allzermalmer

Truth suffers from too much analysis

Posts Tagged ‘Biconditional’

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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3 Value Logic

Posted by allzermalmer on May 10, 2013

I am going to use Polish Notation in expressing these truth tables of 3 value logic and 2 value logic. Lower case letters are variables: x, y, z, …
Capital Letters represent logical operators: N, A, K, E, C

Nx = ~x
Axy = x v y
Kxy = x & y
Cxy = x → y
Exy = x ↔ y

This notation is explicated in the text book Formal Logic by logican A.N. Prior in the late 1950’s to early 1960’s. The notationw as used by Polish logican Jan Lukasiewicz. Lukasiewicz was one of the first logicans to formally organize a three value logic. The logical matrix or logical matrices of both 2 value logic and 3 value logic are presented.

2 value logic uses 1 and 0.
3 value uses 1, 1/2, and 0.

1 stand for true.
1/2 stands for indeterminate.
0 stands for false.

I have put in bold those portions of truth tables in 3 value logic that do not have a similar truth table in 2 value.

Affirmation & Negation (x & Nx) : 2 value

  • (1) If x = 1 then Nx = 0.
    (2) If x = 0 then Nx = 1.

Affirmation & Negation (x & Nx) : 3 value

  • (1) If x = 1 then Nx = 0.
  • (2) If x = 1/2 then Nx = 1/2
  • (3) If x = 0 then Nx = 1

Conditional (Cxy) : 2 value

  • (1) If x = 1 and y = 1, then Cxy = 1
  • (2) If x = 1 and y = 0, then Cxy = 0
  • (3) If x = 0 and y = 1, then Cxy = 1
  • (4) If x =0 and y = 0, then Cxy = 1

Conditional (Cxy) : 3  value

  • (1) If x = 1 and y = 1, then Cxy = 1
  • (2) If x = 1 and y = 1/2, then Cxy = 1/2
  • (3) If x = 1 and y = 0, then Cxy = 0
  • (4) If x = 1/2 and y = 1, then Cxy = 1
  • (5) If x = 1/2 and y = 1/2, then Cxy = 1
  • (6) If x = 1/2 and y = 0, then Cxy = 1/2
  • (7) If x = 0 and y = 1, then Cxy = 1
  • (8) If x = 0 and y = 1/2, then Cxy = 1
  • (9) If x = 0 and y = 0, then Cxy = 1

Conjunction (Kxy) : 2 value

  • (1) x = 1 and y = 1, then Kxy = 1
  • (2) x = 1 and y = 0, then Kxy = 0
  • (3) x = 0 and y = 1, then Kxy = 0
  • (4) x = 0 and y = 0, then Kxy = 0

Conjunction (Kxy) : 3 value

  • (1) If x = 1 and y = 1, then Kxy = 1
  • (2) If x = 1 and y = 1/2, then Kxy = 1/2
  • (3) If x = 1 and y = 0, then Kxy = 0
  • (4) If x = 1/2 and y = 1, then Kxy = 1/2
  • (5) If x = 1/2 and y = 1/2, then Kxy = 1/2
  • (6) If x = 1/2 and y = 0, then Kxy = 1/2
  • (7) If x = 0 and y = 1, then Kxy = 0
  • (8) If x = 0 and y = 1/2, then Kxy = 0
  • (9) If x = 0 and y = 0, then Kxy = 0

Disjunction (Axy) : 2 value

  • (1) If x = 1 and y = 1, then Axy = 1
  • (2) If x = 1 and y = 0, then Axy = 1
  • (3) If x = 0 and y = 1, then Axy = 1
  • (4) If x = 0 and y = 0, then Axy = 0

Disjunction (Axy) : 3 value

  • (1) If x = 1 and y = 1, then Axy = 1
  • (2) If x = 1 and y = 1/2, then Axy = 1
  • (3) If x = 1 and y = 0, then Axy = 0
  • (4) If x = 1/2 and y = 1, then Axy = 1
  • (5) If x = 1/2 and y = 1/2, then Axy = 1/2
  • (6) If x = 1/2 and y = 0, then Axy = 1/2
  • (7) If x = 0 and y = 1, then Axy = 1
  • (8) If x = 0 and y = 1/2, then Axy = 1/2
  • (9) If x = 0 and y = 0, then Axy = 0

Biconditional (Exy) : 2 value

  • (1) If x = 1 and y = 1, then Exy = 1
  • (2) If x = 1 and y = 0, then Exy = 0
  • (3) If x = 0 and y = 1, then Exy = 0
  • (4) If x = 0 and y = 0, then Exy = 1

Biconditional (Exy) : 3 value

  • (1) If x = 1 and y = 1, then Exy = 1
    (2) If x = 1 and y = 1/2, then Exy = 1/2
    (3) If x = 1 and y = 0, then Exy = 0
    (4) If x = 1/2 and y = 1, then Exy = 1/2
    (5) If x = 1/2 and y = 1/2, then Exy = 1
    (6) If x = 1/2 and y = 0, then Exy = 1/2
    (7) If x = 0 and y = 1, then Exy = 0
    (8) If x = 0 and y = 1/2, then Exy = 1/2
    (9) If x = 0 and y = 0, then Exy = 1

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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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