The Savior of all

3 This is good, and it is pleasing in the sight of God our Savior, 4 who desires all people to be saved and to come to the knowledge of the truth. 5 For there is one God, and there is one mediator between God and men, the man Christ Jesus, 6 who gave himself as a ransom for all (1 Tim 2:3-5). 

10 For to this end we toil and strive, because we have our hope set on the living God, who is the Savior of all people, especially of those who believe (1 Tim 4:10). 

11 For the grace of God has appeared, bringing salvation for all people (Tit 2:11).

This comprises a set of popular Arminian prooftexts. They accuse Calvinists of tampering with the plain sense of these passes. 

But, of course, the Arminian appeal isn't that straightforward. Because Arminians think human freewill can overrule God's will, they don't think God actually saves everyone, or that God is really the Savior of everyone. So they have to qualify their prooftexts by interjecting "potential" into the passages.

How do Calvinists construe them? Let's consider two interpretive approaches:

i) Tom Schreiner notes that in the Pastorals, Paul is combatting a Jewish heresy which seems to restrict access to salvation to those with the right bloodlines. So Paul would be correcting that heresy by explaining that access to salvation cuts across ethnic and genealogical distinctives. Pedigree can't save you or condemn you. Cf. Paul: Apostle of God's Glory in Christ, 184-85.

ii) Another explanation is that pagans believed in tribal or national gods. Local patron gods. But if there is only one God, then that's the only God you can turn to for salvation. One God is the source of salvation for anyone. That nicely explains Paul's one-to-all correlation, which has its background in OT monotheism.