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endobj ~B → (B → A). (3.5 The Soundness of Proofs) Deductive Proofs of Predicate Logic Formulas In this chapter, we will develop the notion of formal deductive proofs for Predicate Logic. (A → B) → (~B → ~A). << /S /GoTo /D (subsection.2.1) >> << /S /GoTo /D (subsection.3.5) >> endobj Think about the simple example of the profit of a company, which equals revenue minus costs. B,
Proof. (1 Example of a Proof) Proof. 47 0 obj 65 0 obj 11 0 obj (3.3 Universal Generalization \(UG\) Lines) Apply the Deduction Theorem one more time to get. Lemma 4a. endobj For any well-formed formulas A and
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7 0 obj 55 0 obj (A → B) → ((~A → B) → B). This is a demo of a proof checker for Fitch-style natural deduction systems found in many popular introductory logic textbooks. ~B → (B → A). Therefore, for any well-formed formula A and B, ~B →
→ (A → B) is theorem of L. Let us swap the variable in the Lemma 4 and see
→ B) for A. 40 0 obj ˚ ¬˚ Œ ¬e L The proof rule could be called Œi. endobj B,
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Q5FJ��D*R,'�-�&�A~@�a����V�ห����W�vq��Ȧ��,��:�w6N�87�ávV�*�>B�D���O8��l?g�܍~�uF(�%��#�����|��n�F �d�P���8̳L.��cqNJQQ����KΑC��fV�jᐌT��a�@R�=I�F��^v�Ҡ���1¾�(I��~O$��|����*��.,̾;�.�3�� Deductive logic is concerned with the structure of the argument more than the argument's content. (2.1 Templates Constants Names) Proof. We shall construct a proof in L of
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(2.4 Handling Parametrized Formulas) endobj endobj (~B → ~A) → (A → B). Lemma 5. Therefore, for any well-formed formula B, (~~B →
we get. ~A → (A → B). Given: w = x, x = y, y = z Prove: w = z ~~B → B. B → ~~B. << /S /GoTo /D (subsection.2.3) >> (A → B) → (~B → ~A). Proof. For example, take a gander at the following formal proof. endobj B) → (~B → ~A) is theorem of L. Lemma 7. → B)
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~~B→ B. << /S /GoTo /D (section.4) >> (2 Schemas) Lemma 8. B → ~~B. 35 0 obj endobj And apply the Deduction Theorem one more time and
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