classical logic with indeterminate sentences · see table ii which includes at lines 3 and 4 the...

Metalogicon (2009) XXII, 2

107

Classical Logic with Indeterminate Sentences

Michele Malatesta

Summary

Lukasiewcz’s n-valued logics (n≥3) do not allow the foundation of probability calculus because of the truth value of the contradictory of an indeterminate sentence which gives rise to difficulties when we value the well formed formulas in which the principles of non-contradiction and excluded middle are formulated. A paradoxical situation emerges: logics which should found the probability calculus cannot achieve such a purpose. In order to solve the problem it is convenient to put that if the truth value of a sentence is indeterminate then its contradictory is so, but under the condition that its degree of indetermination is the complement of the degree of indetermination of the first. Such a viewpoint preserves the Boolean conditions a + a’ = 1 and a · a’ = 0, not only in the case of contradictory sentences but also in the case of two different atomic sentences, one of which has an indeterminate value and the other has an indeterminate value which is the complement of the first. In conclusion we do not refuse any condition of Boolean algebra and eliminate the danger of a sure wreck. 1. Lukasiewicz’s 3-valued logic and its semantic difficulties

The starting point of Lukasiewic’s 3-valued logic is well known: his wish to refute philosophical determinism in order to preserve contingency, free will, and therefore human responsibility. The Polish logician writes: «I can assume without contradiction that my presence in Warsaw at a certain moment of the next year, e.g. at noon on 21 December, is at the present time determined neither positively nor negatively. Hence it is possible, but not necessary, that I shall be present in Warsaw at the given time. On this assumption the proposition: “I shall be at Warsaw at noon on 21 December of next year” can at the present time neither true nor false. For if were true now, my future presence in Warsaw would have to be necessary, which is contradictory to the

108

assumption. It were false now, on the other hand, my future presence in Warsaw would to be impossible, which is also contradictory to the assumption. Therefore the proposition considered is at the moment neither true nor false and must possess a third value, different from ‘0’ or falsity and ‘1’ or truth. This value we can designate by 1/2». See Lukasiewicz (1930).

Lukasiewicz’s purpose is noble from the metaphysical and moral standpoint but dangerous from the scientific and epistemological one, because, by denying the universal validity of the principles of non contradiction and excluded middle, the collapse of probability calculus is inescapable.

There are two approaches to found the probability calculus: the axiomatic set theoretic and the computing sentence rules one. Both are valid insofar as both are interpretations of Boolean algebra. Let us chose the latter. See Skyrms (1966).

The first two rules of such a foundation are respectively: Rule 1. If a well formed formula is a tautology, then its

probability is equal to 1. Rule 2. If a well formed formula is a contradiction, then its

probability is equal to 0. Let the complete comparative classical sentence logic truth

table be: Boolean

algebra a b 1 + ≥ ≤ ! = a' b'

Lukasiewicz p q V A B C D E F G

Standard logic

p q

!

"

!

"

!

"

!

"

!

"

|

!

"

!

"

!

¬p

!

¬q

1 1 1 1 1 1 0 1 0 0 1 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 1 0 1 0 0 0 1 0 1 1 1 1 1 1


109

Boolean algebras

a b 0 ↓ ‹ › ·

€

/ ≡ a b

Lukasiewicz p q O X M L K J I H

Standard logic

p q ∨

€

/ ⊂

€

/ ←

€

/ ⊃

€

/ →

€

• &

€

/ ≡

€

/ ↔

p q

1 1 0 0 0 0 1 0 1 1 1 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0

Let us select only the truth tables of classical logic normally utilized

A K C E p Np p \ q 1 0 1 0 1 0 1 0

1 0 1 1 1 1 0 1 0 1 0 0 1 0 1 0 0 0 1 1 0 1

and the corresponding of Lukasiewicz’s 3-valued logic

p Np

1 0 ½ ½ 0 1

110

A K C E p/ q

1 ½ 0 1 ½ 0 1 ½ 0 1 ½ 0

1 1 1 1 1 ½ 0 1 ½ 0 1 ½ 0 ½ 1 ½ ½ ½ ½ 0 1 1 ½ ½ 1 ½ 0 1 ½ 0 0 0 0 1 1 1 0 ½ 1

Let the common formulations of the first principles in weak form be: Cpp (principle of identity) NKpNp (principle of non contradiction) ApNp (principle of excluded middle)

It easy to see that whereas the above well formed formulas are

tautologies in classical logic

Cpp NKpNp ApNp C11 = 1 NK1N1 = NK10 = N0 = 1 A1N1 = A10 = 1 C00 = 1 NK0N0 = NK01 = N0 = 1 A0N0 = A01 = 1

only the first of them is a tautology in Lukasiewicz’s 3-valued one: Cpp NKpNp ApNp C11 = 1 NK1N1 = NK10 = N0 = 1 A1N1 = A10 = 1 C½½= 1 NK½N½ = NK½/½ = N½ = ½ A½N½ = A½½ = ½ C00 = 1 NK0N0 = NK01 = N0 = 1 A0N0 = A01 = 1

Consequently the probability that “I shall be present at Warsaw

at noon of 21 December of next year or I shall not be present at Warsaw at noon of 21 December of next year” would be not equal to 1 in the case of indetermination of the atomic sentence “I shall


111

be present at Warsaw at noon of 21 December of next year”. We may see the same ting as far as the principle of non contradiction is concerned. But this is absurd! Given that before tossing a coin the probability that the head will appear is indeterminate, it does not follow that at the next tossing the probability that the head will or will not appear is indeterminate too. This is true, not indeterminate. Neither thing goes better with Lukasiewicz’s n-valued logics.

2. Lukasiewicz’s n-valued logics and their semantic difficulties

In the early 1930s Lukasiewicz made an infinitary many-valued

generalization of his 3-valued logic by putting the following conditions:

/Np/ = 1 – /p/ /Apq/ = max [/p/, /q/] /Kpq/ = min [/p/, /q/]

/Cpq/ = {

€

11− / p /+ / q /

}according as{

€

≤ / q /> / q /

/Epq/ = /KCpqCqp/ = 1 – |/p/ – /q/|

Such a numerical development of many-valued logics is readily generalised as follows. See Rescher (1969). See TABLE I.

Let the truth tables of Lukasiewicz’s 4-valued logic be:

p Np 1 0

€

23

€

13

€

13

€

23

0 1

112

n Truth-values

2 3 4 . .

€

11 ,

€

01

€

22 ,

€

12 ,

€

02

€

33,

€

23,

€

13,

€

03

.

.

(1, 0) (1,

€

12 , 0)

(1,

€

23,

€

13,

€

03, 0)

.

.

n 1 =

€

n −1n −1

,

€

n − 2n −1

, ...,

€

2n −1

,

€

1n −1

,

€

0n −1

= 0

TABELLA I

Let the truth tables of Lukasiewicz’s 4-valued logic be:

p Np 1 0

€

23

€

13

€

13

€

23

0 1


113

A K

p\q 1

€

23

€

13 0 1

€

23

€

13 0

1 1 1 1 1 1

€

23

€

13 0

€

23 1

€

23

€

23

€

23

€

23

€

23

€

13 0

€

13 1

€

23

€

13

€

13

€

13

€

13

€

13 0

0 1

€

23

€

13 0 0 0 0 0

C E

p\q 1

€

23

€

13 0 1

€

23

€

13 0

1 1

€

23

€

13 0 1

€

23

€

13 0

€

23 1 1

€

23

€

13

€

23 1

€

23

€

13

€

13 1 1 1

€

23

€

13

€

23 1

€

23

0 1 1 1 1 0

€

13

€

23 1

Let us value the well formed formulas of the three principles in weak form: Cpp C11 = 1 C

€

23

€

23 = 1

C

€

13

€

13 = 1

C00 = 1

NKpNp NK1N1 = NK10 = N0 = 1 NK

€

23N

€

23= NK

€

23

€

13 = N

€

13 =

€

23

NK

€

13N

€

13 = NK

€

13

€

23 = N

€

13 =

€

23

NK0N0 = NK01 = N0 = 1 ApNp A1N1 = A10 = 1

114

A

€

23N

€

23 = A

€

23

€

13 =

€

23

A

€

13N

€

13 = A

€

13

€

23 =

€

23

A0N0 = A01 = 1

Once again we note that whereas the principle of identity preserves its quality of being a logical law the other two principle do not.

If we extend the number of truth values then we obtain for n values (n≥3) the truth value

€

1n (read 1 computed n times) as far as the principle of identity is concerned and the following truth values as far as the remaining principles is concerned.

See TABLE II which includes at lines 3 and 4 the evaluation of principles of Excluded middle and Non-contradiction just examined.

We conclude that the situation has become more and more complicated. In order to avoid such difficulties we are compelled to change direction.

3. A Classical logic with indeterminate sentences From the above complete Polish language of classical logic let

us select the following connectives:

N “not” negation

A “either …or” logical sum

B “then…if” inverse conditional

C “if…then” conditional

D “nand” incompatibility

E “if and only if” biconditional

J “either…or” exclusive disjunction

K “and” logical product

X “nor” binary rejection


115

n (n ≥3) Evaluation of Excluded middle and Non-

contradiction principles

3 1

€

12 1

4 1

€

23

€

23 1

5 1

€

34

€

24

€

34 1

6 1

€

45

€

35

€

35

€

45 1

7 1

€

56

€

46

€

36

€

46

€

56 1

8 1

€

67

€

57

€

47

€

47

€

57

€

67 1

9 1

€

78

€

68

€

58

€

48

€

58

€

68

€

78 1

10 1

€

89

€

79

€

69

€

59

€

59

€

69

€

79

€

89 1

. .

€

n (odd)

. .

1 =

€

n −1n −1

,

€

n − 2n −1

,…,

€

n −12

n −1 ,…,

€

n − 2n −1

,

€

n −1n −1

,

= 1

€

n (even)

1 =

€

n −1n −1

,

€

n − 2n −1

,…,

€

n2

n −1,

€

n2

n −1 ,…,

€

n − 2n −1

,

€

n −1n −1

, = 1

TABLE II

116

Let the complete list of all the formulations of the first principles be

Identity Non contradiction Excluded middle Cpp (1st weak formulation)

NKpNp (1st formulation)

ApNp (1st weak formulation)

Bpp (2nd weak formulation)

NKNpp (2nd formulation)

ANpp (2nd weak formulation)

Epp (strong formulation)

DpNp (3rd weak formulation)

DNpp (4th weak formulation)

JpNp (1st strong formulation)

JNpp (2nd strong formulation)

Let the following truth tables be, where x is the indeterminate value:

p Np 1 0 x 1 - x 1 - x x 0 1


117

A K p\q 1 x 1-x 0 1 x 1-x 0

1 1 1 1 1 1 x 1-x 0 x 1 x 1 1 x x 0 0

1-x 1 1 1-x 1-x 1-x 0 1-x 0 0 1 x 1-x 1-x 0 0 0 0

C B p\q 1 x 1-x 0 1 x 1-x 0

1 1 x 1-x 0 1 1 1 1 x 1 1 1-x 1-x x 1 x 1

1-x 1 x 1 x 1-x 1-x 1 1 0 1 1 1 1 0 1-x x 1

D X p\q 1 x 1-x 0 1 x 1-x 0

1 0 1-x x 1 0 0 0 0 x 1-x 1-x 1 1 0 1-x 0 1-x

1-x x 1 x 1 0 0 x x 0 1 1 1 1 0 1-x x 1

E J p\q 1 x 1-x 0 1 x 1-x 0

1 1 x 1-x 0 0 1-x x 1 x x 1 0 1-x 1-x 0 1 x

1-x 1-x 0 1 x x 1 0 1-x 0 0 1-x x 1 1 x 1-x 0

118

The evaluation of the above formulations of first principles is very easy

Cpp Bpp Epp

C11 = 1 B11 = 1 E11 = 1 Cxx = 1 Bxx = 1 Exx = 1 C1–x 1–x = 1 B1–x 1–x = 1 E1–x 1–x = 1 C00 = 1 B00 = 1 E00 = 1

Principle of Identity (1st weak formulation)

Principle of Identity (2nd weak formulation)

Principle of Identity (strong formulation)

NKpNp NNKpp

NK1N1 = NK10 = N0 = 1 NKN11 = NK01 = N0 = 1 NKxNx = NKx1–x = N0 = 1 NKNxx = NK1–xx = N0 = 1 NK 1–x N1–x = NK1–x x = = N0 = 1

NKN1–x 1–x = NK x1–x = = N0 = 1

NK0N0 = NK01 = N0 = 1 NKN00 = NK10 = N0 = 1

Principle of Non-Contradiction (1st formulation)

Principle of Non-Contradiction (2nd formulation)

ApNp ANpp

A1N1 = A10 = 1 A1N1 = A10 = 1 AxNx = Ax1–x = 1 AxNx = Ax1–x = 1 A1–xN1–x = A1–xx = 1 A1–xN1–x = A1–xx = 1 A0N0 = A01 = 1 A0N0 = A01 = 1

Principle of Excluded Middle (1st weak formulation)

Principle of Excluded Middle (2nd weak formulation)


119

DpNp DNpp

D1N1 = D10 = 1 DN11 = D01 = 1 DxNx = Dx1–x = 1 DNxx = D1–xx = 1 D1–xN1–x = D1–xx = 1 DN1–x1–x =Dx1–x = 1 D0N0 = D01 = 1 DN00 = D10 = 1

Principle of Excluded Middle (3rd weak formulation)

Principle of Excluded Middle (4th weak formulation)

JpNp JNpp

J1N1 = J10 = 1 JN11 = J01 = 1 JxNx = Jx1–x = 1 JNxx = J1–xx = 1 J1–xN1–x = J1–xx = 1 JN1–x1–x = Jx1–x = 1 ÅJ0N0 = J01 = 1 JN00 = J10 = 1

Principle of Excluded Middle (1st strong formulation)

Principle of Excluded Middle (2nd strong formulation)

4. Toward Sheffer’s unique function

It is known that Henry Maurice Sheffer (1913) proved that there exists a unique algebraic function (stroke operation) interpretable either as “nand” or “nor”, which has the power to define not only all the binary logical functions of classical logic but also the unary logical function (negation). Now, let us call D-formulas each expression made up of the only operator D, and X-formulas each expression made up of the only operator X. In the year 1981 I demonstrated two metatheorems: 1. Every D-.formula or X-formula expresses the sense of one of the three principles (however weak or strong the formulation) if and only if it expresses the sense of the remaining two. 2. The sense of three principles is one and only one and the same. See Malatesta (1981).

120

Classical logic with indeterminate sentences corroborates both metatheorems.

DpDpp DDppp D1D11 = D10 = 1 DD111 = D01 = 1 DxDxx = Dx1-x = 1 DDxxx = DD1–xx = 1 D1–xD1–x1–x = D1–xx = 1 DD1–x1–x1–x = Dx1–x = 1 D0D00 = D01 = 1 DD000 = D10 = 1

First Principles unique formulation

(1st D-formulation)

First Principles unique formulation

(2nd D-formulation)

XXpXppXpXpp

XX1X11X1X11 = XX10X10 = X00 = 1 XXxXxxXxXxx = XXx1–xXx1–x = X00 = 1 XX1–xX1–x1–xX1–xX1–x1–x = XX1–xxX1–xx = X00 = 1 XX0X00X0X00 = XX01X01 = X00 = 1

First Principles unique formulation (1st X-formulation)

XXXpppXXppp

XXX111XX111 = XX01X01 = X00 = 1 XXXxxxXXxxx = XX1–xxX1–xx = X00 = 1 XXX1–x1–x1–xXX1–x1–x1–x = XXx1–xXx1–x = X00 = 1 XXX000XX000 = XX10X10 = X00 = 1

First Principles unique formulation (2nd X-formulation)


121

XXpXppXXppp

XX1X11XX111 = XX10X01 = X00 = 1 XXxXxxXXxxx = XXx1–xX1–xx = = X00 = 1 XX1–xX1–x1–xXX1–x1–x1–x = XX1–xxXx1–x = X00 = 1 XX0X00XX000 = XX01X10 = X00 = 1

First Principles unique formulation (3rd X-formulation)

XXXpppXpXpp XXX111X1X11 = XX01X10 = X00 = 1 XXXxxxXxXxx = XX1–xxXx1–x = X00 = 1 XXX1–x1–x1–xX1–xX1–x1–x = XXx1–xX1–xx = X00 = 1 XXX000X0X00 = XX10X01 = X00 = 1

First Principles unique formulation (4th X-formulation)

5. Conclusion

We have proved that unlike Lukasiewicz’s many-valued logics the above classical logic with indeterminate sentences is suitable to found the probability calculus. What now? Will we reject Lukasiewicz’s n-valued logics? Not at all! Linke (1948) and Scholz (1957) proved that such logics are rigorous mathematical calculi but they are valid only outside alethic logical contexts. They are important like fuzzy logics in several fields of informatics, robotic and industrial machinery.

122

References. Feigl H. (1950); De Principiis non disputandum? On the Meaning and the Limits of Justification; in M. Blak (ed.), Philosophical Analysis, Cornell University Press, Ithaca (pp. 113-147) Linke P. F. (1948); Die mehrwertigen Logiken und das Wahrheitsproblem; “Zeitschrift für philosophische Forschung” (pp. 378-398, 530-536) Lukasiewicz, J. (1963); Elements of Mathematical Logic; Pergamon Press; Oxford-London-New York Lukasiewicz, J. (1930); Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalkülus; “Comptes rendus de la Société des Sciences et Lettres de Varsovie”, Class III, vol. XXIII (pp. 51-57) Malatesta M. (1977); On the Principles of Identitry, Non-contradiction and Excluded middle; “Rassegna di Scienze Filosofiche”, XXIX, Vol. 1 (pp. 85-89) Malatesta M. (1981); Sulla formulazione unica dei principi di identità, non contraddizione e terzo escluso, “Quaderni dell’Istituto di Filosofia Teoretica dell'Università di Napoli”, Parte terza: Napoli, Giannini. Malatesta M. (1982); Sul senso dei principi d’identità, non contraddizione e terzo escluso; Atti degli incontri di logica matematica; “Incontri di logica matematica”, Scuola di Specializzazione in Logica Matematica, Dipartimento di Matematica, Università di Siena (pp. 93-94) Rescher N. (1969); Many-valued Logic; McGraw-Hill Book Company, New York. Scholz H. (1957); In memoriam J. Lukasiewicz; “Archiv für mathematische Logik und Grundlagenforschung”, No. 3 (pp. 3-18) Sheffer H. M. (1913); A Set of Five Independent Postulates for Boolean Algebras, with Application to Logical Constants; “Transactions of the American Mathematical Society”, XIV (pp. 481-488) Skyrms B. (1966); Choice and Change; Dickenson Publishing Company, Inc. Belmont, Cal.

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