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Proposition to Table - Constructing tables

Prop2Table #6 :: 18-05-2018

Contents
 -Proposition to Table - Preface
 -Proposition to Table - Parsing input (SLR parser)
 -Proposition to Table - Parser F# code
 -Proposition to Table - Well-formed syntax
 -Proposition to Table - Semantics of propositional logic
 @Proposition to Table - Constructing tables
 -Proposition to Table - The application

We now both have the semantics and the syntax of propositional logic. With this in place we only need to evaluate every permutation of atoms. We need to delve into what a model is: a concrete assignment of atoms from the set {true,false}. Say we have the contingent proposition p \Rightarrow q \Rightarrow r. We can list this in a truth table as

pqrp ⇒ q ⇒ r
TTTT
FTTT
TFTT
FFTT
TTFF
FTFT
TFFT
FFFT

Here every ordered set of atomic values describes a model. This we have as the set of ordered sets: {{T,T,T},{F,T,T},{T,F,T} ... {F,F,F}}. Each of these models are used to evaluate more and more complex propositions. The simplest way I found to create these models is to alternate. First for n different atomic variables we have 2^{n} permutations or models. The left most column, the one under p, we alternate for every new model. For the next column we alternate for every 2 models. For the next for every 4 models and so on. We just double the number for which we alternate.

Now for every model we evaluate the parse tree to obtain a truth value for the whole expression.

And we have our application. Simple as that.

How to proof with this application

If we have some proposition, we can proof its validity by creating the truth table. If the truth table results in a tautology, the proposition is proofed.

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