Are there any examples of deductive arguments used to defend a doctrine?

An Example

One example (out of many, I am sure):

In the Summa Theologiae, Thomas Aquinas argues for the Filioque clause, as follows; I have added some formatting and numbering to bring out the analytic structure of the argument:

It must be said that the Holy Ghost is from the Son. For if He were not from Him, He could in no wise be personally distinguished from Him; as appears from what has been said above [i.e. elsewhere: see here, and here].

1. ...
   • it cannot be said that the divine Persons are distinguished from each other in any absolute sense;
   • for it would follow that there would not be one essence of the three persons: since everything that is spoken of God in an absolute sense, belongs to the unity of essence. Therefore

• it must be said that the divine persons are distinguished from each other only by the relations.

2. Now the relations cannot distinguish the persons except forasmuch as they are opposite relations;
3. • which appears from the fact that the Father has two relations, by one of which He is related to the Son, and by the other to the Holy Ghost; but these are not opposite relations, and therefore they do not make two persons, but belong only to the one person of the Father.

• If therefore in the Son and the Holy Ghost there were two relations only, whereby each of them were related to the Father [e.g. if the Filioque was rejected], these relations would not be opposite to each other, as neither would be the two relations whereby the Father is related to them.

4. Hence, as the person of the Father is one, it would follow that the person of the Son and of the Holy Ghost would be one, having two relations opposed to the two relations of the Father.
5. But this [the identification of the Son with the Spirit] is heretical since it destroys the Faith in the Trinity.
6. Therefore the Son and the Holy Ghost must be related to each other by opposite relations.
7. Now there cannot be in God any relations opposed to each other, except relations of origin, as proved above [i.e. elsewhere].
8. And opposite relations of origin are to be understood as of a "principle," and of what is "from the principle."
9. Therefore we must conclude that it is necessary to say that either the Son is from the Holy Ghost; which no one says; or that the Holy Ghost is from the Son, as we confess.

Let's look at the structure of the argument in brief:

1. (Premise A.i) It must be said that the divine persons are distinguished from each other only by the inter-personal relations,
2. (Premise A.ii) moreover they can be distinguished only by relations which are opposite.
3. (Premise B) The Spirit and the Son are only individually related to the Father, if and only if the Holy Spirit and the Son are not related with opposite relations.
4. (Conclusion W) If the Spirit and the Son were only individually related to the Father, the Holy Spirit and the Son would not be distinguished.
5. (Premise C) But the Holy Spirit and the Son are distinguished [because of the Trinity], so
6. (Conclusion X) The Holy Spirit and the Son are related with opposite relations.
7. (Premise D) All opposite relations in God are relations of origin.
8. (Premise E) Relations of origin involve both a 'to' and a 'from'.
9. (Conclusion Y) Among Holy Spirit and the Son are to be identified one as a 'to' and the other a 'from'. That is to say, either the Holy Ghost is from the Son, or the Son is from the Holy Ghost.
10. (Premise F) The Son is not from the the Holy Ghost [since nobody says that].
11. (Conclusion Z) The Holy Ghost is from the Son.

A, B gives W. Then, B, C, and W gives X. Next X, D, and E gives Y. Finally, Y and F gives Z.

Although Aquinas did not present the argument in this precise Premise-Conclusion format, it should be fairly clearly from the original text that his line of argument is deductive in nature. In fact, this is his style for the most of the Summa.

Validity, Soundness

The argument is valid, because the conclusions follow from the premises.

The argument is only sound if you accept the premises. That's going to be more controversial. Premises B, C, E and F are probably less controversial compared to Premises A and D, which is why Aquinas attempts to defend those two elsewhere: see the links in the original quoted text above.

General Remarks

Christian theology is not much different from other disciplines: deductive reasoning is a tool to clarify thinking and to persuade. Different schools of thought rely on this tool more than others.

The Scholastics, such as Aquinas, were characteristically fond of deductive reasoning (albeit sometimes based on controversial or speculative premises), but this is by no means exclusive to them. In fact, except for some of the theology of the 19th and 20th century, historically deductive reasoning was arguably the norm for settling doctrinal disputes. In more modern times, a trend toward deductive reasoning lies in Analytic Theology, following trends in its namesake: Analytic Philosophy.

It should be noted that deductive reasoning does not always take the form of Premise-Conclusion tabulated propositions, or formal logic. Indeed, this is usually overkill and fairly uncommon in ordinary theological discourse for reasons of brevity. However, the reasoning employed may still be deductive.