Tutorial Contents Tutorial Two: Propositional Calculus: Language - Truth Functors - The Propositional Language - "Implicate" - Stand-alone sentences - Scope More Tutorials One Three Four Five Six Seven

#### [j«y] (material equivalence; the biconditional)

[j«y] has the following truth-table:

 j  y [j«y] T  T T T  F F F  T F F  F T

One can read it as "j double arrow y"; or as "j if and only if y".

Using it one can translate "John likes rugby if and only if he is Welsh" as "[John likes rugby « John is Welsh]". When translating "if and only if" with "[j«y]" one needs, of course, to exercise the same caution as when translating "if" with "[j®y]".

Notice that we already had a way of translating "j if and only if y" without using "[j«y]". We could have paraphrased it as "j if y, and j only if y",  and translated that as "[[y®j]Ù[j®y]]". The translations come to the same thing. "[j«y]" is false just when "j" and "y" have different truth-values. "[[y®j]Ù[j®y]]" is false just when "[y®j]" is false or "[j®y]" is false; that is, it is false just when either "y" is true and "j" is false or when "y" is true and "j" is false; that is, just when "j" and "y" have different truth-values.

This is a rather a wordy way of arguing. One could represent the same facts neatly by using a truth-table:

 j  y [j«y] [[y®j] Ù [j®y]] T  T T T T T T  F F T F F F  T F F F T F  F T T T T

From this it can be seen that the two translations will be true in exactly the same circumstances. (To draw a truth table, one works from the less complicated expressions to the more complicated expressions; one writes the truth-values under the corresponding symbol. So the right hand column of values represents the values that "[j®y]" takes in each case; and the column to its left represents the values that "[[y®j]Ù[j®y]]" takes; and so on.)

##### We noticed earlier that one way of translating "j OR y", where the "or" is intended to be exclusive would be to use "[[jÚy]Ù¬[jÙy]]". A neat way of saying the same thing would be "[j«¬y]".

Let us use a truth-table to check this.

 j  y [j« ¬y] [[yÚj] Ù ¬ [jÙy]] T  T F F T F F T T  F T T T T T F F  T T F T T T F F  F F T F F T F

From this we can see that both are true just when the constituents take different truth-values.