# Logical Formula and Arithmetic Order

Logical Formula
and
Calculation Order

There has been some introduction to the formulas that connect T and F with the logical arithmetic unit. However, more complicated formulas would result if those formulas are combined. For example, without a defined order, the formula below could have a truth value of either true or false (in other words, it results in =T or =F). The following shows the order in which to find the truth value.

( ) > NOT > AND > OR > XOR > IMP > EQV

In short, what is inside the parentheses should be calculated first. The calculation should then progress from the highest to the lowest priority. Calculations proceed from left to right when the propositions of the same priority are next to each other.

The calculation order for the formula below:

 T OR (NOT F) EQV T IMP F IMP T XOR F AND F (3) (1) (7) (5) (6) (4) (2)

Let's examine the calculation order:

(1) NOT F = T
(2) F AND F = F
(3) T OR T = T The second T is the result of 1.
(4) T XOR F = T The second F is the result of 2.
(5) T IMP F = F
(6) F IMP T = T The first F is the result of 5.
(7) T EQV T = T The first T and the second T are results of 3 and 6.

The value of the whole formula becomes T from the result of 7. The following shows the formula and the answer:

T OR(NOT F) EQV T IMP F IMP T XOR F AND F = T