1. A contingent being is one the nonexistence of which is possible, whereas a necessary being is one the nonexistence of which is impossible. (At play in these definitions is broadly logical possibility which is between narrowly logical and nomological possibility.)
2. Framing a definition is one thing, showing that something answers to it is another. Are there any necessary beings? Since a necessary being could be either abstract or concrete, I can show that there are necessary beings by showing that there is at least one abstract necessary being. To convey the senses of 'concrete' and 'abstract' by example one could say that God and Socrates are concrete while the proposition 7 is prime and Socrates' singleton -- {Socrates} -- are abstract. All and only concreta are causally active/passive whereas abstracta are not. Please avoid the mistake of thinking that x is concrete if and only if x is physical.
3. Some truths are necessary, others are contingent. 'I am now blogging' is contingently true: it is true, but it might not have been true. I might have been doing something inconsistent with blogging now, sleeping for instance. By contrast, 'If I am blogging, then I am writing' is necessarily true. To see this, negate the sentence in question. The result is a sentence expressing a broadly logical impossibility: 'I am blogging and it is not the case that I am writing.' Consider also, 'If I am blogging, then it is not the case that I am not blogging.' This too is necessarily true, except that the negation expresses a narrowly logical impossibility: 'I am blogging and I am not blogging.'
I don't see how any reasonable person can deny that there are necessary truths. Another example: '7 is a prime number' expresses a necessary truth. This doesn't just happen to be true in the way that it just happens to be true that there are seven cans of Dr. Pepper left in the reefer. It is necessarily true: true in all (BL)-possible worlds.
4. A truth is a true truth-bearer. Now I don't understand how ink on paper, or chalk on a blackboard, or any physical modification of any physical medium, no matter how complex the modification and how complex the medium, could be true or false. I don't understand how anything physical could, qua physical, be a truth-bearer or truth-vehicle, i.e., an item capable of being either true or false. Marks on paper cannot be either true or false. They just exist. But suppose you think they -- or complex modifications of the stuff between your ears -- can be either true or false. Still, the marked-up paper exists contingently. Consequently, the sentence-token '7 is prime' scratched onto the paper exists contingently. Similarly for anything inscribed in your brain. Your brain and its 'inscriptions' are contingent.
5. But then how could any truth be necessarily true? How could any truth be necessarily true if no truth-bearer is necessarily existent? There is no possible world in which 7 is not prime, but there are worlds in which there are no material things. Material things are contingent. How could the proposition in question be true in those worlds if there is nothing in those worlds to serve as truth-bearer? Let's spell this out.
If an item has a property, then, pace Meinong, the item exists: existence is a necessary condition of property-possession. So if an item such as a truth-bearer has the property of being necessarily true, then that truth-bearer necessarily exists. For if the truth-bearer is true in every world, then it exists in every world. Therefore, if there are necessary truths, then there are necessary beings. Now there are necessary truths. Therefore, there are necessary beings. Given that everything physical is contingent, these necessary beings are nonphysical. So they are either mental (accusatives of mental acts) or abstract. For present purposes, it doesn't matter which of these they are. The present point is that there is good reason to believe in (i.e., believe that there are) necessary beings.
6. But I hear an objection coming: An item can have a property essentially without having it necessarily. Thus Socrates is essentially human, but not necessarily human. He is human in every world in which he exists, but he does not exist in every world. So he is essentially but not necessarily human. Why can't the proposition expressed by '7 is prime' be like that? Why can't it be essentially (as opposed to accidentally) true, true in every world in which it exists, but neither true nor false in the worlds in which it does not exist? If this is the way it is, then your argument from necessary truths to necessary beings collapses.
The objector is suggesting that truth-bearers are contingent beings. But this is problematic as Alvin Plantinga argues (Warrant and Proper Function, Oxford UP, 1993, p. 119.) Suppose that truth-bearers are brain inscriptions, and consider the proposition
1. There are brain inscriptions.
(1) is such that it could not have been false. You read that right and I wrote it right. For in a possible world in which there are no brain inscriptions, there are no truth-bearers, which implies that (1) in those words is neither true nor false, hence not false. And in every world in which there are brain inscriptions, (1) is of course true. So (1) is true in every world in which it exists, and not false in every world in which it does not exist. So (1) could not have been false. But this is bizarre. Surely there might have been no brains and no brain-inscriptions. It is not necessarily true that there are brains. If it is not necessarily true that there are brains, then it is possibly true that there are no brains. Now what is this possibility of there being no brains? It is plausibly identified with the possibly being true of the proposition, There are no brains. But then this proposition must exist in those possible worlds in which it is not true.