Tutorial Contents Tutorial Two: Propositional Calculus: Language - Truth Functors - The Propositional
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[jy] (material equivalence; the biconditional)

 

[jy] has the following truth-table:

 

 
j y

[jy]

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 "[jy]" one needs, of course, to exercise the same caution as when translating "if" with "[jy]".

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

[jy]
[[yj]
[jy]]

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 "[jy]" takes in each case; and the column to its left represents the values that "[[yj][jy]]" 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 "[[jy][jy]]". A neat way of saying the same thing would be "[jy]".

 

Let us use a truth-table to check this.

 

 
j y

[j
y]
[[yj]
[jy]]

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.

 

 

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