Leftow's 'Latin Trinity' standard demonstrates that numerical identity, not mere cooperation, is necessary for orthodox Trinitarian doctrine, and Swinburne's model explicitly abandons numerical identity.

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Latin Trinity(as a specific approach to explaining the Trinity): A traditional way of understanding the Christian Trinity (God as Father, Son, and Holy Spirit) that comes from medieval theology, emphasizing how three persons can be one God.
Leftow(as a philosopher being cited for his views on the Trinity): Brian Leftow is a contemporary philosopher who specializes in philosophy of religion and has written influential work on how to understand the Christian Trinity.
Mere cooperation(as contrasted with numerical identity in Trinity doctrine): When separate things work together or coordinate their actions without actually being the same thing—like how three musicians playing together are still three different people.
Orthodox Trinitarian doctrine(as the standard Christian view being discussed): The traditional Christian theological position that God exists as three distinct persons (Father, Son, Holy Spirit) while remaining one God.
Swinburne(in philosophy of religion): Richard Swinburne, a famous British philosopher who wrote about God, religion, and the problem of evil—he argued that God's existence can be rationally defended despite the existence of evil in the world.
numerical identity(Distinguished from qualitative similarity when discussing whether a tailed cat and a tailless cat are the same individual.): The relation an entity bears to itself and nothing else; being one and the same individual entity rather than merely qualitatively similar.

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