Deduction theorem of propositional logic
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The document discusses propositional logic, focusing on the deduction theorem, which states that if a statement g logically follows from statements f1, f2, ..., fn, then the implication is always true. It highlights the importance of this theorem in determining logical consequences and its application in the synthesis of flow sheeting. The proof demonstrates the theorem's validity by using de Morgan’s law and establishes its significance in practical scenarios.
Deduction theorem of propositional logic
- 2. Copyright2013-2014 What is propositional logic? Propositional logic deals with statements which can either be true or false. Example: The statement “Distillation column separate miscible liquids exploiting the difference in their volatilities” is a scientific fact which is always true. (with the exception of azeotropes) Thursday, February 23, 2017THAPAR UNIVERSITY, PATIALA 2
- 3. Copyright2013-2014 Desired products can be manufactured if one has suitable reactors, equipment for unit operations and necessary raw materials. If R, U, RM and P are defined as: Thursday, February 23, 2017THAPAR UNIVERSITY, PATIALA 3 Background R = one has suitable reactors U = one has equipment to carry out unit operations RM = one has necessary raw materials P = one manufactures desired product
- 4. Copyright2013-2014 Background Thursday, February 23, 2017THAPAR UNIVERSITY, PATIALA 4 Considering the above, P would be a logical consequence of R, U and RM if and only if: (R.U.RM) → P R, U and RM need not be atoms but can also be equations themselves. In the above equation, with the knowledge of RM and P, it is possible to find the flowsheet using deduction theorem.
- 5. Copyright2013-2014 The deduction theorem of propositional logic states that if G is a logical consequence of statements 𝐹1, 𝐹2, … … … … , 𝐹𝑛 then: 𝐹1 ∙ 𝐹2 ∙ . … … … ∙ 𝐹𝑛 → 𝐺 is valid, that is, true always. This is equivalent to saying: 𝐹1 ∙ 𝐹2 ∙ . … … … ∙ 𝐹𝑛 . 𝐺, is inconsistent. Thursday, February 23, 2017THAPAR UNIVERSITY, PATIALA 5 Statement
- 6. Copyright2013-2014 Since 𝐹1 ∙ 𝐹2 ∙ … … … … ∙ 𝐹𝑛 → 𝐺 is always true, 𝐹1 ∙ 𝐹2 … … … … ∙ 𝐹𝑛 → 𝐺 will always be false (or inconsistent). Since we know, 𝐴 → 𝐵 = 𝐴 + 𝐵, thus, 𝑋 = 𝐹1 ∙ 𝐹2 ∙ … … … … ∙ 𝐹𝑛 → 𝐺 = 𝐹1 ∙ 𝐹2 ∙ … … … … ∙ 𝐹𝑛 + 𝐺 Thursday, February 23, 2017THAPAR UNIVERSITY, PATIALA 6 Proof
- 7. Copyright2013-2014 Using De Morgan’s Law on ‘X’ we get: 𝑋 = 𝐹1 ∙ 𝐹2 ∙ … … … … ∙ 𝐹𝑛 . 𝐺 = 𝐹1 ∙ 𝐹2 ∙ … … … … ∙ 𝐹𝑛 . 𝐺 As, X is inconsistent, 𝐹1 ∙ 𝐹2 ∙ … … … … ∙ 𝐹𝑛 . 𝐺 is inconsistent. Hence, the theorem is proved. Thursday, February 23, 2017THAPAR UNIVERSITY, PATIALA 7 Proof
- 8. Copyright2013-2014 It is a very important relation and is the basis of reduction based synthesis procedure of flow sheeting. Thursday, February 23, 2017THAPAR UNIVERSITY, PATIALA 8 Importance
- 9. Copyright2013-2014 Thursday, February 23, 2017THAPAR UNIVERSITY, PATIALA 9