The Trinity is logically incoherent — you cannot have one God who is also three Gods.

Why people find it persuasive

'Three-in-one' sounds arithmetically impossible on its face, and centuries of failed lay explanations (clover leaves, water/ice/steam) that theologians themselves reject as misleading only reinforce the sense that it's incoherent nonsense dressed up as mystery.

Important distinctions

• • One what vs. three what — the crucial question of which term 'one' and 'three' each modify.
• • Person (a center of consciousness and relation) vs. essence/nature (what a thing fundamentally is).
• • Logical contradiction (A and not-A in the same respect) vs. a claim that is merely surprising or hard to fully picture.
• • Popular folk analogies vs. the technical doctrine actually defined at Nicaea and Chalcedon.

Christian response

• • State the doctrine precisely rather than in caricature, because the objection as posed attacks a claim no creed makes. Christianity does not say one God is also three Gods. It says God is one in respect of essence or nature — what God is — and three in respect of persons — who God is, understood as centers of relation within that one nature. A contradiction requires affirming and denying the same predicate of the same subject in the same respect at the same time. 'One nature, three persons' does not do that, any more than 'one triangle, three angles' does.
• • Then show why the distinction is not a verbal trick by testing it. 'Nature' and 'person' plainly are not synonyms in ordinary usage: you and your neighbor share one nature, humanity, while being two persons, which already establishes that counting natures and counting persons are different operations. The analogy breaks down at the crucial point, since you and your neighbor each have a separate instance of human nature while the divine persons share numerically one and the same being — but its limited job is to establish that person-count and nature-count come apart, and it does that job.
• • Explain what the doctrine's technical vocabulary was invented to do. The Cappadocian Fathers used ousia for the one being and hypostasis for each of the three, defining the persons by their relations of origin — the Father unbegotten, the Son begotten, the Spirit proceeding — rather than by separate substances or separate parts. Nicaea's homoousios was chosen precisely to rule out the two errors people usually assume the doctrine commits: modalism, where the persons are masks of one actor, and subordinationism, where the Son is a lesser created being.
• • It also helps to say openly that the popular analogies are wrong and that theologians say so first. The shamrock and the three states of water teach modalism; the man who is father, son, and employee teaches it more clearly still. Their failure is evidence of the doctrine being policed for precision, not evidence that it is empty. Contemporary philosophers of religion have offered more careful models — Brian Leftow's Latin Trinitarianism, William Hasker's and Richard Swinburne's social Trinitarian accounts, Peter van Inwagen's work on relative identity — which is what a live and defensible doctrine looks like under analysis.
• • Finally, concede the right thing while refusing the wrong one. The Trinity is a mystery in the sense of exceeding exhaustive comprehension, and Augustine's fifteen books on it end in worship rather than in a diagram. But mystery is not contradiction: quantum superposition and the relativity of simultaneity resist mental picturing without violating logic, and we do not conclude that physicists are speaking nonsense. The Christian claim is not that the Trinity is understood but that it is not incoherent — and the burden is on the critic to identify a genuine violation of the law of non-contradiction rather than a difficulty of imagination.

Christian counterresponse

Trinitarian theologians (from the Cappadocian Fathers onward) respond that 'person' in this technical sense was developed precisely to name something real but sui generis — a mode of subsisting in relation — and while it resists a tidy independent definition, the same is true of many foundational metaphysical categories (like 'substance' itself), so the difficulty is one of conceptual novelty rather than demonstrated incoherence.