How can I make a tautological conclusion from these premises?

[ETuten]

I am in needof help with the following problem:

Premises : Cube(a) v Cube(b)
Dodec(c) v Dodec(d)
~Cube(a) v ~Dodec(c)

Conclusion: Cube(b) v Dodec(d)


I need to add a sentence to the proof that is tautalogical consequence of two of the premises. I just can't see how to make such a deduction. Any help would be much appreciated

 

[MathematicalPhysicist]

well it's quite immediate if take into account material conditional and destructive dillema:
~cube(a)->cube(b)
~dodec(c)->dodec(d)
~Cube(a) v ~Dodec(c)
so from DD you get the conclusion.
but let's say for the sake of argument that you can't use it here, so:
so what about ((Cube(a)vCube(b))->(Dodec(c)vDodec(d))->(~Cube(a)v~Dodec(c))->(Cube(b)vDodec(d))
you only need to check that then next thing is a tautology, quite long:
((~P->Q)->(~R->S)->(P->R)->(~Q->~S))
but as you might see it's all equivalent.

 

[Architeuthis Dux]

.

One way to make a tautological conclusion from these premises is by using the rule of disjunctive syllogism. This rule states that if we have two disjunctions (statements joined by "or") and the negation of one of the disjuncts, then we can conclude the other disjunct. In this case, we have the premises "Cube(a) v Cube(b)" and "~Cube(a) v ~Dodec(c)", and by applying the rule of disjunctive syllogism, we can conclude "Cube(b) v Dodec(c)". This is a tautological conclusion because it is always true, regardless of the truth values of the premises.