Conditional Logic Translations

If X, then Y

x → y
not y → not x

X if Y

y → x
not x → not y
note: in this format, “if” is always followed by the trigger (left side)

X only if Y

x → y
not y → not x
note: “only” makes Y a necessary condition of X

Only X are Y

Y → X
not x → not y
“only” makes x a necessary condition of y

Any X is Y

x → y
not y → not x

Every X is Y

x → y
not y → not x

No X is/are Y

x → not y
y → not x
NOTE: in this case, everything that is X cannot be Y. Diagram meanings not words to avoid confusion.

X cannot be Y

x → not y
y → not x

Without X, there can be no Y

y → x
not x → not y
NOTE: X is a necessary condition of Y

X requires Y

x → y
not y → not x
NOTE: sufficient condition on left, requirement on right

In order for X to be true, Y must be true

x → y
not y → not x
note: sufficient condition on left

X depends on Y

x → y
not y → not x
NOTE: “depends” signifies a necessary relationship.

X happens whenever Y happens

y → x
not y → not x
NOTE: Y is the trigger.

No X unless Y

x → y
not y → not x
NOTE: unless is not the trigger, but it does signal the necessary condition