarence Irving Lewis, C.I. Lewis. This system of logic plays off of the ideas of possible and necessary discussed about by Gottfried Wihelm von Leibniz, G.W. Leibniz.

Possible and Necessary are interchangeable, or we may define one based on the other. We may define them as so:

(1) Necessarily so if and only if Not possibly not so
(2) Possibly so iff Not necessarily not x so
(3) Not possibly so iff Necessarily not x so
(4) Possibly not so iff Not necessarily so

Truth is defined based on Necessary and Possible, which is done by Possible Worlds. A statement is Necessary if it is true in every possible world. A statement is Possible if it is true in some possible world.

There are some axioms in Modal Logic, and one of them is what I shall call NP: Whatever is necessarily so is actually so. It is necessarily so implies it is actually so. If it is necessarily so then it is actually so.

NA, in conjunction with some other axioms of modal logic and some rules of inference, is a theorem derived in modal logic. This theorem I shall call AP: Whatever is actually so is possibly so. It is actually so implies that it is possibly so. If it is actually so then it is possibly so.

One inference of Modal Logic is what I shall call GR: Whatever is provably so is necessarily so. It is provably so implies it is necessarily so. If it is provably so then it is necessarily so.

One comment is required of GR. Pythagorean Theorem is provably so, and in fact has been proved to be so, so it is necessarily so. It was proved based on a formal system known as Euclidean Geometry, which has its own definitions, axioms, and rules of inference. From these we are able to prove some statements. These proved statements show that it’s negation is not possible, and so the processes of elimination leads to that proved statement necessarily being so.

(GR) Whatever is provably so is necessarily so; (NP)Whatever is necessarily so is actually so; Thus Whatever is provably so is actually so. This in turn means that AP is actually so since it was proved like the Pythagorean Theorem was proved. Since AP being provably so implies AP is necessarily so. And since AP is necessarily so, AP is actually so.

All that is logically possible to be the case is actually the case

Max Tegmarks hypothesis is the converse of AP. We may call this MH: Whatever is possibly so is actually so. It is possibly so implies it is actually so. If it is possibly so then it is actually so.

We may thus assume MH is true and assume that AP is true. Since both of these are assumed true, they form a logical equivalence. We may call this *MH*: Whatever is actually so is possibly so if and only if Whatever is possibly so is actually so. If it is actually so implies it is possibly so then  it is possibly so implies it is actually so.

Max Tegmark presents his hypothesis, similar to how Albert Einstein presented Special Relativity, by his hypothesis being based on two assumptions. One of these assumptions, as already previously stated is MH. The second hypothesis of Max Tegmark is what we may call EW: There exists an external physical reality and it is independent of human observers.

So Tegmark’s two assumptions are as follows:

EW: There exists an external physical reality and it is independent of human observers.
MH: Whatever is possibly so is actually so.

EW is an existential statement and MH is a universal statement. This is very important to keep in mind, as shall be shown later on.

Mr. Tegmark prefers to talk about MH being something like this, “Our external physical reality is a mathematical structure”. A mathematical structure, or mathematical existence, is “merely freedom from contradiction.” I use MH as I do because the definition of mathematical existence is the same as possible. For something to be possible it must not contain a contradiction. For something to be impossible it must contain a contradiction.

Euclid’s geometry, for example, is a mathematical structure, and also has a mathematical existence. This means that Euclid’s geometry is “free from contradiction”. One cannot derive a contradiction within Euclid’s geometry.

We may say that there are two categories. There is what is possible and there is what is impossible. What is possible contains two sub-categories. These are Necessary and Contingent. Something is necessary because it not being actual is impossible. Something is contingent because it not being actual is possible and it being actual is possible. For example, it is necessary that all bachelors are unmarried males and it is contingent that all like charges repel.

Mathematics and Logic, at least, deal with what is Necessary. Metaphysics and Science deal with what is Contingent. The Criterion of Demarcation, or Line of Demarcation, between Metaphysics and Science, or Metaphysical Arguments and Empirical Arguments, is Falsifiability. Falsifiability was first laid out by Karl Popper in his book The Logic of Scientific Discovery, and throughout his other writings.

Some Criterion of Falsifiability for Empirical Hypothesis

There is one thing that all hypothesis must conform to, which is that of consistency, i.e. don’t allow contradictions. Necessary statements obviously conform to this, and Contingent statements are also suppose to follow consistency.

“The requirement of consistency plays a special role among the various requirements which a theoretical system, or an axiomatic system, must satisfy. It can be regarded as the first of the requirements to be satisfied by every theoretical system, be it empirical or non-empirical…Besides being consistent, an empirical system should satisfy a further condition: it must be falsifiable. The two conditions are to a large extent analogous. Statements which do not satisfy the condition of consistency fail to differentiate between any two statements within the totality of all possible statements. Statements which do not satisfy the condition of falsifiability fail to differentiate between any two statements within the totality of all possible empirical basic statements.” Karl Popper

Karl Popper points out, basically, that both metaphysics and science must adhere to consistency. One of the ways to refute a hypothesis is to show that it leads to a contradiction, which is known as a Reductio Ad Absurdum. You assume the opposite of a statement, and from this assumption you deduce a contradiction from the assumption. This proves the statement derived to be necessarily true, since its negation is impossible.

One tests of Scientific hypothesis is to make sure it is consistent with all other scientific hypothesis (generally, unless a new hypothesis that alters the edifice of science like Galileo and Einstein did). Another test is to show that the hypothesis is internally consistent.

Max Tegmark’s hypothesis, which contains both EW and MH are contradictory to one another. This is because MH allows for, what I shall call IW: There exists world and it is not independent of human observers. IW does not state how many human observers there are. There could be only one human observer, which is solipsism, or there can be infinitely many human observers, i.e. Human observer + 1 or N+1. MH allows for these possibilities, since there is no contradiction in such a situation. This implies that there exists a possible world where I am the only human observer, and it also implies that you,the reader, exists in a possible world where you are the only human observer. This also implies there exists a possible world in which only you the reader and I are the only inhabitants of a possible world where we are only human observers, and etc and etc.

Instead of accepting MH itself, which means both accepting EW and IW, Max Tegmark accepts only a part of MH by accepting only EW and denying IW. MH is both being affirmed and denied since denying a part of MH and accepting a part of MH. This would also follow by a simple example of Modus Tollens.

(1) All logically possible worlds exist implies there exists an external physical reality and it is independent of human observers and there exists a world and it is not independent of human observers.
(2) There doesn’t exist a world and it is not independent of human observers. (Because of EW)
(3) Thus, not all logically possible worlds exist. (Thus, Not MH)

The general point is that it is logically possible that there exists a world and it is dependent on human observers. But it is also possible that there exists a world and it is not dependent on human observers. Both of these are contained in MH, and Tegmark denies one but accepts the other, while also accepting MH. This would be similar to holding to the Theory of Special Relativity (which would be MH here) as a whole and accepting the first postulate (which would be EW here) and denying the second postulate (which would be IW). This is impossible since the Theory of Special Relativity is defined by both postulates together.

“A theoretical system may be said to be axiomatized if a set of statements, the axioms, has been formulated which satisfies the following four fundamental requirements. (a) The system of axioms must be free from contradiction (whether self-contradiction or mutual contradiction). This is equivalent to the demand that not every arbitrarily chosen statement is deducible from it. (b) The system must be independent, i.e. it must not contain any axiom deducible from the remaining axioms. (In other words, a statement is to be called an axiom only if it is not deducible within the rest of the system.) These two conditions concern the axiom system as such;” Karl Popper (Bold is my own emphasis and Italics are Popper’s own emphasis.)

It has already been shown that Tegmark’s hypothesis already violates (a). But Tegmark’s hypothesis also violates (b). This means that the two axioms of Tegmark’s hypothesis (MH & EW) are not independent of each other. We may deduce EW from MH, which means that EW is not independent of MH. It would be charitable to believe that Tegmark doesn’t hold that EW is not possible, which means that Tegmark doesn’t believe that EW is impossible.  But MH deals with everything that is possible. And so EW would be possible and thus be part of MH.

These two “proofs” don’t assume that Max Tegmark’s hypothesis aren’t an empirical hypothesis, but they are consistent with Max Tegmark’s hypothesis not being an empirical hypothesis, i.e. consistent with Max Tegmark’s hypothesis being a metaphysical hypothesis. These are also theoretical proofs, not practical or “empirical proofs” themselves.

There are two steps at falsifiability. One of them is that we show that the theoretical structure of the hypothesis is not itself contradictory. If the theoretical structure is not found to be contradictory, then we try to show that the theoretical structure is contradictory with empirical observations. If the theoretical structure is contradictory with the empirical observations, then the theoretical structure is falsified. First we try to show that the theoretical structure is contradictory or we try to show that the theoretical structure is contradicted by the empirical observations.

There will always be partial descriptions

The paper “A Logical Analysis of Some Value Concepts” was written by the logican Frederic B. Fitch, and appeared in the peer-review journal called The Journal of Symbolic Logic, Vol. 28, No. 2 (Jun., 1963), pp. 135-142.In this paper, a formal system was created for dealing with some “Value Concepts” like “Truth”, “Provability”, “Knowledge”, “Capability”, and “Doing”, to name a few. This deals with an abstract relationship, one as usually described by formally consistent systems like S-5 Modal logic.

What Frederic Fitch presents in the paper is what Tegmark would call a “Mathematical Structure”. This “Mathematical Structure” also has some Theorems that are proved within it. Like AP was a Theorem in a “Mathematical Structure” known as S-5 Modal Logic and the Pythagorean Theorem is a “Mathematical Structure” in Euclidean Geometry, so too are there two specific Theorems that are counter-intuitive, and can both respectively be called the “Knowability Paradox” and “Provability Paradox”. These are, respectively, Theorem 5 and Theorem 6 in Fitch’s paper.

Being Theorems, by the rule of inference GR, they are proved to be the case then they are necessarily the case. Whatever is provably so  is necessarily so. By MP, whatever is necessarily so is actually so. So Theorem 5 and Theorem 6 are actually so, which is also consistent with the hypothesis of Tegmark with MH, i.e. whatever is possibly so is actually so. Which in turn means that Tegmark would have to accept that Theorem 5 and Theorem 6 are true if they accept that their hypothesis MH is true.

Theorem 5, the “Knowability Paradox”, states that “If there is some true proposition which nobody knows (or has known or will know) to be true, then there is a true proposition which nobody can know to be.”

Some equivalent ways of stating Theorem 5 is such as: It is necessary that it isn’t known that both “P is true” & it isn’t known that “P is true”. It isn’t possible that it is known that both “P is true” & it isn’t known that “P is true”. The existence of a truth in fact unknown implies the existences of a truth that necessarily cannot be known. There exists such a true statement that both statement is true & for every agent no agent knows that statement is true implies there exists a true statement that both statement is true and for every agent it isn’t possible agent knows that statement is true.

Theorem 6, the “Provability Paradox”, states that “If there is some true proposition about proving that nobody has ever proved or ever will prove, then there is some true proposition about proving that nobody can prove.”

Some equivalent ways of stating Theorem 6 is such as: It is necessary that it isn’t provable that both “P is true” & it isn’t provable that “P is true”. It isn’t possible that it is provable that both “P is true” & it isn’t provable that “P is true”. The existence of truth in fact unproven implies the existence of a truth that necessarily cannot be proven.There exists such a true statement that both statement is true & for every agent no agent proves that statement is true implies there exists a true statement that both statement is true and for every agent it isn’t possible agent proves that statement is true.

These two Theorems show that it is necessary that agents, like human observers, know everything that can be known by those agents and proved everything that can be proven by those agents. This implies that Goldbach’s Conjecture, which hasn’t been proven to be true by human observers or proven false, cannot possibly be proven true or proven false. It will forever remain unprovable to human observers. It also implies that MH, or  cannot possibly be known and will forever remain unknown. This would also hold for all agents, which are not omniscient agents. These is necessarily so and means it is actually so, especially by MH and GR.

This is interesting because MH is presented as a hypothesis that is possibly true and it is not known that it is true or false. But since it is not known to be true and it is not known to be false, it cannot known to be true or false. MH, in conjunction with GR and Fitch’s Theorems, tells us that it cannot be known to be true or false and that it also isn’t provable that it is true or false, i.e. unprovable that it is true or false.

The Knowability Paradox and Provability Paradox also attack one of the aspects of Tegmark’s hypothesis, which is that of EW. EW implies that other agents that are not human observers, which can be supercomputers or aliens, would also fall for these paradoxes as well. This shows that we can never have a complete description of the world, but can only have a partial description of the world. This means that human observers, supercomputer observers, or alien observers, all cannot have a complete description of the world. We, the agents of EW, will never have a complete description.

What is interesting is that both paradoxes are very closely aligned with IW, or lead one to accept IW as true. Sometimes pointed out that the Knowability Paradox leads to Naive Idealism, which is part of IW and is thus not part of EW. This, in some sense would appear to imply that MH again implies another contradiction.

Must a Mathematical Structure be Free from Contradiction? 

“Mathematical existence is merely freedom from contradiction…In other words, if the set of axioms that define a mathematical structure cannot be used to prove both a statement and its negation, then the mathematical structure is said to have [Mathematical Existence].” Max Tegmark

Does mathematical existence really have to be freedom from contradiction? It is possible to develop formal systems that allow for both violations of non-contradiction and violations of excluded middle. A formal system of such a sort was developed by Polish logical Jan Lukasiewicz. This logic was created by using three values for logical matrices than two values.

Lukasiewicz three value logic has been axiomatized, so that there axioms, definitions, and logical relationships between propositions. And from this three value logic one may obtain violations of non-contradiction and violations of excluded middle. If there is a violation of non-contradiction then there is a violation of mathematical existence.

As Tegmark points out, A formal system consists of (1) a collection of symbols (like “~”, “–>”, and “X”) which can be strung together into strings (like “~~X–>X” and “XXXXX”), (2) A set of rules for determining which such strings are well-formed formulas, (3) A set of rules for determining which Well-Formed Fomrulas are Theorems.

Lukasiewicz three value logic satisfy all three of these criterion for a formal system.

The primitives of Lukasiewicz’s three valued calculus is negation “~”, implication “–>”, and three logical values “1, 1/2, and 0”. 1 stands for Truth, 1/2 stands for Indeterminate, and 0 stands for False. From negation and implication, with the three values, we can form a logical matrices of both negation and implication. And from these primitive terms we may define biconditional, conjunction, and disjunction as follows:

Disjunction “V” : (P–>Q)–>Q ; Conjunction “&” : ~(~P–>~Q) ; Biconditional “<—>” : (P–>Q) & (Q–>P)

“&” is symbol for Conjunction, “V” is symbol for Disjunction, “<—>” is symbol for Biconditional. Lukasiewicz’s Three-value calculus have the following truth tables:
Luk Truth TLukasiewicz’s axioms are as follows:
[Axiom 1] P –>(Q –>P)
[Axiom 2] (P –>Q ) –>(( Q–>R) –>(P –>R))
[Axiom 3](~Q –>~P ) –> (P –>Q)
[Axiom 4] ((P –>~P) –>P) –>P

Lukasiewicz’s rule of inference was Modus Ponens, i.e. Rule of Detachment:
(Premise 1) P –> Q
(Premise 2) P
(Conclusion) Q

From this it becomes obvious that formal systems do not need to be free from contradictions. This formal system allows for both (P & ~P) to have a truth value of neither True nor False. This is because, as the Conjunction Truth table shows, P= 1/2 or Indeterminate and ~P= 1/2 or Indeterminate is a well formed formula that is itself Indeterminate.

Does this mean that mathematical structures must be free from contradiction? It appears that Lukasiewicz’s formal system, and there are some others that can be created, show that mathematical structures and thus mathematical existence, do not need to follow the being free from contradiction. Lukasiewicz’s formal system can be expanded to allow for infinite number of truth values.

One important part of Tegmark’s idea of MH, which implies EW, is that it prohibits Randomness. He states that “the only way that randomness and probabilities can appear in physics (by MH) is via the presence of ensembles, as a way for observers to quantify their ignorance about which element(s) of the ensemble they are in.” Now Lukasiewicz’s logic can be the way our actual world is. This would mean that the world is random or indeterminate. Lukasiewicz’s even himself says that his three value logic is based on the position of indeterminacy, which is contradictory to determinacy.

[This post will be updated at sometime in the future….with more to come on this subject.]

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