[3.5] Telescoping implications; transitive property) If a first logical statement implies the second, and the second in turn implies the third, then the first logical statement implies the third.
Notation: For any logical statements p,q,r if p ==> q and q ==> r then p ==> r. This statement is true for any truth values of the variables.
A truth table will be used to prove [3.5]. The expressions in parentheses indicate which columns to the left to get truth values.
1 2 3 4 5
6
7
8
(1+2)
(2+3)
(4+5)
(1+3)
(6+7)
p q r p ==> q q ==> r
(p ==> q) and (q ==> r) p ==> r
no T under 6 and F under 7 (in same row)
T T T T T
T
T
T
T T F T
F
F
F
T
T F T F T
F
T
T
T F F F T
F
F
T
F T T T T
T
T
T
F T F T F
F
T
T
F F T T T
T
T
T
F F F T T
T
T
T
It is important to remember the only way that an implication can be false: the hypothesis is T but the conclusion is F. In all other situations, the implication is true. These facts are used to get truth values for columns 4,5 and 7. The truth values for column 6 are determined by the definition of a conjunction: a conjunction is T if and onliy if both participating statements are T Otherwise the conjunction is false. Column 8 is obtained merely looking at columns 6 and 7. The fact that column 8 has no F's means that the expression under 6 does imply the expression under 7. This proves [3.5].