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Posts Tagged ‘Modal Logic’

Knowability Paradox and Modal Realism

Posted by allzermalmer on August 17, 2013

Equivalency is defined this way: (p–>q)<–>(q–>p)

So if we assume (p–>q) is true and (q–>p) is true, then it necessarily follows that (p–>q) if and only if (q–>p).

(1) p–><>p
(2) <>p–>p [Modal Realism]
(3) (p–><>p)<–>(<>p–>p)

There is one thing that must be made clear.

(1) is necessarily true. It is not possible that it is not true. It is an axiom of modal logic. Now (2) isn’t necessarily true. It is possible that it is not true. It is not an axiom or theorem of modal logic.

(1) can be substituted with variable of x. So x if and only if p–><>p. (1) being necessarily true implies necessarily x. (1) and x, are analytically true.

(2) can be substituted with variable of y. So y if and only if <>p–>p. (2) being not necessarily true implies possibly not y. (2) or y, are contingently true.

(3), based on substitutions of (1) & (2), takes on the form of x<–>y. Or we can say analytically true if and only if contingently true.

x is analytic implies either necessarily x or necessarily not x. [](p–><>p) v []~(p–><>p).
y is contingent implies possibly y and possibly not y. <>(<>p–>p) & <>~(<>p–>p).

(3) shows that we have collapsed any modal distinction between possibility and actuality. There is no modal difference between possibly true and actually true. This specific proposition presents that possible if and only if actual.

(4) p–>Kp [Fitch’s Theorem]
(5) Kp–>p
(6) (p–>Kp)<–>(Kp–>p)

There is one thing that must be made clear.

(5) is necessarily true. It is not possible that it is not true. It is an axiom of epistemic logic. Now (4) is also necessarily. It is not possible that it is not true. It is a theorem of epistemic logic.

(4) can be substituted with variable of x. So x if and only if p–>Kp. (4) being necessarily true implies necessarily x. (4) and x are necessarily true.

(5) can be substituted with variable of y. So y if and only if Kp–>p. (5) being necessarily true implies necessarily y. (4) and y are necessarily true.

(6), based on substitutions of (4) & (5), takes on the form of x<–>y. Or we can say analytically true if and only if analytically true

x is analytic implies either necessarily x or necessarily not x. [](p–>Kp) v []~(p–>Kp).
y is analytic implies either necessarily y or necessarily not y. [](Kp–>p) v []~(Kp–>p).

(6) shows that we have collapsed any epistemic distinction between knowledge and truth. There is no difference between knowing something is true and something is true. What this specific proposition presents is that truth if and only if knowledge

Equivalency: ECpqCqp or ECNqNpCNpNq

(1) CpMp
(2) CMpp
(3) ECpMpCMpp

(4) CpKp
(5) CKpp
(6) ECpKpCKpp

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The Logic of Discussions

Posted by allzermalmer on June 7, 2013

This blog will be based on a logical system developed by Polish logician Stanislaw Jakowski. It was published in the journal Studia Logica: An International Journal for Symbolic Logic, T. 24 (1969), pp. 143-1960

Implication (–> or C), Conjunction (& or K), Disjunction (v or A), Equivalence (<–> or E), Negation (~ or N, Possibility (<> or M), Necessity ([] or L),  and Variables (P or p, Q or q, R or r).

P = <>P or p = Mp
<>P = ~[]~P or Mp = NLNp

Discussive Implication (D): P–>Q = <>P–>Q or Dpq = CMpq
Discussive Equivalence (T): P<–>Q = (<>P–>Q) & (<>Q–><>P) or Tpq = KCMpqCMqMp

D1: P–>P = Dpp
D2: (P<–>Q) <–> (Q<–>P) = TTpqTqp
D3: (P–>Q) –> ((Q–>P)–>(P<–>Q)) = DDpqDDqpTpq
D4: ~(P&~P) = NKpNp [Law of Contradiction]
D5: (P&~P) –>Q = DKpNpq [Conjunction Law Overfilling]
D6: (P&Q) –>P = DKpqp
D7: P –> (P&Q) = DpKpq
D8: (P&Q) <–> (Q&P) = TKpqKqp
D9: (P&(Q&R)) <–> ((P&Q)&R) = TKpKqrKKpqr
D10: (P–>(Q–>R)) –> ((P&Q)–>R) = DDpDqrDKpqr [law importation]
D11: ((P–>Q)&(P–>R)) <–> (P–>(Q&R)) = TKDpqDprDpKqr
D12: ((P–>R)&(Q–>R)) <–> ((PvQ)–>R) = TKDprDqrDApqr
D13: P <–> ~~P = TpNNp
D14: (~P–>P) –> P = DDNppp
D15: (P–>~P) –>~P = DDpNpNp
D16: (P<–>~P) –> P = DTpNpp
D17: (P<–>~P) –>~P = DTpNpNp
D18: ((P–>Q)&~Q) –>~P = DKDpqNqNp

D19: ((P–>)&(P–>~Q)) –>~P = DKDpqDpNqNp
D20:  ((~P–>Q)&(~P–>~Q)) –> P = DKDNpqDNpNqp
D21:  (P–>(Q&~Q)) –>~P = DDpKqNqNp
D22:  (~P–>(Q&~Q)) –> P = DDNpKqNqp

D23: ~(P<–>~P) = NTpNp
D24: ~(P–>Q) –> P = DNDpqp
D25: ~(P–>Q) –> ~Q = DNDpqNq
D26: P–> (~Q–>~(P–>Q)) = DpDNqNDpq

A formulation of Aristotle’s Principle of Contradiction would be: “Two contradictory sentences are not both true in the same language” or “Two contradictory sentences are not both true, if the words occurring in those sentences have the same meanings”.

In Two Valued Logic, there is a Theorem known as the Law of Overfilling, or Implicational Law of Overfilling, or Dun Scotus Law, or L2 Theorem 1.

L2 Theorem 1: P—> (~P—>Q)

If an assertion implies its contradiction, then that assertion implies any and all statements.

“A deductive system…is called inconsistent, if its theses include two such which contradict one another, that is such that one is the negation of the other, e.g., (P) and (~ P) . If any inconsistent system is based on a two valued logic, then by the implicational law of overfilling one can obtain in it as a thesis any formula P which is meaningful in that system. It suffices…to apply the rule of modus ponens twice[ to P—> (~P—>Q)]. A system in which any meaningful formula is a thesis shall be termed overfilled.”

  1. Assume: P—> (~P—>Q)
  2. Modus Ponens: P
  3. Conclusion: ~P—>Q
  4. Modus Ponens: ~P
  5. Conclusion: Q

“[T]he problem of the logic of inconsistent systems is formulated here in the following manner: the task is to find a system of the sentential calculus which: (1) when applied to the inconsistent systems would not always entail overfilling, (2) would be rich enough to enable practical inference, (3) would have an intuitive justification. “

This means that Discussive Logic does not have the theorem of implicational law of overfilling. The theorem is not always true in Discussive Logic. Discussive Logic does not entail that a contradiction does not always entail any and all assertions. So Discussive Logic rejects the truth of the theorem P—> (~P—> Q), which is a theorem is two value logic, i.e. been proven true under conditions of two value logic.

Kolmogorov’s System

Here are Four axioms from Hilbert’s positive logic, and one axiom introduced by Kolmogorov.

K 1: P—> (Q—>P)
K 2: (P—> (P—>Q))—> (P—>Q)
K 3: (P—> (Q—>R))—> ((Q—> (P—>R))
K 4: (Q—>R)—> ((P—>Q)—> (P—>R))
K 5: (P—>Q)—> ((P—>~Q)—>~P)

Under these axioms, Two valued logic cannot be proved. Implicational Law of Overfilling not being provable in Discussive Logic implies that Two Valued logic cannot be proved in Discussive Logic. This entails that there might be overlap between Two Valued logic and Discussive Logic, but there is not a total overlap between Two Valued logic and Discussive Logic. Not all theorems of Two Valued logic will be theorems in Discussive logic (like law of overfilling), but some theorems of two valued logic are theorems in Discursive logic.

From these Axioms and the rule of inference known as Modus Ponens, there is one theorem which has some similarities to implicational law of overfilling.

K 9: P—> (~P—>~Q)

It is not the only Theorem that can be derived from the Axioms and Modus Ponens. Here is a list of some Theorems that can be derived by using  Modus Ponens on the Axioms.

K 6 : (P—>Q)—> ((Q—R)—> (P—>R))
K 7:  ((Q—>P)—>R)—> (P—>R)
K 8:  P—> ((Q—>~P)—> ~Q)
K 9:  P—> (~P—>~Q)

Proof of how K 6 – K 9 are derived are ignored for here. All that needs to be known is that K3 and applications of Modus Ponens is equal to If K4 then K 6. K 6 and applications of Modus Ponens is equal to If K 1 then K 7. K 7 and applications of Modus Ponens is equal to If K 5 then K 8. K 6 and applications of Modus Ponens is equal to If K 8 then K 7 implies K 9.

This forms Kolmogorov’s System.

Lewis System of Strict Implication

Strict Implication is defined by modal operator of “it is possible that P” or <>P. So “P strictly implies Q” is equal to “It is not possible that both P and not Q”. But taking the conditional statement —> as strict implication means that the implicational law of overfilling is not a theorem.

Material Implication as a conditional is usually defined by the logical relationship of a conjunction.

Material Conditional: P—>Q = ~(P & ~Q)
“P implies Q” is equal to “Not both P and not Q”
Strict Conditional: P—» Q = ~<>(P & ~Q)
“P strictly implies Q” is equal to “It is not possible that both P and not Q”

Under Strict Implication, Law of Overfilling is not a theorem. Under Material Implication, Law of Overfilling is a theorem. And set of theorems which include only strict implication and not material implication is very limited.

Many Valued Logics

Based on a certain Three Value logical matrix, which shall be ignored, the Law of Overfilling is not a theorem. But there is another theorem in the Three Value logic which has some similarity to the Law of Overfilling.

L 1: P—> (~P—> (~~P—> Q))

Based on the theorem (stated above) of this specific three valued logic, it holds the overfilling of a system when it includes the inconsistent thesis of P, ~P, and ~~P. And the implicational theses of two valued calculus remains valued in the three valued logic. But the three valued logic also holds other theorems that are not in two valued logic, which are as follows.

L 2: P—> ~~P
L 3: ~~~P—> P
L 4: ~P—> ~(P—> P)

So in the three valued logic, which is ignoring the logical matrix of this three valued logic, we cannot obtain the Law of Overfilling. The Law of Overfilling will thus be a theorem in two valued logic but not a theorem in this three valued logic. But the three valued logic has a theorem that is similar to the Law of Overfilling but is not equivalent to the Law of Overfilling. This three value logic also has some theorems that are not theorems in two valued logic. Besides the Law of Overfilling not being a theorem in the three valued logic, the rest of implicational theorems in two valued logic are theorems in the three valued logic.

Calculus of Modal Sentences (M2)

The Modal Sentences of (M2) will assume that modal assertions are either true or false, or simply that the Modal sentences are two valued. But now suppose that there are factors that do not allow for the assertion P to be determined strictly to be either true or false.

For example: Suppose that you are flipping a coin. Suppose that you make the assertion that “During the game heads will turn up more times than tails will” and this is represented by the variable of P. There will be certain sequences that turn up so that P is true, and there will be certain sequences that turn up so that P is false. So P may take on both true and false.

“It is necessary that P” = []P

Taking the example above, we can say that “P occurs for all possible events”.

Q is any formula that includes (1) operators —>, V, &, <—>, ~ and [], (2) and variables p,q,r,s..etc. R is any formula that is already a Q formula and is replacements of variables in Q by interpreting them as P(x), Q(x), R(x), S(x)…etc, and interpreting [] by universal quantifiers “for every x”. Every Q satisfies (1) and (2) and every R satisfies (1) and (2), and additionally satisfying (3).

The operators are implication, disjunction, conjunction, equivalence, and necessity. These are applied to variables or connects variables. When those conditions are met, then it is a formula of Q. The replacement of the variables and [] are formula of R. (1) and (2) can be recognized as P–>Q, or []P—>Q, or []P—>P. We can replace those variables to formulas in R: P(x) —> Q(x), or For every x, P(x) —> Q(x), or For every x, P(x) —> P(x).

“It is possible that P” = <>P

<>P can be taken as “it is not necessary that not P”.

<>P = ~[]~P

Like we could change []P into “for every x”, we may also change <>P into “for some x”.

Definition of Discussive Implication and Discussive Equivalence

As is known, even sets of those inscriptions which have no intuitive meaning at all can be turned into a formalized deductive system. In spite of this theoretical possibility, logical researches so far have been taking into consideration such deductive systems which are symbolic interpretations of consistent theories, so that theses in each such system are theorems in a theory formulated in a single symbolic language free from terms whose meanings are vague.

But suppose that theses which do not satisfy those conditions are included into a deductive system. It suffices, for instance, to deduce consequences from several hypotheses that are inconsistent with one another in order to change the nature of the theses, which thus shall no longer reflect a uniform opinion. The same happens if the theses advanced by several participants in a discourse are combined into a single system, or if one person’s opinions are so pooled into one system although that person is not sure whether the terms occurring in his various theses are not slightly differentiated in their meanings. Let such a system which cannot be said to include theses that express opinions in agreement with one another, be termed a discussive system. (Italics is authors and Bold is mine)

Each the theses in discussive logic are preceded so that each thesis has the speaker has the reservation such that each assertion means  “in accordance with the opinion of one of the participants in the discussion” or “for a certain admissible meaning of the terms used”. So when you add an assertion to a discussive system, that assertion will have a different intuitive meaning. Discussive assertions have the implicit condition of the equivalence to <>P.

King Solomon having to decide between two harlots claiming to be the mother of a baby. Woman A claimed to be the mother of the baby and not the mother of the dead baby, and Woman B claimed to be the mother of the baby and not the mother of the dead baby. King Solomon being the arbitrator, under Discussive assertions, would have taken each Woman’s claim as having the prefix of possibility, or “it is possible that Woman A is the mother” or “it is possible that Woman A is not the mother”.

Discussive logic is not based on ordinary two valued logic. Discussive logic would not hold Modus Ponens in all cases if it did.

Take the statement P—>Q is asserted in a discussion. It would be understood to mean “It is possible that If P, then Q”. P is asserted in the same discussion. It would be understood to mean “It is possible that P”. Q would not follow from the two assertions in the discussion. For by Q would not follow in the discussion because Q stands for “It is possible that Q”. So it is invalid to infer from “It is possible that if P, then Q” and “It is possible P” that “It is possible that Q”. But people might assume the normal two value logic in which Modus Ponens holds in all cases.

For Discussive Logic, Discussive Implication is defined as such:

Definition of Discussive Implication: P—>Q = <>P—>Q

There is a theorem of M2 based on Discussive Implication.

M2 Theorem 1: <>(<>P—>Q) —> (<>P—><>Q)

So Modus Ponens may be used in Discussrive Logic when we understand that from (<>P—><>Q) and <>P, we may infer that <>Q by Modus Ponens.

For Discussive Logic, Discussive Equivalence is defined as such:

Definition of Discussive Equivalence: P <—> Q = (<>P—>Q) & (<>Q—><>P)

There is a theorem of M2 based on Discussive Implication.

M2 Theorem 2: <> (P<—>Q) —> (<>P—> <>Q)
M2 Theorem 3: <> (P <—> Q) —> (<>Q —> <>P)

Two valued Discussive System of Sentential Calculus: D2

The system of D2 (i.e. Discussive Logic) of two valued discussive sentential calculus is marked by the formula T, and are marked by the following properties: (1) Sentential variables and functors of Discussive Implication, Discussive Equivalence, Disjunction, Conjunction, and Negation. (2)  precedening T with the symbol of <> yields a theorem in two valued sentential calculus of modal sentences M2.

As the author says, “The system defined in this way is discussive, i.e., its theses are provided with discussive assertion which implicitly includes the functor <>/ This is an essential fact, since even such a simple law as P—>P, on replacement of —> with —-> (i.e. discussive implication leads) to a new theorem.”

D2 Theorem 1: P—>P

D2 is not a theorem in M2, specifically because M2 did not have discussive implication. But in order to make D2 theorem 1 into a theorem in M2, you have to add <> to D2 theorem 1 like this:

M2 Theorem 4: <>(P—>P)

System M2 is decidable, so the discussive sentential calculus D2, defined by an interpretation in M2 is decidable too.

Methodological Theorem 1: “Every thesis T in two valued sentential calculus which does not include constant symbols of —>, <—>, V, becomes a thesis in T(d) in discusive sentential calculus D2 when in T the implication symbols is replaced by the [discurssive implication], and the equivalence symbols are replaced by [discusrive equivalence]. “

“Proof. Consider a formula T(d) constructed so as the theorem to be proved describes. It is to be demonstrated that <>T(d) is a thesis in M2. It is claimed that <>T(d) is equivalent to some other formulae; the equivalences will be proved gradually.”

Here are a couple more M2 Theorems.

M2 Theorem 5: <>(P—>Q) <—> (<>P—><>Q)
M2 Theorem 6: <>(P <—> Q) <—> (<>P <—> <>Q)
M2 Theorem 7: <>(P v Q) <—> (<>P v <>Q)

These theorems are about the distribution of <> over the variables. For example, M2 Theorem 5 distributes <> over implication, and M2 Theorem 6 distributes <> over equivalence, and M2 Theorem 7 distributes over Disjunction. M2 Theorem 5 and M2 Theorem 6 have Discussive Implication and Discussive Implication as the antecedents, respectively.

This shows how we can replace Discurssive Implication with regular implication and how we can replace Discurssive Equivalence with regular equivalence. So from <>(P—>Q), which contains Discurssive Implication, can be replaced with regular implication as <>P—><>Q. The form <>(P<—>Q), which contains Discurssive Equivalence, can be replaced with regular implication as <>P,—><>Q. Discurssive assertion like <>(PvQ) has the equivalence in M2, or Modal Logic, as <>P v <>Q.

The procedure yields the formula W, which is equivalent to <>T(d) and includes (1) only the symbols —>, <—>, and V, (2) variables, and (3) symbols <> in certain special positions, like each variable is directly preceded by <> and each symbol <> directly precedes a variable. Forming T(d) from the thesis T belonging to two value logic is possibly be seen that W can be obtained from T by preceding each variable by <>. For example, precede the variable P by <>P or precede the variable Q by <>Q. This procedure would yield the following theorems in M2.

(a) W is a result of the substitution in T
(b) <>T(d) is equivalent to W.
Hence T(d) is a thesis of D2

The theorems just listed above, immediately yields these theorems in Discussive Logic:

D2 Theorem 2: (P<—>Q) <—> (Q<—>P)
D2 Theorem 3: (P—>Q) —> ((Q—>P) —> (P<—>Q))

Each of the connectives in the D2 theorem just listed are Discurssive Equivalence for D2 Theorem 2 and Discurssive Implication for D2 Theorem 3.

Methodological Theorem 2: If T is a thesis in the two valued sentential calculus and includes variables and at the most the functors V, &, ~, then (1) T and (2) ~T —> q, are thesis in D2. The implication of (2) is Discurssive Implication.

Proof: The symbols V, &, and ~ retain respective meanings in M2 and D2, and that (3) []T is a thesis in M2. The symbols V, &, and ~ retain respective meanings in M2 and D2 and that (3) []T is a thesis in M2. Hence (1) by M2 Theorem 8 []P—><>P and (2) by M2 Theorem 9 []P—><>(<>~P—>Q).

M2 Theorem 8: []P—><>P
M2 Theorem 9: []P—> <>(<>~P—>Q)

We may apply the Methodological Theorem 2 to the Two Valued Logic Theorem of ~(Pv~P), i.e. Aristotle’s Principle of Contradiction.

L2 Theorem 3: ~(P&~P)

“Methodological Theorem 2 and Law of Contradiction in Two valued logic yieleds – in view of the law of double negation- the following theorem of Discussive logic.”

D2 Theorem 4: ~(P&~P) [Law of Contradiction]
D2 Theorem 5: (P&~P) —> Q [Conjunctional Law of Overfilling]

What these two theorems are basically stating is this: Suppose that we have an individual in a discussion, and this individual holds to the Discussive assertion of (P&~P), this individual would hold inconsistent opinions. And in Discussive logic, when an individual holds to inconsistent opinion, the persons opinion implies any and all discussive assertions. This basically forbids an individual from holding to discussive assertions that are contradictory to one another by D2 Theorem 4, and if we do hold to contradictory discussive assertions then any discussive assertion follows from the conjunction of contradictory discussive assertions. This is similar to Law of Overfilling in two value logic but not exactly the same.

We also have the following theorems in Discussive Logic.

D2 Theorem 6: (P&Q) —> P
D2 Theorem 7: P—> (P&P)
D2 Theorem 8: (P&Q) <—> (Q&P)
D2 Theorem 9: (P& (Q&R)) <—> ((P&Q) & R)
D2 Theorem 10: (P—> (Q—>R)) —> ((P&Q) —>R) [Law of Importation]
D2 Theorem 11:((P–>Q) & (P—>R)) <—> (P—>(Q & R))
D2 Theorem 12: ((P—>R) & (Q—>R)) <—> ((PvQ) —> R)
D2 Theorem 13:P <—> ~~P
D2 Theorem 14:(~P—>P) —>P
D2 Theorem 15:(P—>~P)>~P
D2 Theorem 16:(P<—>~P) —>P
D2 Theorem 17: (P<—>~P) —>~P
D2 Theorem 18: ((P—>Q) & ~Q) —> ~P

There are laws of inference by reductio ad absurdum that remain valid in Discurssive Logic.

D2 Theorem 19:((P—>Q) & (P—>~Q)) —> ~P
D2 Theorem 20:((~P—>Q) & (~P—>~Q)) > P
D2 Theorem 21:(P> (Q&~Q)) —> ~P
D2 Theorem 22: (~P —> (Q&~Q)) —>P

Here are some other theorems of Discurssive Logic.

D2 Theorem 23:~(P <—> ~P)
D2 Theorem 24:~(P—>Q) —>P
D2 Theorem 25: ~(P—>Q) —>~Q
D2 Theorem 26: P —> (~Q—>~(P—>Q))

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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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