SEMANTIC QUBE

Visualising Logic

Modified Truth Functions

Welcome to the Semantic-Qube. This project tries to understand in what sense modal logic and predicate logic are truth functional. The aim is to represent these logics using the Boolean one and zero and offer ways to visualise them.

On this page we show how enlarged and modified truth functions express modal propositions. For predicate logic start at General Truth Functions

Here is the truth function for 'p and q' in the T and F format and again as Boolean 1 and 0. 

(FFFT)(p, q) = p & q           (0001)(p, q) = p & q

Truth functional modal logic entails partitioning truth. We expand the Boolean format calling each additional Boolean 1 a partition. Instead of 1 = True there may be two partitions 11 = True or three partitions 111 = True and so on. (Additional partitions extend Boolean logic. The extension should not be confused with non standard logics). 

The following two examples (which increase the number of partitions) are equivalent to the previous examples FFFT and 0001. 

(00 00 00 11)(p, q) = p & q                             11 = T

(000 000 000 111)(p, q) = p & q                     111 = T

An expanded truth function is a standard truth function. However, in the expanded form operations more complex than the standard Boolean operators are possible. When an operation rewrites an expanded truth function the result is a modified truth function. Modification allows a truth function to express an enlarged vocabulary. With a suitable semantic modification is interpreted modally.

A plausible alethic logic is described by the simplest two partition extension (see Łukasiewicz Matrices). This is the four valued Boolean set {00, 10, 01, 11} we call B4. As before 11 = True and 00 = False. Whilst 10 and 01 also indicate truth and deny false their meaning is nuanced and the beginnings of a fuller account is found on the page Primary Color Principle. The two new values modify the meaning of an expanded truth function. For example: 

(00 00 11 11)(p) = p (00 00 00 01)(p, q) = □p & q, p & □q

(00 11 00 11)(q) = q (00 00 00 10)(p, q) = p & q & ~□q

(00 00 01 01)(p) = □p (00 00 01 00)(p, q) = □p & ~q

(00 01 00 01)(q) = □q                      (00 00 10 00)(p, q) = p & ~q & ~□p

(01 01 11 11)(p) = ◊p                       (00 01 00 00)(p, q) = □q & ~p

(01 11 01 11)(q) = ◊q                     (00 10 00 00)(p, q) = ~p & q & ~□q

(00 00 00 11)(p) = p & q             (01 00 00 00)(p, q) = ◊p & ~p & ~q

(01 01 01 11)(p) = ◊p & ◊q           (10 00 00 00)(p, q) = ~◊p & ~◊q 

Many valued modal logic is not a new idea. Modal matrices trace back to the early years following the First World War. However more may be done to understand the relationship between many valued logic and truth tables. We attempt to clarify the conceptual framework. To do this we strip down a truth table to its basics and offer an interpretation standard accounts often pass over. Go to Truth Tables for that account. We also provide an important rule for interpreting many valued logic. Go to Primary Valued Principle to see that rule. There is also the question of the plausibility of this modal system. How does it compare to S5 and other well known modal logics? We begin to answer these questions in the section Łukasiewicz Matrices.

 

Garry              

contact: semantic-qube@outlook.com

Last edited 2015-12-5