The question that flows from that, even if we accept that there is objective truth, is, How can we know the truth when we find it? Can we rely on our consciences, our "moral compasses," so to speak, to tell us when we have found it?
Well, first the short answer: no, we can't. Our compasses have been contaminated and no longer reliably point toward north.
Of course that answer deserves some explanation. Our intellects have been darkened and wills weakened by original sin to the point that they are unreliable when it comes to identifying truth on our own. In fact, not only are we not able to recognize truth on our own, we also cannot accurately tell how far from the truth we have strayed (i.e. we cannot even recognize the existence or degree of our own sin). This is a concept that is largely lost on our society today, which tends to discount the possibility of sin altogether and explain our suffering under the human condition "as merely a developmental flaw, a psychological weakness, a mistake, or the necessary consequence of an inadequate social structure, etc." (CCC 387).
Saint Augustine, in his Confessions, admitted that even once he recognized and conceded that there was evil, he could not discern where it came from. This "mystery of lawlessness" as he referred to it, was only resolved by his conversion to the "mystery of faith."
So then, if we cannot recognize our own sin, what good is our conscience? Doesn't it play a role? Isn't our ultimate responsibility not to violate our conscience?
I tend to think of our conscience as being like one of those old cassette tapes (you remember those, don't you?). The quality of the recording depended on two things: the quality of the cassette receiving the recording, and the clarity with which the original recording was transmitted to it. In the same way, our consciences are only as good as we allow them to be by: (1) disposing ourselves--through an attitude of humility and obedience--to the source of truth and the grace by which we receive it; and (2) Seeking out that authentic truth as He has revealed himself to us, and not as we would intend him to be. Just like a cassette tape whose fidelity quickly fades when it is re-recorded from itself or other tapes instead of the master, so it is with our consciences; their fidelity quickly fades when not frequently and properly oriented to the source of righteousness and goodness.
So, yes we do have a grave responsibility not to sin by violating our conscience, but that is only half of the equation. The other half lies in our responsibility to form our consciences to be like the mind of Christ; in other words to have no tolerance to what he has defined as sin and to act as he has commanded us to act.
Then Cardinal Joseph Ratzinger explained this idea in his essay, Conscience and Truth, which is included in the compilation Values in A Time of Upheaval:
It is true on this level of judgment (conscientia in the narrower sense) that an erring conscience obligates. The rational tradition of Scholasticism makes this proposition absolutely clear. As Paul had affirmed (Rom 14:23), no one may act against his own convictions. But the fact that one's conviction is naturally binding at the moment one acts does not mean a canonization of subjectivity. One who follows the conviction at which he has arrived, never incurs guilt. Indeed, one must follow such a conviction. But guilt may very well consist in arriving at such perverse convictions by trampling down the protest made by the anamnesis of one's true being. The guilt would then like on a deeper level, not in the act itself, but in that neglect of my own being that has dulled me to the voice of truth and made me deaf to what it says within me.
So, the ability to recognize the truth and live according to it--wisdom--is not something that comes organically from within us, at least in our fallen state. Rather it has existed from the beginning of creation (cf Proverbs 8:22-36) and is given to us as a gift, to correct, untwist, and reorient us back to the purpose for which we were created, to know, love, and serve God. To go with the early compass analogy, the grace and gift of wisdom frees our compasses to swing back toward north, so that we are able to live lives oriented toward the Truth--toward Christ--and not toward ourselves and society. It cannot be obtained by grasping at it, only by submitting ourselves to His loving authority, as has been handed down to us in His Church. Then we are able to do as St. Paul commends us in his Second Letter to the Thessalonians, To avoid the deception of the "man of lawlessness" (cf 2 Thess 2:3-10) and to "stand firm and hold fast to the traditions which you were taught by us, either by word of mouth or by letter" (2 Thess 2:15).