-
Infix
Polish
Boolean
Np
Apq
Kpq
Cpq
Epq
By
John W. Williams
Notation
Statements in the propositional calculus can be expressed in Boolean notation using the following guide:
|
Infix |
Polish |
Boolean |
|---|---|---|
|
|
Np |
|
|
|
Apq |
|
|
|
Kpq |
|
|
|
Cpq |
|
|
|
Epq |
|
Method
Translate the premises and the conclusion into Boolean notation.
Multiply the product of all the premises by the negation (complement) of the conclusion, and reduce this product to disjunctive normal form.
If the above product vanishes, then the argument is valid; otherwise, each nonzero term in the disjunctive normal form represents a counterexample to the argument.
Explanation
Denoting
premises by
and
conclusion by c, an argument has the form
.
The argument is valid if and only if this implication is a
tautology:
.
Equivalently, the argument is valid if and only if its negation is a contradiction:
.
Essential knowledge
These fundamental laws and theorems of Boolean Algebra allow for the transformation of any proposition into disjunctive normal form. The short name (in brackets) can be used when a law is invoked to justify a step in a proof.
|
Commutation (Com) |
Association (Assoc) |
|---|---|
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Distribution (Dist) |
Tautology (Taut) |
|---|---|
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Absorption (Absn) |
Complementation (Comp) |
|---|---|
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De Morgan (DM) |
Disjoint Partition (DP) |
|---|---|
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One |
Zero |
|---|---|
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