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Aquinas proves the Trinity

October 23, 2011

This was taken from W. Hodges’ Model Theory.

According to Thomas Aquinas, God is a structure G with the elements ‘pater‘, ‘filius‘ and ‘spiritus sanctus‘, in a signature consisting of one asymmetric binary relation (‘relatio opposita‘) R , read as ‘relatio originis‘. Aquinas asserts also that the three elements can be uniquely identified the terms of R^G . We are to deduce – as Aquinas did – that if the pairs (pater, filius) and (pater, spiritus sanctus) lie in R^G, then exactly one of the pairs (filius, spiritus santus) and (spiritus sanctus, filius) lies in R^G.

Proof:
According to Aquinas, (pater, filius) \in R^G. Likewise, (pater, spiritus sanctus)\in R^G. Then there are only two possibilities left that need to be dealt with in the relation, that of filius and spirtitus sanctus. Either the case is (filius, spiritus\, santus) \in R^G or it is (spiritus\, sanctus, filius) \in R^G, or both. Assume it is (filius, spiritus\, santus)\in R^G, then (spiritus\, sanctus, filius) \not \in R^G, because R is asymmetric. On the other hand, if it is the case that (spiritus\, sanctus, filius) \in R^G then (filius, spiritus\, santus) \not \in R^G, since R^G is asymmetric.
\therefore only one of the two lies in R^G.
Q.E.D.

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