If any truth-functional argument is valid, we have the tools to prove it valid; and if it is invalid, we have the tools to prove it invalid. For each of the following arguments determine whether it is valid or invalid using the STTT. If it is valid, prove it valid using the nineteen rules of inference. These proofs will be more difficult to construct than in preceding exercises, but they will offer greater satisfaction.

If the president cuts Social Security benefit payments, he will lose the support of the senior citizens; and if he cuts defense spending, he will lose the support of the conservatives. If the president loses the support of either the senior citizens or the conservatives, then his influence in the Senate will diminish. But his influence in the Senate will not diminish. Therefore the president will not cut either Social Security benefits or defense spending. ( B, S, D, C, I )