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.

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DEDUCTION THEOREM
OF PROPOSITIONAL
LOGIC
Saurabh Gupta

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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)
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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:
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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

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Background
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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.

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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.
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Statement

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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
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Proof

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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.
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Proof

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It is a very important relation and is the basis of
reduction based synthesis procedure of flow
sheeting.
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Importance

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