Use the method of conditional proof to explain in words why the sentence {(P Q)

Question

Use the method of conditional proof to explain in words why the sentence {(P Q) [(P => R) (Q => S)]} => (R S) is a tautology.^12 Be explicit about discharging assumptions. Note that it would be very tedious to do the preceding exercise by means of a truth table. Since 4 propositional variables are involved (namely P, Q, R, and S), the truth table would have 2^4 = 16 rows. Also, it would have 11 columns. Thus there would be a total of 16 times 11 = 176 entries in the truth table! The explanation in words can be given in just a few lines and is much more enlightening than the truth table would be.

Explanation / Answer

given proposition can be simplified into three TRUE parts as all are joined by AND operator.

PVQ    ...(1)
P=>R   ...(2)
Q=>S   ...(3)

PVQ means either P or Q is true

if P is true then
P, applying Modus Ponens on (2) gives R is TRUE.    ...(4)
if Q is true then
Q, applying Modus Ponens on (3) gives S is TRUE.    ...(5)

combining (4) and (5) gives either R or S is TRUE
Hence
RVS is TRUE.

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