# In Predicate Logic, for a Conditional Proof, why can you directly assume the Statement Function?

Abbreviate Conditional Proof to CP, Statement Function to SF, and Universal Quantifier to UQ.

Source: p 483, A Concise Introduction to Logic (12 Ed, 2014) by Patrick Hurley

The next example differs from the previous one in that the antecedent of the conclusion is a statement function, not a complete statement. With arguments such as this, only the statement function is assumed as the first line in the conditional sequence. The quantifier is added after the sequence is discharged. Earlier, p 464 stated the necessity of removing all quantifiers before applying any Rule of Inference. So how is the bold correct? Why can you correctly assume only the SF?

Ax is an assumption (see note : ACP) : thus, it can be whatever we want.

ACP is used to "prepare" you to apply the MP in line 5, in order to derive Cx, under assumption Ax.

Then, by CP, we derive : Ax ⊃ Cx.

As you can see in line 4, UI is used to "unpack" the leading UQ of premise 1.

The UQ is already "there" in 4: thus we have no need to assume the UQ again in the ACP, in order for us to apply UI.

• Thank you. I should have better written my question: Because you can assume whatever you wish, why is it easier or advised to assume only the SF for ACP? – Greek - Area 51 Proposal Jan 5 '16 at 21:27