Syādvāda Logic; Open Edition

https://syad.etud.io ; Copyright “2022”; CC-BY-NC-SA; Alex Márquez Pérez Muñíz Díaz Puras Thaureaux Fagúndo RIvera <mw@etud.io> (گیسپر גִּזְבָּר(‎𑀆𑀚𑀻𑀯))

Syādvāda is a system of argumentation developed by Jaina philosophers to account for the complexity and multifaceted nature of reality. It is based on the idea that any given statement or claim about an object or phenomenon can be viewed from multiple perspectives, each of which may be true or false depending on the context and the assumptions being made.

This seven predicate theory consists in the use of seven claims about sentences, each preceded by "arguably" or "conditionally", concerning a single object and its particular properties, composed of assertions and denials, either simultaneously or successively, and without contradiction. These seven claims are the following:

Arguably, it (that is, some object) exists (syad asty eva).

Arguably, it does not exist (syan nasty eva).

Arguably, it exists; arguably, it doesn't exist (syad asty eva syan nasty eva).

Arguably, it is non-assertible (syad avaktavyam eva).

Arguably, it exists; arguably, it is non-assertible (syad asty eva syad avaktavyam eva).

Arguably, it doesn't exist; arguably, it is non-assertible (syan nasty eva syad avaktavyam eva).

Arguably, it exists; arguably, it doesn't exist; arguably it is non-assertible (syad asty eva syan nasty eva syad avaktavyam eva).

There are three basic truth values, namely, true (t), false (f) and unassertible (u). These are combined to produce four more truth values, namely, tf, tu, fu, and tfu. Though, superficially, it appears that there are only three distinct truth values, a deeper analysis of the Jaina system reveals that the seven truth values are indeed distinct. This is a consequence of the conditionalizing operator "arguably" denoted in Sanskrit by the word syat. This Sanskrit word has the literal meaning of "perhaps it is", and it is used to mean "from a certain standpoint" or "within a particular philosophical perspective".

In this discussion the term "standpoint" has been used in a technical sense. Consider a situation in which a globally inconsistent set of propositions, the totality of philosophical discourse, is divided into sub-sets, each of which is internally consistent. Any proposition might be supported by others from within the same sub-set. At the same time, the negation of that proposition might occur in a distinct, though possibly overlapping subset, and be supported by other propositions within it. Each such consistent sub-set of a globally inconsistent discourse, is what the Jainas call a "standpoint" (naya). A standpoint corresponds to a particular philosophical perspective.

In this terminology, it can be seen that the seven predicates get translated to the following seven possibilities. Each proposition p has the following seven states:

p is a member of every standpoint in S.

Not-p is a member of every standpoint in S.

p is a member of some standpoints, and Not-p is a member of the rest.

p is neutral with respect to every standpoint.

p is a member of some standpoints, the rest being neutral.

Not-p is a member of some standpoints, the rest being neutral.

p is a member of some standpoints and Not-p is a member of some other standpoints, and the rest are neutral.

Consider a 7-value PARACONSISTENT modal formal logic using the following 7 identifiers to represent each of those claims as a separate logic value: T F TF U TU FU TFU

This is the truth table for the unary operator NOT:

p NOT p

T F

F T

TF TF

U U

TU FU

FU TU

TFU TFU

Here is the table for the binary operator AND:

| T F TF U TU FU TFU

---|-----|-----|-----|-----|-----|-----|-----

T | T F TF U TU FU TFU

F | F F F F F F F

TF | TF F TF TF TU FU TFU

U | U F TF U U FU U

TU | TU F TU U TU FU TU

FU | FU F FU FU FU FU FU

TFU| TFU F TFU U TU FU TFU

Here is NAND's table:

| T F TF U TU FU TFU

---|-----|-----|-----|-----|-----|-----|-----

T | F T FU U FU FU FU

F | T T T U TU TU TU

TF | FU T FU U TU FU FU

U | U U U U U U U

TU | FU TU FU U FU TU FU

FU | FU TU FU U TU FU TU

TFU| FU TU FU U TU TU FU

Here is OR's table:

| T F TF U TU FU TFU

---|-----|-----|-----|-----|-----|-----|-----

T | T T T T T T T

F | T F TF U TU FU TFU

TF | T TF TF TF T TF TFU

U | T U TF U TU U U

TU | T TU T TU TU TU TU

FU | T FU TF U TU FU TFU

TFU| T TFU TFU U TU TFU TFU

NOR's table:

| T F TF U TU FU TFU

---|-----|-----|-----|-----|-----|-----|-----

T | F F F F F F F

F | F T FU U FU FU FU

TF | F FU F F FU F F

U | F U F U F U F

TU | F FU FU U F FU FU

FU | F FU F U FU F FU

TFU| F FU F F FU FU F

XOR table:

| T F TF U TU FU TFU

---|-----|-----|-----|-----|-----|-----|-----

T | F T TU U TU TU TU

F | T F TF U TF TF TF

TF | TU TF TU U TFU TU TU

U | U U U U U U U

TU | TU TF TFU U TU TU TU

FU | TU TF TU U TU TU TU

TFU| TU TF TU U TU TU TFU

XNOR table:

| T F TF U TU FU TFU

---|-----|-----|-----|-----|-----|-----|-----

T | T F FU U FU FU FU

F | F T TU U TU TU TU

TF | FU TU FU U TU FU FU

U | U U U U U U U

TU | FU TU TU U FU TU FU

FU | FU TU FU U TU FU TU

TFU| FU TU FU U TU TU TU

The IMPLIES table:

| T F TF U TU FU TFU

---|-----|-----|-----|-----|-----|-----|-----

T | T F TU U TU FU TU

F | T T T U TU TU TU

TF | TU TU TU U TFU TU TU

U | U U U U U U TFU

TU | TU TU TFU U TU TU TU

FU | TU TU TU TU TU TU TU

TFU| TU TU TU TFU TU TU TFU

NONIMPLIES table:

| T F TF U TU FU TFU

---|-----|-----|-----|-----|-----|-----|-----

T | F T FU U FU TU FU

F | F T TU U FU TU TU

TF | FU FU FU U FU FU FU

U | U U U U U U U

TU | FU TU FU U FU TU FU

FU | TU FU TU U TU FU TU

TFU| FU FU FU U FU FU FU

BICOND table:

| T F TF U TU FU TFU

---|-----|-----|-----|-----|-----|-----|-----

T | T F TU U TU FU TU

F | F T T U TU TU TU

TF | TU TU TU U TFU TU TU

U | U U U U U U U

TU | TU TU TFU U TU TU TU

FU | TU TU TU U TU TU TU

TFU| TU TU TU U TU TU TU

Note that BICOND != IMPLIES(A,B) & IMPLIES(B,A)

EQV table:

| T F TF U TU FU TFU

---|-----|-----|-----|-----|-----|-----|-----

T | T F TU U TU FU TU

F | F T TU U FU TU TU

TF | TU TU TU U TU TU TU

U | U U U U U U U

TU | TU FU TU U TU FU TU

FU | FU TU TU U FU TU TU

TFU| TU TU TU U TU TU TU

NEQV table:

| T F TF U TU FU TFU

---|-----|-----|-----|-----|-----|-----|-----

T | F T FU U FU TU FU

F | T F FU U TU FU FU

TF | FU FU FU U FU FU FU

U | U U U U U U U

TU | FU TU FU U FU TU FU

FU | TU FU FU U TU FU FU

TFU| FU FU FU U FU FU FU

Now some of the ternary operators:

MAJORITY: At least 2 arguments are T

MAJORITY(A, B, C) = (A & B) | (B & C) | (A & C)

MINORITY: No more than 1 argument is T

MINORITY = NOT MAJORITY

XMAJORITY: Exactly 2 arguments are T

XMAJORITY(A, B, C) = (A & B & NOT C) | (A & NOT B & C) | (NOT A & B & C)

XMINORITY: Exactly 1 argument is T

XMINORITY(A, B, C) = (A & NOT B & NOT C) | (NOT A & B & NOT C) | (NOT A & NOT B & C)

PARITY: The number of T's is odd

PARITY(A, B, C) = XMINORITY(A, B, C) | (A & B & C)

IMPLIES3: Formulated -- respectively -- not requiring transitivity, requiring circularity, and further entirely requiring transitivity as:

N-IMPLIES3(A,B,C) := IMPLIES(A,B) & IMPLIES(B,C)
C-IMPLIES3(A,B,C) := N-IMPLIES3(A,B,C) & IMPLIES(C,A)
T-IMPLIES3(A,B,C) := C-IMPLIES3(A,B,C) & IMPLIES(A,C)

NONIMPLIES3: Formulated -- respectively -- not requiring transitivity, requiring circularity, and further entirely requiring transitivity as:

N-NONIMPLIES3(A,B,C) := NOT (IMPLIES(A,B) | IMPLIES(B,C))
C-NONIMPLIES3(A,B,C) := NOT (IMPLIES(A,B) | IMPLIES(B,C) | IMPLIES(C,A))
T-NONIMPLIES3(A,B,C) := NOT (IMPLIES(A,B) | IMPLIES(B,C) | IMPLIES(C,A) | IMPLIES(A,C))

TRICOND: Formulated -- respectively -- not requiring transitivity, and requiring transitivity as:

N-TRICOND(A,B,C) := BICOND(A,B) & BICOND(B,C)
T-TRICOND(A,B,C) := N-TRICOND(A,B,C) & BICOND(C,A)

As a reminder, these ternary operators are equivalent to their binary counterparts:

AND3 = (A & B) & C = A & (B & C)
OR3 = (A | B) | C = A | (B | C)
NAND3 = NOT AND3
NOR3 = NOT OR3

Also:

XOR3 = XMINORITY
XNOR3 = NOT XMINORITY

TRINEQV:

N-TRINEQV(A,B,C) := NEQV(A,B) & NEQV(B,C)
T-TRINEQV(A,B,C) := NEQV(A,B) & NEQV(B,C) & NEQV(A,C)