Formal logic is the a priori study of statements called propositions and of deductive arguments by way of identifying structures or logical forms in these elements and expressing them in symbolic notation in order to test their validity. Therefore, formal logic is not empirical study because it does not depend on a posteriori observations for data.
A deductive argument is one in which the conclusion, a proposition, follows necessarily from the premises, another proposition or set of propositions such that denying the conclusion would be inconsistent or contradictory.
Conditions of proof for a sound argument
In order to prove the truth of the conclusion of a deductive argument,
- the premises must be true;
- the deduction must be logically correct.
If a deductive argument meets these conditions, it is called sound.
Whilst formal logic will deal with the first condition, it cannot determine the truth or falsity of the premises where the propositions are a posteriori, contingent or synthetic. The truth or falsity of the premises, in this case, rests with the empiricist.
Therefore in proving the soundness of the deductive argument for the existence of God I shall use both formal logic to prove the deduction and both innate and empirical evidence to prove the truth of the premises from which the existence of God as a conclusion necessarily follows.
If, however, only the first condition, that the conclusion is logically deducible from its premises, the argument is said to be deductively valid even though the premises are false or not known to be true. However, the argument would be unsound.
For example, the argument that:
- Every dog is a mammal.
- Some quadrupeds are dogs.
- Therefore, some quadrupeds are mammals.
is valid, because they can expressed in the same valid logical inference form:
- Every X is a Y.
- Some Z’s are X’s.
- Therefore, some Z’s are Y’s.
However, the soundness of each argument depends also on whether the premises are true or false, and this is outside the scope of formal logic if the propositions a posteriori, contingent or synthetic.