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What is truth anyway?

Previously we have been performing purely symbolic moves to generate our proofs. How does this relate to a given mathematics proof or debate topic?

To motivate our rules we have often replaced our symbols $A,B,C,$… by actual statements, eg “the sun is out”.

“The sun is out” can either be true or false – we just look outside. So at a very basic level this provides a way to assign either T or F to a propositional variable. We could of course willy-nilly assign T and F for each our propositional variables. But all of our connectives have familiar interpretation in the context of T/F values.

  • Not true better be false and vice-versa. So \(\neg T \mapsto F \ , \ \neg F \mapsto T.\) We can put this into a table:
$A$$\neg A$

This expresses the values of $\neg A$ given the values of $A$.

  • Suppose I know that $A \to B$ and we know that is $A$ is $T$. It would make a lot sense to conclude that $B$ should also be assigned $T$. But if we allow ourselves to look at possible assignments it makes sense to assign T/F values to each expression using T/F and $\to$. We can make another table where we list the values assigned to $A$ along the first column and the values assigned to $B$ in the first row.
$A$$B$$A \to B$

One can read this as saying the implication is itself is true if either assumption is false or both the assumption and conclusion are true.

  • We also have a table for $\land$
$A$$B$$A \land B$

which says that $A \land B$ should only be true if both $A$ and $B$ are.

  • And we have a table for $\lor$
$A$$B$$A \lor B$

which says $A \lor B$ is false only when both $A$ and $B$ are.

  • We also want \(\top \mapsto T \\ \bot \mapsto F\)

Using these rules, once we have a chosse of T/F, we can assign T/F to any propositional formula. Let’s look at a more complicated formula.

Example. Let’s take the formula \(\neg A \lor B \to C \land \neg D\) and the truth assignment \(A, C \mapsto T, \ B, D \mapsto F\)

It convenient notation to “plug in” to the formula the values of $T$ and $F$. This gives \(\neg T \lor F \to T \land \neg F\) Then use our rules above to simplify down to a single value. First the negation \(F \lor F \to T \land T\) Then the $\lor$ and $\land$ \(F \to T\) Finally for $\to$ we reduce to $T$.

Different assignments for $A,B,C,D$ can yield a different value for our formula. For example, if \(B, C, D \mapsto T, \ A \mapsto F\) Then \(\neg A \lor B \to C \land \neg D \mapsto F\)

We can think of the possible truth value assignments to our collection of propositional variables as different possible universes. For example, if $A$ is standing for “the sun is out”, then $A \mapsto T$ is saying we know the sun it out while $A \mapsto F$ is saying the sun is not out. So any real-world or mathematical possibility can be found by listing out all the possible T/F assignments and the values taken by the formula given that assignment. Below is a table for our example

The truth table for our example. (Expand to view)
$A$$B$$C$$D$$\neg A \lor B \to C \land \neg D$