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Truth suffers from too much analysis

Posts Tagged ‘Truth’

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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Desiring, Believing, Knowing, Obligation, and Fitch’s Paradox

Posted by allzermalmer on April 26, 2013

Assume that Striving, Doing, Believing, & Knowing (SDBK) have some fairly simple properties.
Assume that (SDBK) Striving, Doing, Believing, & Knowing are two-termed relations between an Agent and a Possible State of Affairs.

It shall be a convention to treat Possible State of Affairs as Propositions. So, Φ is assumed to be an agent and  p is assumed to be a proposition.
(i) “Φ strives for p” is equivalent to saying “Φ strives to bring about or realize the (possible) state of affairs expressed by the proposition p.”
(ii) “Φ does p” is equivalent to saying “Φ brings about the (possible) state of affairs expressed by the proposition p.”
(iii) “Φ believes p” is equivalent to saying “Φ believes about or realize the (possible) state of affairs expressed by the proposition p.”
(iv) “Φ knows p” is equivalent to saying “Φ knows about or realize the (possible) state of affairs expressed by the proposition p.”

It shall be a convention to ignore the Agent and treat concepts to be considered, i.e. Striving, Doing, Believing, & Knowing (SDBK), as a “Class of Propositions” instead of Two-Termed relation.
(i) ‘Striving’ means the Class of Propositions striven for (that is striven to be realized).
(ii) ‘Believing’ means the Class of Propositions believed.
(iii) ‘Doing’ means the Class of Propositions doing (that is doing to be realized).
(iv) ‘Knows’ means the Class of Propositions known.

Simplification and Adjunction

Here are two basic rules of Logical Inference in propositional logic. They are known, respectively, as Simplification and Adjunction.

Simplification: Because both components of a conjunctive argument are true, it is permissible to infer that either of it conjuncts is true.
(Premise): Germany Lost World War One & Germany Lost World War Two
(Conclusion): Germany Lost World War One

(Premise): Germany Lost World War One & Germany Lost World War Two
(Conclusion): Germany Lost World War Two

Adjunction: Because both premises are presumed true, we can infer their conjunction.
(Premise): Germany Lost World War One
(Premise): Germany Lost World War Two
(Conclusion): Germany Lost World War One & Germany Lost World War Two

Assume p and q are variables for two different propositions. Assume Ω stand for Class of Propositions, which can be either Striving, Believing, Obligation, and Knowing, or etc. Assume → stands for Strict Implication, Logical Entailment, Entailment, Logical Consequence.

Closed with Respect to Conjunction Elimination

Class of Propositions are Closed with Respect to Conjunction Elimination whenever a conjunctive proposition is in the Class, then those two propositions themselves are in that Class. Closed with Respect to Conjunction Elimination follows the logical inference of Simplification, but it uses one the relation of (SDBK).

Class of Propositions Closed with Respect to Conjunction Elimination:
(p)(q)[(Ω[p & q]) → [(Ωp) & (Ωq)]]

Assume that Ω stands for the Class of Propositions “know”. So the Class of Propositions (know) Closed with Respect to Conjunction Elimination means that “If (know both p & q) then logically entails (know p) & (know q)”. We can replace Ω with “Believe”, “Striving”, “Doing”, or the others listed.

Class of Propositions covered by Closed with Respect to Conjunction Elimination are: Striving (for), Doing, Believing, Knowing, Proving, Truth, Causal Necessity (in the sense of Burks), Causal Possibility ( in the sense of Burks), (Logical) Necessity, (Logical) Possibility, Obligation (Deontic Necessity), Permission (Deontic Possibility), Desire (for),

Closed with Respect to Conjunction Elimination

Class of Propositions Closed with Respect to Conjunction Introduction whenever two propositions are in the class, then so is the conjunction of the two propositions. Closed with Respect to Conjunction Introduction follows the logical inference of Adjunction, but it uses one of the relations of (SDBK).

Class of Propositions Closed with Respect to Conjunction Introduction:
(p)(q)[[(Ωp) & (Ωq)] → (Ω[p & q])]

Assume Ω stands for the Class of Propositions “know”. So the Class of Propositions (know) Closed with Respect to Conjunction Introduction means that “If (know p) and (know q), then logically entails (Know both p & q).” We can replace Ω with “Believe”, “Strive”, “Doing”, or others listed.

Class of Propositions covered by Closed with Respect to Conjunction Introduction are:
Truth, Causal Necessity (in the sense of Burks), Logical Necessity, Obligation (Deontic Necessity).

Class of Propositions possibly covered by Closed with Respect to Conjunction Introduction are: Striving (for), Doing, Believing, Knowing, Proving, Desire (for).

Class of Propositions not covered by Closed with Respect to Conjunction Introduction are: Causal possibility (in the sense of Burks), Logical Possibility, and Permission (Deontic Possibility).

Truth Class

Class of Propositions are a Truth Class if every member of it is true.

Class of Propositions Truth Class:
(p)[(Ωp) → p]

Assume Ω stands for Class of Propositions Truth Class “knows”. So the Class of Propositions Truth Class (knows) means “If (knows p) then logical entails p.”

Class of Propositions Truth Class are: Truth, Causal Necessity (in the sense of Burks), Logical Necessity, Knowing, Done, and Proving.

Theorems about Truth Classes

Theorem 1: If (Class of Propositions) is a Truth Class which is Closed with Respect to Conjunction Elimination, then the proposition, [p & ~(Ωp)], which asserts that p is true but not a member of (Class of Propositions) (where p is any proposition), is itself necessarily not a member of (Class of Propositions).

Proof: Suppose the contrary, [p & ~(Ωp)], is a member of (Class of Propositions), i.e. suppose that (Ω[p & ~(Ωp)]) is a member of (Class of Propositions). Since (Class of Propositions) are Closed with Respect to Conjunction Elimination, the propositions p and ~(Ωp) must both be members of (Class of Propositions), so that the propositions (Ωp) and (Ω(~(Ωp))) must both be true. But the fact that (Class of Propositions) is a truth class and has ~(Ωp) is true, and this contradicts the result that (Ωp) is true. Thus from the assumption that [p & ~(Ωp)] is a member of (Class of Propositions) we have derived contradictory results. Hence, that assumption is necessarily false.

Theorem 2: If (Class of Propositions) is a Truth Class which is Closed with Respect to Conjunction Elimination, and if p is a true proposition which is not a member of (Class of Proposition), then the proposition, [p & ~(Ωp)], is a true proposition which is necessarily not a member of (Class of Propositions).

Proof: The proposition [p & ~(Ωp)] is clearly true, and by Theorem 1 it is necessarily not a member of (Class of Propositions).

Omnipotent and Fallibility

Theorem 3: If an Agent is all-powerful in the sense that for each situation that is the case, it is logically possible that that situation was brought about by that Agent, then whatever is the case was brought about (done) by that Agent.

Proof: Suppose that p is the case but was not brought about by the agent in question. Then, since (doing) is a Truth Class Closed with Respect to Conjunction Elimination, we conclude from Theorem 2 that there is some actual situation which could not have been brought about by that Agent, and hence that Agent is not all-powerful in the sense described. Hence, that assumption is necessarily false.

Theorem 4: For each Agent who is not omniscient, there is a true proposition which that Agent cannot know.

Proof: Suppose that p is true but not known by the Agent. Then, since (knowing) is a Truth Class Closed with Respect to Conjunction Elimination, we conclude from Theorem 2 that there is some true proposition which cannot be known by the Agent.

Knowability Paradox

Theorem 5: 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 true.

Proof: Similar to proof in Theorem 4.

Proved True Never Proved

Theorem 6: 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.

Proof: Similar to the proof in Theorem 4, using the fact that if p is a proposition about proving, so is [p & ~(Ωp)].

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Paradox of Knowability

Posted by allzermalmer on April 12, 2013

Theorem 5: 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 true

“There are truths that cannot be known. For suppose that all truths can be known. Then all truths actually are known. Otherwise, we may suppose for some p that p but it is not known that p. Then it can be known that p but it is not known that p. But when it is known that thus and such, it is known that thus and it is known that such. So it could be known that p and known that it is not known that p. But what is known is true. So it could be known that p and not known that p. But that is a contradiction, and no contradiction can be true. So all truths are actually known.” W.D. Hart

(1) Assume that if X is true then possible to know that X is true. (2) Then, if X is true & do not know that X is true, then possible to know that both X is true & do not know X is true. (3) But, not possible to know that both X is true & do not know X is true. (4) Not both X is true & do not know X is true. (5)  If X is true then do not not know that X is true. (6) If X is true then know that X is true.

What if the World is non-omniscient? This would mean that nobody knows all truths, and nobody ever will. Therefore, there are unknowable truths. If some truth is unknown, then that it is unknown is itself unknowable; Because the world is non-omniscient, there is some unknowable truth. If there at exists at least one Truth, such that Truth is true and Truth is unknown, then there exists at least one Truth, such that Truth is unknown and Truth is unknowable. If there does not exist at least one Truth, such that Truth is unknown and Truth is unknowable, then there does not exist at least one Truth, such that Truth is true and Truth is unknown.

It is possible that it is known by someone at some time that both X is true & It is not known by someone at some time that X is true. It is possible that both It is known by someone at some time that X is true & It is not known by someone at some time that X is true (reduction ad absurdum)

Non-Omniscience: X is true & It is not known by someone at some time that X is true.

Verdicality (KV): If it is known by someone at some time that X is true, then X is true.

Distribution (KC): If it is known by someone at some time that both X is true & Y is true, then both it is known by someone at some time that X is true & It is known by someone at some time that Y is true.

Non-Contradiction (LNC): It is not possible that both X is true & X is not true.

Clousure (CP): If X is true implies Y is true & it is possible that X is true, then it is possible that Y is true.

Knowability (KP): If X is true then it is possible that it is known by someone at some time that X is true.

(1) Assume that X is true & It is not known by someone at some time that X is true

(2) It is possible that it is known by someone at some time that both X is true & It is not known by someone at some time that X is true. (By KP & (1).

(3) It is known by someone at some time that both X is true & It is not known by someone at some time that X is true. It is known by someone at some time that X is true & It is known by someone at some time that it is not known by someone at some time that X is true.

(4) It is known by someone at some time that both X is true & it is not known by someone at some time that X is true. It is known by someone at some time that X is true & It is not known by someone at some time that X is true. (By Simp, VK, and Adjunction (and Transitivity implication))

(5) It is possible that both It is known by someone at some time that X is true & It is not known by someone at some time that X is true. (by CP)

(6) It is not possible that both It is known by someone at some time that X is true & It is not known by someone at some time that X is true. (by LNC)

(7) It is necessary that not both X is true & X is not true.

*(8) X is true & It is known by someone at some time that X is true. (by Reduction Ad Absurdim)

Thus, If X is true, then it is known by someone at some time that X is true:: If it is not known by someone at some time that X is true, then X is not true.

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Logically Valid Arguments

Posted by allzermalmer on April 8, 2013

Categorically Valid Syllogisms

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

Figure 1

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

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

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

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

Figure 2

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

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

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

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

Figure 3

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

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

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

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

Figure 4

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

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

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

Propositional Logic

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

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

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

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

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

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

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

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

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

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

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

 

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Aristotle’s Sea Battle and Future Contingents

Posted by allzermalmer on December 18, 2012

This comes from Aristotle’s book “On Interpretation” and is Part 9. It deals with future contingents and statement either being true or not being true, i.e. law of excluded middle.

“In the case of that which is or which has taken place, propositions, whether positive or negative, must be true or false. Again, in the case of a pair of contradictories, either when the subject is universal and the propositions are of a universal character, or when it is individual, as has been said,’ one of the two must be true and the other false; whereas when the subject is universal, but the propositions are not of a universal character, there is no such necessity. We have discussed this type also in a previous chapter.

When the subject, however, is individual, and that which is predicated of it relates to the future, the case is altered. For if all propositions whether positive or negative are either true or false, then any given predicate must either belong to the subject or not, so that if one man affirms that an event of a given character will take place and another denies it, it is plain that the statement of the one will correspond with reality and that of the other will not. For the predicate cannot both belong and not belong to the subject at one and the same time with regard to the future.

Thus, if it is true to say that a thing is white, it must necessarily be white; if the reverse proposition is true, it will of necessity not be white. Again, if it is white, the proposition stating that it is white was true; if it is not white, the proposition to the opposite effect was true. And if it is not white, the man who states that it is making a false statement; and if the man who states that it is white is making a false statement, it follows that it is not white. It may therefore be argued that it is necessary that affirmations or denials must be either true or false.

Now if this be so, nothing is or takes place fortuitously, either in the present or in the future, and there are no real alternatives; everything takes place of necessity and is fixed. For either he that affirms that it will take place or he that denies this is in correspondence with fact, whereas if things did not take place of necessity, an event might just as easily not happen as happen; for the meaning of the word ‘fortuitous’ with regard to present or future events is that reality is so constituted that it may issue in either of two opposite directions. Again, if a thing is white now, it was true before to say that it would be white, so that of anything that has taken place it was always true to say ‘it is’ or ‘it will be’. But if it was always true to say that a thing is or will be, it is not possible that it should not be or not be about to be, and when a thing cannot not come to be, it is impossible that it should not come to be, and when it is impossible that it should not come to be, it must come to be. All, then, that is about to be must of necessity take place. It results from this that nothing is uncertain or fortuitous, for if it were fortuitous it would not be necessary.

Again, to say that neither the affirmation nor the denial is true, maintaining, let us say, that an event neither will take place nor will not take place, is to take up a position impossible to defend. In the first place, though facts should prove the one proposition false, the opposite would still be untrue. Secondly, if it was true to say that a thing was both white and large, both these qualities must necessarily belong to it; and if they will belong to it the next day, they must necessarily belong to it the next day. But if an event is neither to take place nor not to take place the next day, the element of chance will be eliminated. For example, it would be necessary that a sea-fight should neither take place nor fail to take place on the next day.

These awkward results and others of the same kind follow, if it is an irrefragable law that of every pair of contradictory propositions, whether they have regard to universals and are stated as universally applicable, or whether they have regard to individuals, one must be true and the other false, and that there are no real alternatives, but that all that is or takes place is the outcome of necessity. There would be no need to deliberate or to take trouble, on the supposition that if we should adopt a certain course, a certain result would follow, while, if we did not, the result would not follow. For a man may predict an event ten thousand years beforehand, and another may predict the reverse; that which was truly predicted at the moment in the past will of necessity take place in the fullness of time.

Further, it makes no difference whether people have or have not actually made the contradictory statements. For it is manifest that the circumstances are not influenced by the fact of an affirmation or denial on the part of anyone. For events will not take place or fail to take place because it was stated that they would or would not take place, nor is this any more the case if the prediction dates back ten thousand years or any other space of time. Wherefore, if through all time the nature of things was so constituted that a prediction about an event was true, then through all time it was necessary that that should find fulfillment; and with regard to all events, circumstances have always been such that their occurrence is a matter of necessity. For that of which someone has said truly that it will be, cannot fail to take place; and of that which takes place, it was always true to say that it would be.

Yet this view leads to an impossible conclusion; for we see that both deliberation and action are causative with regard to the future, and that, to speak more generally, in those things which are not continuously actual there is potentiality in either direction. Such things may either be or not be; events also therefore may either take place or not take place. There are many obvious instances of this. It is possible that this coat may be cut in half, and yet it may not be cut in half, but wear out first. In the same way, it is possible that it should not be cut in half; unless this were so, it would not be possible that it should wear out first. So it is therefore with all other events which possess this kind of potentiality. It is therefore plain that it is not of necessity that everything is or takes place; but in some instances there are real alternatives, in which case the affirmation is no more true and no more false than the denial; while some exhibit a predisposition and general tendency in one direction or the other, and yet can issue in the opposite direction by exception.

Now that which is must needs be when it is, and that which is not must needs not be when it is not. Yet it cannot be said without qualification that all existence and non-existence is the outcome of necessity. For there is a difference between saying that that which is, when it is, must needs be, and simply saying that all that is must needs be, and similarly in the case of that which is not. In the case, also, of two contradictory propositions this holds good. Everything must either be or not be, whether in the present or in the future, but it is not always possible to distinguish and state determinately which of these alternatives must necessarily come about.

Let me illustrate. A sea-fight must either take place to-morrow or not, but it is not necessary that it should take place to-morrow, neither is it necessary that it should not take place, yet it is necessary that it either should or should not take place to-morrow. Since propositions correspond with facts, it is evident that when in future events there is a real alternative, and a potentiality in contrary directions, the corresponding affirmation and denial have the same character.

This is the case with regard to that which is not always existent or not always nonexistent. One of the two propositions in such instances must be true and the other false, but we cannot say determinately that this or that is false, but must leave the alternative undecided. One may indeed be more likely to be true than the other, but it cannot be either actually true or actually false. It is therefore plain that it is not necessary that of an affirmation and a denial one should be true and the other false. For in the case of that which exists potentially, but not actually, the rule which applies to that which exists actually does not hold good. The case is rather as we have indicated.”

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Truth of Reasoning and Truth of Fact

Posted by allzermalmer on October 26, 2012

“All that which implies contradiction is impossible, and all that which implies no contradiction is possible.” G.W. Leibniz

“I assume that every judgement (i.e. affirmation or negation) is either true or false and that if the affirmation is true the negation is false, and if the negation is true the affirmation is false; that what is denied to be true-truly, of course- is false, and what is denied to be false is true; that what is denied to be affirmed, or affirmed to be denied, is to be denied; and what is affirmed to be affirmed and denied to be denied is to be affirmed. Similarly, that it is false that what is false should be true or that what is true should be false; that it is true that what is true is true, and what is false, false. All these are usually included in one designation, the principle of contradiction.” G.W. Leibniz

“There are . . . two kinds of truths, those of reasoning and those of fact. Truths of reasoning are necessary and their opposite is impossible; truths of fact are contingent and their opposite is possible. When a truth is necessary, its truth can be found by analysis, resolving it into more simple ideas and truths, until we come to those which are primary. It is thus, that in Mathematics speculative Theorems and practical Canons are reduced by analysis to Definitions, Axioms, and Postulates. In short, there are simple ideas, of which no definition can be given; there are also axioms and postulates, in a word primary principles, which cannot be proved, and indeed have no need of proof, and these are identical propositions, whose opposite involves an express contradiction.” G.W. Leibniz

 So Leibniz obtains all knowable propositions or statements to be divided based on the principle of contradiction. The truth of statements is divided into two realms. This also deals with what people can know, or knowability. It basically says that
“For each statement, if statement is knowable, then statement is either truth of reasoning or truth of fact. For each statement, if statement is truth of reasoning, then statements affirmation is logically possible and statements negation is logically impossible. For each statement, if statement is truth of fact, then statements affirmation is logically possible and statements negation is logically possible.”

A truth of reasoning is always true and not possible it is false. It is logically impossible that it is false. The negation of a truth of reasoning is an impossible statement or impossible proposition. It is self-contradictory. A truth of fact is not always true and possible it is false. It is logically possible that it is true or logically possible it is false. Truth of Reasoning is Logically Necessary and Truth of Fact is Logically Contingent.

“For each statement, if statement is Truth of Fact, then statement is an empirical claim. For each statement, if statement is Truth of Reasoning, then statement is not an empirical claim. For each statement, if statement is Truth of Reasoning, then statement is non-empirical claim. For each statement, if statement is Truth of Fact, then statement is not non-empirical claim.”

What also happens to come from this is that Truth of Facts do not entail or lead to Truth of Reasoning, and Truth of Reasoning do not entail or lead to Truth of Fact. This means that Truth of Facts do not imply or entail non-empirical claims and Truth of Reasoning do not imply or entail empirical claims. This means that statements of experience are not non-empirical claims and means statements of experience are empirical claims.

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Chuang Tzu’s Theory of Truth

Posted by allzermalmer on June 13, 2011

This blog comes from an article called Chuang-Tzu’s Theory of Truth. The article appeared in the journal Philosophy East and West, Vol. 3, No. 2 (Jul., 1953), pp. 137-146. The article is by Siao-Fang Sun.

The concept of truth has its problems. They are usually divided into two sorts. They are Absolute Truth and Relative Truth. Absolute Truth is usually based on a statement is identical with reality or the real. Relative Truth is usually based the property of the statements, and outside of language there is no truth. When we say, in Relative Truth, that something is true, we say that this or that statement is true. Thus, we come to a distinction of truth. Absolute Truth is based a concept of  metaphysics, while Relative Truth is based on a concept of semantics.

Chuang Tzu accepted both theories of truth. “Truth is one and many at the same time. It is one when it is considered as reality itself. It is many when it is considered as a property of our knowledge of things.” (Italics is my emphasis). Thus, Chuang Tzu held to two theses of truth:
[1.] There is the absolute truth, and this is the goal or ideal of our life.
[2.] We only have relative truth.

Relative Truth: We come to find that we have relative truth. All events in the world are relative to one another. One thing is bigger than another, and one thing is better looking than another. Distance between each other are relative to their location to each other. What we see is relative to us, and where we are. What is big to one thing is small to another. However, ignoring these relations, the thing is neither big or small. And if we leave out this relation of a thing, we find that it is not a thing at all, or at least not as we know it.

“Everything has infinite relations with other things, and it is impossible, therefore, for men to have complete knowledge of what a thing is. A man can know some aspects of the nature of what a thing is, but he can never know all the aspects of a thing…All the predicates in our language by which we describe things are by nature relative.”

We find that human knowledge is relative, and that there is also differing opinions within the human framework of knowledge. What is true for one system of thought is false under another system of thought. We not only find this within religion, politics, but also in science. There are different systems of thought in science, and some claim that theirs correlate to reality while the other side says that their does. “[F]or the truth of our knowledge depends upon the objects, the external things, as well as the subject, the knower.”

The article, as a footnote, quotes a portion of Chuang Tzu’s books. Here is the portion that is quoted from Chuang Tzu’s book.

“Now I would ask you this, If a man sleeps in a damp place, he gets lumbago and dies. But how about an eel? And living up in a tree is precarious and trying to the nerves; but how about monkeys? Of the man, the eel, and the monkey, whose habitat is the right one, absolutely? Human beings feed on flesh, deer on grass, centipedes on snake’s brains, owls and crows on mice. Of these four, whose is the right taste, absolutely? Monkey mates with monkey, the buck with the doe; eels consort with fishes, while men admire Mao Ch’iang and Li Chi (beauties of the fifth and seventh centuries B.C. respectively), at the sight of whom fishes plunge deep down in the water, birds soar high in the air, and deer hurry away. Yet who shall say which is the correct standard of beauty? In my opinion, the standard of human virtue, and of positive and negative, is so obscured that it is impossible to actually know it is as such.”

In fact, within this quote, we can see how Chuang Tzu is using some of the ten trops of the skeptic, and also the criterion argument that we find within Sextus Empiricus. Thus, we see that there is some skeptical attitude within Chuang Tzu on knowledge.

In our experiences, perhaps, we find that some people are optimist. Some people are pessimist. We find that they take a very different outlook on things, since they are different observers. They see the world through different lenses, through different eyes. They come to a different understanding on the world. “A frog in the well can never know the grandeur of heaven, because it is limited by the place where it lives.”

But now that we find that there is a controversy between different relative truths, we find that our controversy would never end. Chuang Tzu tries to show us how we could try to escape this, but never does help us escape.

“Granting that you and I argue. If you beat me, and not I you, are you necessarily right and I wrong? Or if I beat you, and not you me, am I necessarily right and you wrong? Or are we both partly right and partly wrong? Or are we both wholly right and wholly wrong? You and I cannot know this, and consequently the world will be in ignorance of the truth.

Who shall I employ as arbiter between us? If I employ some one who takes your view, he will side with you. How can such a one arbitrate between us? If I employ someone who takes my view, he will side with me. How can such a one arbitrate between us? And if I employ some one who either differs from you, or agrees with both of us, he will be equally unable to decide between us. Since then you, and I, and man, cannot decide, must we not depend on Another?”

Many different theories of knowledge, or systems of thought, have different specific frameworks that they work within. Thus, different systems have different frameworks. Thus, what fits for one framework does not work in another framework. It is like taking a fish out of water, and place it in the desert. The only way we can accommodate something from framework X into framework Y, is to modify or change framework Y. We, in short, find that there is no absolute system of truth that humans construct.

However, this does not mean that each system does not contain some truth within them. They contain a truth, but they do not contain the truth. Each system can have from one truth, to many, but not all the truth. The truth is not found in any system, since all our truths will be relative to us the knower, which is human beings. We are limited to what we can know, and how we can know it.”The truth cannot be a matter of knowledge…”.

The search for the absolute truth is what many of the systems of thought are after. They are after the way things are, and is an eternal quest of the human mind and species. However, each person gets their own relative truth, and relative knowledge, but now the absolute truth. But, this does not stop us from searching, and trying to obtain it.

Absolute Truth: Chung Tzu is famous for one story, or parable. This is the parable of the Butterfly.This is taken from Zhuangzi, book by Chuang Tzu

“One day about sunset, Zhuangzi dozed off and dreamed that he turned into a butterfly.

He flapped his wings and sure enough he was a butterfly…

What a joyful feeling as he fluttered about, he completely forgot that he was Zhuangzi.

Soon though, he realized that that proud butterfly was really Zhuangzi who dreamed he was a butterfly, or was it a butterfly who dreamed he was Zhuangzi!

Maybe Zhuangzi was the butterfly, and maybe the butterfly was Zhungzi?”

Now this shows that we have reason to believe that our knowledge is unreliable, besides it being relative, or that even our knowledge is an illusion. In modern day Western philosophy, there is the problem of being a Brain-In-A-Vat. “Not only is what we perceive and do merely dream, but even when I am conscious that I am dreaming, I do not go a step beyond the dream. Only the degree of dreaming is less when I am conscious of it than when I am not conscious of it. but the consciousness of dreaming does not change the fact that we are dreaming.”

Chuang Tzu points out to us that everything is changing, and that everything is change itself, which is related to the Greek philosopher Heraclitus. Yet we think that there is something underneath this change that is unchangeable, or that which we experience as changing. For without this, change becomes unthinkable. For how can something change if there is not something that is changing, which is itself unchanging? We just experience the manifestation of change itself.

Chuang Tzu is an empiricist, and takes knowledge of the world to come from the senses. Empirical knowledge is the way in which we come to have knowledge of the world. Thus, “All knowledge of the world is based upon our experience. And as we have reason to believe ourselves in a dream and as experience is most unreliable, our knowledge is unreliable.” We also find, through experience, that “everything merely happens to be” and that there is no necessity in anything that we experience. Thus, laws of nature have no universal validity. In fact, these are the very things that we try to use to explain and predict our experiences. But in fact, laws of nature are just universals, and we only experience particulars. We find that we do not experience laws of nature, but just our particular experiences which just happen. The empiricist position is the skeptical position.

But Chuang Tzu does come to one conclusion in our skepticism. We find that (1.) there is harmony in the universe, and (2.) the concept of transcendentalism.

We find there is harmony in the way things are arranged, and these events do not occur in chaos. This harmony is good enough to secure the relative certainty of our knowledge.

For the transcendental, we have this:

“[F]or Chuang Tzu it is true that everything in the world is relative and that our knowledge of a thing is also relative. But with the totality of all the relative things, the case is entirely different. While the individual things are relative, the totality of all things is not relative. The totality of all things is itself not a thing. It is, to use a familiar term in Western philosophy, a transcendental concept. It transcends all relativities. It is one and it is absolute.”

Thus, we find that there are many relative truths, but that all of these things together come to form the one thing. This one thing becomes the transcendental. There is even more to this, which is suppose to show how Chuang Tzu came to accept that there is an absolute truth.

“Since everything in the world is not only in a process of change but also is change itself, the reality of every individual thing is doubtful. From moment to moment change occurs and an individual thing appears and then disappears. Once an individual thing disappears, it disappears forever. There is never a repetition of the same individual thing. what looks like the same thing is in fact a different thing. The so-called identity of things does not exist. Therefore, from the standpoint of the individual thing, we find that the reality of a thing is questionable. Not only can we not grasp a thing with absolute certainty at all, since in every moment the thing is changing, but also we cannot grasp our own bodies, for we are changing things, too. But, looking from the standpoint of the totality of things, we find that there is the change which we cannot doubt. For the ultimate change we may imagine that there is something in it which sustains the change. We do not know what this something is, but we imagine there is something there underlying the changes, just as there is a fundamental form in accordance with which every change occurs, though every individual change has also its specific form. This something-we-do-not-know and this fundamental form are identified. They are different aspects from which we see the whole change itself. They constitute the totality of phenomena are real.”

The position of Chuang Tzu is very similar to that Immanuel Kant. The Absolute Truth would be the Noumena, and Relative Truth would be the Phenomena. In fact, the Phenomena does not show that there are other minds and that there is an external world. These things are contained within the Noumena, and are hidden from us, and even relative. The Absolute is the glue that holds all change together, and holds all the relative truths together. It is the foundation of all change, and of all relative truth. We also come to think that there are things that are true if we were not existing, but this can only be contained within the Noumena, and is not contained within the Phenomena. It is not found in experience, and our only way out is to either reject the Noumena, and thus that there are things that are true independent of humans and our point of view, or to accept it and that there are things that are true independent of the human point of view.

But since we are humans, we can only know things relative to us. But there is the sub-set of humans having different points of views within the human point of view. So what about animals, like, for example, a bat? We assume, but is beyond the phenomena of our expeirence, that bats have experience. Thus, there would be the bat experience, and the relative experience of a bat.

So take the category of human, bat, cat, dog, dolphin, amoeba, spider, and etc. Each categories experience is different from that of another categories, like the experience of the category of human is different from that of cat. Each categories experience is relative to that of another categories experience. But, each category has particulars in it. And each particular object in that category has its experiences relative to each other particular object in that category. Thus, the absolute contains each point of view of the category, and the particulars that make up the category. It is the glue that holds them all together, categories and particulars of the categories.

Now, to Chuang Tzu, the Absolute Truth is that of the Tao. We can only come to know this through intuition, but it is not something that we can be taught, like we can be taught mathematics. But we can be trained to be receptive to it, and to come to learn it on our own, through training.


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