What evaluates to True/False in R?

This is documented on ?logical. The pertinent section of which is:

Details:

     ‘TRUE’ and ‘FALSE’ are reserved words denoting logical constants
     in the R language, whereas ‘T’ and ‘F’ are global variables whose
     initial values set to these.  All four are ‘logical(1)’ vectors.

     Logical vectors are coerced to integer vectors in contexts where a
     numerical value is required, with ‘TRUE’ being mapped to ‘1L’,
     ‘FALSE’ to ‘0L’ and ‘NA’ to ‘NA_integer_’.

The second paragraph there explains the behaviour you are seeing, namely 5 == 1L and 5 == 0L respectively, which should both return FALSE, where as 1 == 1L and 0 == 0L should be TRUE for 1 == TRUE and 0 == FALSE respectively. I believe these are not testing what you want them to test; the comparison is on the basis of the numerical representation of TRUE and FALSE in R, i.e. what numeric values they take when coerced to numeric.

However, only TRUE is guaranteed to the be TRUE:

> isTRUE(TRUE)
[1] TRUE
> isTRUE(1)
[1] FALSE
> isTRUE(T)
[1] TRUE
> T <- 2
> isTRUE(T)
[1] FALSE

isTRUE is a wrapper for identical(x, TRUE), and from ?isTRUE we note:

Details:
....

     ‘isTRUE(x)’ is an abbreviation of ‘identical(TRUE, x)’, and so is
     true if and only if ‘x’ is a length-one logical vector whose only
     element is ‘TRUE’ and which has no attributes (not even names).

So by the same virtue, only FALSE is guaranteed to be exactly equal to FALSE.

> identical(F, FALSE)
[1] TRUE
> identical(0, FALSE)
[1] FALSE
> F <- "hello"
> identical(F, FALSE)
[1] FALSE

If this concerns you, always use isTRUE() or identical(x, FALSE) to check for equivalence with TRUE and FALSE respectively. == is not doing what you think it is.

T and TRUE are True, F and FALSE are False. T and F can be redefined, however, so you should only rely upon TRUE and FALSE. If you compare 0 to FALSE and 1 to TRUE, you will find that they are equal as well, so you might consider them to be True and False as well.