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<title>Modal Tutorial (Version 2) – The Trinity as a Necessary Structure of a Monistic Modal Ontology</title>
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<h1>📐 The Trinity as a Necessary Structure of a Monistic Modal Ontology <span class="version-badge">Version 2 – Minimalist</span></h1>
<div class="subhead">
An interactive tutorial – Step by step<br>
<a href="index.html">← Back to the entry page</a> |
<a href="modal2eng.pdf">📄 PDF of the treatise</a>
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<div style="margin-top: 2.5rem; padding-top: 1rem; border-top: 1px solid #e0d6c8; font-size: 0.75rem; color: #8b8b8b; text-align: center;">
<p>© Paul Koop – <a href="index.html" style="color: #6b4f8a;">the-last-freedom.org/Projekt_Pompeji</a></p>
<p style="margin-top:0.2rem;">This version of the treatise uses only two axioms – Monism and Existence of a World.</p>
</div>
</div>
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// ---- Step 0 ----
{
id: 0,
title: "What is the goal of this version?",
explanation: `
<p>This version of the treatise is <strong>more minimalist</strong> than the previous one. It uses only <strong>two axioms</strong>:</p>
<ol>
<li><strong>A1 – Monism:</strong> There are no fundamentally separated realms of reality.</li>
<li><strong>A4 – Existence of a World:</strong> There is at least one possible world.</li>
</ol>
<p>All other principles (modal realism, consciousness, reflexivity, etc.) are <strong>derived</strong> from these two axioms.</p>
<p>The concepts <strong>U (Origin)</strong>, <strong>T (Totality)</strong>, and <strong>S (Self-Knowledge)</strong> are not understood as substances, but as <strong>limits</strong> of an open interval.</p>
<div class="formula-box">T := (U, S) – U = lim inf T – S = lim sup T</div>
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<p><strong>What is the main difference of this version from the previous one?</strong></p>
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<label><input type="radio" name="q0" value="3"> It does not use modal logic.</label>
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<label><input type="radio" name="q0" value="5"> It is longer than the previous version.</label>
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// ---- Step 1 ----
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<p><strong>A1 – Monism</strong> states:</p>
<div class="formula-box">□¬∃x∃y FundamentallySeparated(x,y)</div>
<p>This means: <strong>There are no fundamentally separated realms of reality.</strong></p>
<p>Why is this important? If there were two fundamentally separated realms, they could not interact. Then there would be no unified reality.</p>
<p>This axiom prohibits:</p>
<ul>
<li>Cartesian dualism (mind vs. matter)</li>
<li>Any form of "beyond" as a fundamentally separated realm</li>
<li>Absolute separation between knower and known</li>
</ul>
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<p><strong>What does Axiom 1 (Monism) state?</strong></p>
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<label><input type="radio" name="q1" value="1"> There are no fundamentally separated realms of reality.</label>
<label><input type="radio" name="q1" value="2"> Reality consists of three substances.</label>
<label><input type="radio" name="q1" value="3"> There is a separation between mind and matter.</label>
<label><input type="radio" name="q1" value="4"> Reality is arbitrary.</label>
<label><input type="radio" name="q1" value="5"> There are infinitely many worlds.</label>
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// ---- Step 2 ----
{
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<p><strong>A4 – Existence of a World</strong> states:</p>
<div class="formula-box">◇∃w World(w)</div>
<p>This means: <strong>It is possible that a world exists.</strong></p>
<p>This axiom is deliberately <strong>weak</strong>. It does not say that a world actually exists, but only that it is <strong>possible</strong>.</p>
<p>Why is this important? This axiom ensures that the modal universe is not empty. If there were no possible world, the entire ontology would be empty.</p>
<p>From this weak axiom, we will later derive the existence of consciousness, modal realism, and ultimately the Trinity.</p>
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<p><strong>What does Axiom 4 (Existence of a World) state?</strong></p>
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<label><input type="radio" name="q2" value="0"> A world actually exists.</label>
<label><input type="radio" name="q2" value="1"> It is possible that a world exists.</label>
<label><input type="radio" name="q2" value="2"> There is no world.</label>
<label><input type="radio" name="q2" value="3"> There are infinitely many worlds.</label>
<label><input type="radio" name="q2" value="4"> The world is an illusion.</label>
<label><input type="radio" name="q2" value="5"> The world exists necessarily.</label>
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// ---- Step 3 ----
{
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<p><strong>Totality T</strong> is defined as:</p>
<div class="formula-box">T := (U, S)</div>
<p>This means: T is an <strong>open interval</strong> between Origin U and Self-Knowledge S.</p>
<p><strong>What does "open interval" mean?</strong></p>
<ul>
<li>It contains all states <strong>between</strong> U and S.</li>
<li>It contains <strong>neither U nor S</strong> themselves.</li>
<li>U and S are the <strong>limits</strong> (limit points) of the interval.</li>
</ul>
<p><strong>Transferred to ontology:</strong></p>
<ul>
<li>Totality is the set of all <strong>actually realized</strong> states.</li>
<li>These states are ordered – from "minimal structure" to "maximal structure".</li>
<li>Origin U is the <strong>lower limit</strong> – that which comes closest to nothing.</li>
<li>Self-Knowledge S is the <strong>upper limit</strong> – the complete transparency of Totality.</li>
</ul>
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<p><strong>What does T := (U, S) mean?</strong></p>
<div class="qoptions">
<label><input type="radio" name="q3" value="0"> T is a closed interval.</label>
<label><input type="radio" name="q3" value="1"> T is an open interval between U and S.</label>
<label><input type="radio" name="q3" value="2"> T contains U and S.</label>
<label><input type="radio" name="q3" value="3"> T is the set of all impossible things.</label>
<label><input type="radio" name="q3" value="4"> T is empty.</label>
<label><input type="radio" name="q3" value="5"> T is the same as U.</label>
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// ---- Step 4 ----
{
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title: "U as the Lower Limit – Origin",
explanation: `
<p><strong>Origin U</strong> is defined as:</p>
<div class="formula-box">U := lim inf T</div>
<p>This means: U is the <strong>lower limit</strong> of Totality T.</p>
<p>U is what comes closest to <strong>nothing</strong> – but <strong>not nothing</strong> itself.</p>
<p>Why is U not nothing? Because nothing would be a <strong>fundamental separation</strong> – and A1 prohibits fundamental separation.</p>
<p>U is therefore the <strong>minimal existing state</strong> – the state of minimal structure that still exists.</p>
<p><strong>Image:</strong> Imagine a number line. Point 0 would be nothing. But 0 is not allowed (because of A1). The <em>closest</em> point to 0 that still exists is U – the lower limit.</p>
`,
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<p><strong>What does U := lim inf T mean?</strong></p>
<div class="qoptions">
<label><input type="radio" name="q4" value="0"> U is the upper limit of T.</label>
<label><input type="radio" name="q4" value="1"> U is the lower limit of T – that which comes closest to nothing.</label>
<label><input type="radio" name="q4" value="2"> U is the same as nothing.</label>
<label><input type="radio" name="q4" value="3"> U is part of T.</label>
<label><input type="radio" name="q4" value="4"> U is Totality itself.</label>
<label><input type="radio" name="q4" value="5"> U is Self-Knowledge.</label>
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// ---- Step 5 ----
{
id: 5,
title: "S as the Upper Limit – Self-Knowledge",
explanation: `
<p><strong>Self-Knowledge S</strong> is defined as:</p>
<div class="formula-box">S := lim sup T</div>
<p>This means: S is the <strong>upper limit</strong> of Totality T.</p>
<p>S is the <strong>complete self-transparency</strong> of Totality – but <strong>not itself a part</strong> of T.</p>
<p>Why does S not belong to T? If S belonged to T, then T would already be completely transparent – but then it would no longer be open. The openness of T is the precondition for S.</p>
<p><strong>Image:</strong> Imagine a number line. Point 1 would be complete transparency. But 1 is not allowed (because T is open). The <em>closest</em> point to 1 that is not yet reached is S – the upper limit.</p>
`,
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<p><strong>What does S := lim sup T mean?</strong></p>
<div class="qoptions">
<label><input type="radio" name="q5" value="0"> S is the lower limit of T.</label>
<label><input type="radio" name="q5" value="1"> S is the upper limit of T – the complete self-transparency.</label>
<label><input type="radio" name="q5" value="2"> S is the same as nothing.</label>
<label><input type="radio" name="q5" value="3"> S is part of T.</label>
<label><input type="radio" name="q5" value="4"> S is Totality itself.</label>
<label><input type="radio" name="q5" value="5"> S is Origin.</label>
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// ---- Step 6 ----
{
id: 6,
title: "The Trinity as the Unity of the Three Limits",
explanation: `
<p>The <strong>Trinity</strong> is the unity of the three limits:</p>
<div class="formula-box">Tr := U ∧ T ∧ S</div>
<p>Or in the language of limits:</p>
<div class="formula-box">Tr := lim inf T ∧ T ∧ lim sup T</div>
<p>This means: The Trinity is <strong>not three things</strong>, but <strong>three perspectives on the same reality</strong>:</p>
<ul>
<li><strong>U (Origin)</strong> – the perspective of "Whence?"</li>
<li><strong>T (Totality)</strong> – the perspective of "What is?"</li>
<li><strong>S (Self-Knowledge)</strong> – the perspective of "Who knows?"</li>
</ul>
<p>The three limits are <strong>one</strong> – they are different viewpoints on the same one reality.</p>
`,
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<p><strong>What does Tr := U ∧ T ∧ S mean?</strong></p>
<div class="qoptions">
<label><input type="radio" name="q6" value="0"> Tr is true if at least one of the three concepts is true.</label>
<label><input type="radio" name="q6" value="1"> Tr is true if all three concepts (U, T, and S) are true simultaneously.</label>
<label><input type="radio" name="q6" value="2"> Tr is true if none of the three concepts is true.</label>
<label><input type="radio" name="q6" value="3"> Tr is the same as U.</label>
<label><input type="radio" name="q6" value="4"> Tr is the same as T.</label>
<label><input type="radio" name="q6" value="5"> Tr is the same as S.</label>
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answer: 1
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// ---- Step 7 ----
{
id: 7,
title: "Theorem: From A1 follows A3 – No Absolute Nothing",
explanation: `
<p><strong>Theorem:</strong> From A1 (Monism) follows A3 (No Absolute Nothing).</p>
<div class="formula-box">A1 → □(N → ¬◇World)</div>
<p><strong>Proof:</strong></p>
<ol>
<li>Absolute nothing <strong>N</strong> would be a <strong>fundamentally separated realm</strong> from reality.</li>
<li><strong>A1 prohibits fundamental separation.</strong></li>
<li>Therefore: There is no absolute nothing.</li>
<li>Thus: <span class="symbol">□(N → ¬◇World)</span></li>
</ol>
<p><strong>This is important because:</strong> If there is no absolute nothing, then there must be <strong>something</strong>. Reality is not empty.</p>
`,
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<p><strong>What follows from A1 (Monism)?</strong></p>
<div class="qoptions">
<label><input type="radio" name="q7" value="0"> There is an absolute nothing.</label>
<label><input type="radio" name="q7" value="1"> There is no absolute nothing.</label>
<label><input type="radio" name="q7" value="2"> Nothing can bring forth worlds.</label>
<label><input type="radio" name="q7" value="3"> Nothing is the same as Totality.</label>
<label><input type="radio" name="q7" value="4"> Nothing has many properties.</label>
<label><input type="radio" name="q7" value="5"> Nothing is Origin.</label>
</div>
`,
answer: 1
},
// ---- Step 8 ----
{
id: 8,
title: "Theorem: From A1 and A4 follows A5 – Consciousness is possible",
explanation: `
<p><strong>Theorem:</strong> From A1 (Monism) and A4 (Existence of a World) follows A5 – Consciousness is possible.</p>
<div class="formula-box">A1 ∧ A4 → ◇C</div>
<p><strong>Proof:</strong></p>
<ol>
<li>Suppose there is a world: <span class="symbol">∃w World(w)</span>.</li>
<li>Suppose consciousness is impossible: <span class="symbol">¬◇C</span> → <span class="symbol">□¬C</span>.</li>
<li>If there is no consciousness, the world is <strong>unknowable</strong>.</li>
<li>An unknowable world would be <strong>fundamentally separated</strong> from any conscious perspective.</li>
<li><strong>A1 prohibits fundamental separation.</strong></li>
<li>Therefore: <span class="symbol">¬◇C</span> leads to a contradiction.</li>
<li>Thus: <span class="symbol">◇C</span> – consciousness is possible.</li>
</ol>
<p><strong>This is a transcendental argument:</strong> A world without consciousness would be indistinguishable from non-existence.</p>
`,
question_html: `
<p><strong>What follows from A1 and A4?</strong></p>
<div class="qoptions">
<label><input type="radio" name="q8" value="0"> Consciousness is impossible.</label>
<label><input type="radio" name="q8" value="1"> Consciousness is possible.</label>
<label><input type="radio" name="q8" value="2"> Consciousness exists necessarily.</label>
<label><input type="radio" name="q8" value="3"> Consciousness is an illusion.</label>
<label><input type="radio" name="q8" value="4"> Consciousness is the same as Totality.</label>
<label><input type="radio" name="q8" value="5"> Consciousness is impossible because it contradicts A1.</label>
</div>
`,
answer: 1
},
// ---- Step 9 ----
{
id: 9,
title: "Theorem: From A1, A4 and A5 follows A2 – Modal Realism",
explanation: `
<p><strong>Theorem:</strong> From A1, A4 and A5 follows A2 – Modal Realism.</p>
<div class="formula-box">A1 ∧ A4 ∧ A5 → ∀p(◇p → ∃w Realized(w,p))</div>
<p><strong>Proof:</strong></p>
<ol>
<li>Consciousness (C) is the capacity to experience <strong>differences</strong>.</li>
<li>If there were an <strong>unrealized possibility</strong>, it would be <strong>not experienceable by consciousness</strong>.</li>
<li>An unexperienceable possibility would be <strong>fundamentally separated</strong> from the world of consciousness.</li>
<li><strong>A1 prohibits fundamental separation.</strong></li>
<li>Therefore: There can be no unrealized possibilities.</li>
<li>Thus: <span class="symbol">∀p(◇p → ∃w Realized(w,p))</span></li>
</ol>
<p><strong>This means:</strong> Every possibility is realized in some world.</p>
`,
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<p><strong>What follows from A1, A4 and A5?</strong></p>
<div class="qoptions">
<label><input type="radio" name="q9" value="0"> Possibilities need not be realized.</label>
<label><input type="radio" name="q9" value="1"> Every possibility is realized in some world.</label>
<label><input type="radio" name="q9" value="2"> There are no possibilities.</label>
<label><input type="radio" name="q9" value="3"> Possibilities are only thoughts.</label>
<label><input type="radio" name="q9" value="4"> Only necessary possibilities are realized.</label>
<label><input type="radio" name="q9" value="5"> Possibilities contradict monism.</label>
</div>
`,
answer: 1
},
// ---- Step 10 ----
{
id: 10,
title: "Theorem: From A1, A4 and A5 follows A7 – Possibility of Knowledge",
explanation: `
<p><strong>Theorem:</strong> From A1, A4 and A5 follows A7 – complete knowledge is possible.</p>
<div class="formula-box">A1 ∧ A4 ∧ A5 → □(C → ◇V_T)</div>
<p><strong>Proof:</strong></p>
<ol>
<li>If consciousness exists (C), it is <strong>part of the one reality</strong> (A1).</li>
<li>The one reality is <strong>Totality T</strong>.</li>
<li>If consciousness is part of T, it <strong>can</strong> in principle know T – for there is no fundamental separation (A1).</li>
<li>Thus: <span class="symbol">C → ◇V_T</span></li>
<li>With Necessitation: <span class="symbol">□(C → ◇V_T)</span></li>
</ol>
<p><strong>This means:</strong> If there is consciousness, then complete knowledge of Totality is possible.</p>
`,
question_html: `
<p><strong>What follows from A1, A4 and A5 for knowledge?</strong></p>
<div class="qoptions">
<label><input type="radio" name="q10" value="0"> Complete knowledge is impossible.</label>
<label><input type="radio" name="q10" value="1"> If consciousness exists, then complete knowledge is possible.</label>
<label><input type="radio" name="q10" value="2"> Knowledge is always false.</label>
<label><input type="radio" name="q10" value="3"> Knowledge is separated from reality.</label>
<label><input type="radio" name="q10" value="4"> Only God can know.</label>
<label><input type="radio" name="q10" value="5"> Knowledge is impossible because it contradicts A1.</label>
</div>
`,
answer: 1
},
// ---- Step 11 ----
{
id: 11,
title: "Theorem: From A1 follows A8 – Reflexivity of Knowledge",
explanation: `
<p><strong>Theorem:</strong> From A1 follows A8 – Reflexivity of Complete Knowledge.</p>
<div class="formula-box">A1 → □(V_T(x) → V_T(V_T(x)))</div>
<p><strong>Proof:</strong></p>
<ol>
<li>If there is complete knowledge of T (<span class="symbol">V_T(x)</span>), then this knowledge is <strong>part of T</strong> (for everything is part of T, A1).</li>
<li>If this knowledge is part of T, it must also know <strong>itself</strong> – otherwise it would not be <strong>complete</strong>.</li>
<li>Thus: <span class="symbol">V_T(x) → V_T(V_T(x))</span></li>
<li>With Necessitation: <span class="symbol">□(V_T(x) → V_T(V_T(x)))</span></li>
</ol>
<p><strong>This means:</strong> Complete knowledge always knows itself.</p>
`,
question_html: `
<p><strong>What does Reflexivity of Knowledge (A8) state?</strong></p>
<div class="qoptions">
<label><input type="radio" name="q11" value="0"> Complete knowledge need not know that it is complete.</label>
<label><input type="radio" name="q11" value="1"> Complete knowledge also knows itself.</label>
<label><input type="radio" name="q11" value="2"> Knowledge is always false.</label>
<label><input type="radio" name="q11" value="3"> Knowledge is separated from reality.</label>
<label><input type="radio" name="q11" value="4"> Only God can have complete knowledge.</label>
<label><input type="radio" name="q11" value="5"> Complete knowledge is impossible.</label>
</div>
`,
answer: 1
},
// ---- Step 12 ----
{
id: 12,
title: "The Assumption for the Reductio ad absurdum",
explanation: `
<p>The treatise uses a <strong>Reductio ad absurdum</strong> – a proof technique in which one assumes the opposite and shows that it leads to a contradiction.</p>
<p><strong>The assumption is:</strong></p>
<div class="formula-box">¬Tr</div>
<p>This means: <strong>The Trinity does not exist.</strong></p>
<p>Since <span class="symbol">Tr := U ∧ T ∧ S</span>, <span class="symbol">¬Tr</span> means:</p>
<div class="formula-box">¬U ∨ ¬T ∨ ¬S</div>
<p>So there are <strong>three cases</strong>:</p>
<ol>
<li><strong>Case 1:</strong> T exists, but U does not (<span class="symbol">T ∧ ¬U</span>)</li>
<li><strong>Case 2:</strong> U exists, but S does not (<span class="symbol">U ∧ ¬S</span>)</li>
<li><strong>Case 3:</strong> S exists, but T does not (<span class="symbol">S ∧ ¬T</span>)</li>
</ol>
`,
question_html: `
<p><strong>What is the assumption for the Reductio ad absurdum?</strong></p>
<div class="qoptions">
<label><input type="radio" name="q12" value="0"> The Trinity exists necessarily.</label>
<label><input type="radio" name="q12" value="1"> The Trinity does not exist.</label>
<label><input type="radio" name="q12" value="2"> The Trinity exists partially.</label>
<label><input type="radio" name="q12" value="3"> The Trinity is impossible.</label>
<label><input type="radio" name="q12" value="4"> The Trinity is the same as Totality.</label>
<label><input type="radio" name="q12" value="5"> The Trinity is an illusion.</label>
</div>
`,
answer: 1
},
// ---- Step 13 ----
{
id: 13,
title: "Case 1 – Totality without Origin (T ∧ ¬U)",
explanation: `
<p><strong>Case 1:</strong> <span class="symbol">T ∧ ¬U</span> – Totality without Origin.</p>
<p><strong>The contradiction:</strong></p>
<ol>
<li>T exists. So there is at least one state within the open interval.</li>
<li>Since T is an <strong>open interval</strong>, it has <strong>no smallest element</strong>.</li>
<li>The openness of T <strong>guarantees</strong> the existence of the lower limit U (by definition).</li>
<li>If U does not exist (<span class="symbol">¬U</span>), then there is no lower limit.</li>
<li>Then T would no longer be an open interval.</li>
<li><strong>Contradiction to the definition of T.</strong></li>
</ol>
<div class="formula-box">T ∧ ¬U → ⊥ → □(T → U)</div>
`,
question_html: `
<p><strong>What does Case 1 (Totality without Origin) show?</strong></p>
<div class="qoptions">
<label><input type="radio" name="q13" value="0"> T can exist without U.</label>
<label><input type="radio" name="q13" value="1"> T ∧ ¬U leads to a contradiction.</label>
<label><input type="radio" name="q13" value="2"> U is not necessary for T.</label>
<label><input type="radio" name="q13" value="3"> Totality can come from nothing.</label>
<label><input type="radio" name="q13" value="4"> T and U are the same.</label>
<label><input type="radio" name="q13" value="5"> T is its own ground.</label>
</div>
`,
answer: 1
},
// ---- Step 14 ----
{
id: 14,
title: "Case 2 – Origin without Self-Knowledge (U ∧ ¬S)",
explanation: `
<p><strong>Case 2:</strong> <span class="symbol">U ∧ ¬S</span> – Origin without Self-Knowledge.</p>
<p><strong>The contradiction:</strong></p>
<ol>
<li>U exists. This means: There is a lower limit of Totality.</li>
<li>From the derived theorems: <span class="symbol">◇C</span> (consciousness is possible).</li>
<li>From A2 (Modal Realism): There is a world with consciousness.</li>
<li>From A7: In this world, complete knowledge is possible.</li>
<li>From A8: Complete knowledge knows itself – that is precisely S.</li>
<li>Thus: <span class="symbol">U → S</span>.</li>
<li><strong>Contradiction to the assumption</strong> <span class="symbol">¬S</span>.</li>
</ol>
<div class="formula-box">U ∧ ¬S → ⊥ → □(U → S)</div>
`,
question_html: `
<p><strong>What does Case 2 (Origin without Self-Knowledge) show?</strong></p>
<div class="qoptions">
<label><input type="radio" name="q14" value="0"> U can exist without S.</label>
<label><input type="radio" name="q14" value="1"> U ∧ ¬S leads to a contradiction.</label>
<label><input type="radio" name="q14" value="2"> S is not necessary for U.</label>
<label><input type="radio" name="q14" value="3"> Consciousness is impossible.</label>
<label><input type="radio" name="q14" value="4"> Complete knowledge is impossible.</label>
<label><input type="radio" name="q14" value="5"> U and S are the same.</label>
</div>
`,
answer: 1
},
// ---- Step 15 ----
{
id: 15,
title: "Case 3 – Self-Knowledge without Totality (S ∧ ¬T)",
explanation: `
<p><strong>Case 3:</strong> <span class="symbol">S ∧ ¬T</span> – Self-Knowledge without Totality.</p>
<p><strong>The contradiction:</strong></p>
<ol>
<li>S exists. S is defined as "complete self-knowledge of Totality".</li>
<li>If S exists, there is an <strong>act of knowing</strong> directed at T.</li>
<li>If T does not exist (<span class="symbol">¬T</span>), then S is <strong>knowledge without an object</strong>.</li>
<li>From A9 (derived): <span class="symbol">V_T(x) → T</span> – knowledge of T presupposes T.</li>
<li>Thus: <span class="symbol">S → T</span>.</li>
<li><strong>Contradiction to the assumption</strong> <span class="symbol">¬T</span>.</li>
</ol>
<div class="formula-box">S ∧ ¬T → ⊥ → □(S → T)</div>
`,
question_html: `
<p><strong>What does Case 3 (Self-Knowledge without Totality) show?</strong></p>
<div class="qoptions">
<label><input type="radio" name="q15" value="0"> S can exist without T.</label>
<label><input type="radio" name="q15" value="1"> S ∧ ¬T leads to a contradiction.</label>
<label><input type="radio" name="q15" value="2"> T is not necessary for S.</label>
<label><input type="radio" name="q15" value="3"> S is empty knowledge.</label>
<label><input type="radio" name="q15" value="4"> S and T are the same.</label>
<label><input type="radio" name="q15" value="5"> Axiom 1 is false.</label>
</div>
`,
answer: 1
},
// ---- Step 16 ----
{
id: 16,
title: "The Synthetic Conclusion – The Trinity follows necessarily",
explanation: `
<p>The treatise has <strong>refuted all three cases</strong>:</p>
<ul>
<li><strong>Case 1:</strong> <span class="symbol">T ∧ ¬U → ⊥</span> → <span class="boxed">□(T → U)</span></li>
<li><strong>Case 2:</strong> <span class="symbol">U ∧ ¬S → ⊥</span> → <span class="boxed">□(U → S)</span></li>
<li><strong>Case 3:</strong> <span class="symbol">S ∧ ¬T → ⊥</span> → <span class="boxed">□(S → T)</span></li>
</ul>
<p>This yields a circle:</p>
<div class="formula-box">T → U → S → T</div>
<p>In S5, it follows:</p>
<div class="formula-box">□(U ↔ T) ∧ □(T ↔ S) ∧ □(S ↔ U)</div>
<p>With A4 (Existence of a World) and the definition of T:</p>
<div class="formula-box">∃T → ∃U ∧ ∃S</div>
<p>Thus:</p>
<div class="formula-box" style="font-size:1.2rem; background-color:#d4edda; border-color:#6b4f8a;">□(U ∧ T ∧ S)</div>
<p style="font-weight:bold; color:#6b4f8a;">The Trinity exists necessarily.</p>
`,
question_html: `
<p><strong>What follows from the three cases together with A4?</strong></p>
<div class="qoptions">
<label><input type="radio" name="q16" value="0"> The Trinity is impossible.</label>
<label><input type="radio" name="q16" value="1"> The Trinity exists necessarily.</label>
<label><input type="radio" name="q16" value="2"> The Trinity exists only in thought.</label>
<label><input type="radio" name="q16" value="3"> The Trinity exists partially.</label>
<label><input type="radio" name="q16" value="4"> The Trinity is the same as Totality.</label>
<label><input type="radio" name="q16" value="5"> The Trinity is an illusion.</label>
</div>
`,
answer: 1
}
];
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