Truth-tables
We are already familiar with truth-tables from the previous entry on the truth-functional connectives and the relationship between sentences, connectives and the overall truth-value of a sentence. Here we will look in further depth at how to build truth-tables and on their mathematical relation to binary truth-values. We will also look at examples of complex truth-tables for large compound expressions and the systematic steps we follow to derive the truth conditions of compound sentences from their simple constituents.
Formulae for constructing truth-tables
For any truth-table, the number of rows it will contain is equal to \(2n\) where:
- \(n\) stands for the number of sentences
- \(2\) is the total number of possible truth values that the sentence may have: true or false.
When we count the number of sentences, we mean atomic sentences. And we only count each sentence once. Hence for a compound sentence of the form \((\sim B \supset C) & (A \equiv B)\), \(B\) occurs twice but there are only three sentences: \(A\), \(B\), and \(C\).
Thus for the sentence \(P & Q\) ,we have two sentences so \(n\) is 2 which equals 4 rows (2 x 2):
P Q P & Q
T T T
T F F
F T F
F F F
For the sentence \((P \lor Q) & R\) we have three sentences so \(n\) is 3 which equals 8 rows (2 x 2 x 2):
P Q R ( P ∨ Q ) & R
T T T T
T T F F
T F T T
T F F F
F T T T
F T F F
F F T F
F F F F
For the single sentence \(P\) we have one sentence so \(n\) is 1 which equals 2 rows (2 x 1):
P P
T T
F F
This tells us how many rows the truth-table should have but it doesn’t tell us what each row should consist in. In other words: how many Ts and Fs it should contain. This is fine with simple truth-tables since we can just alternate each value but for truth-tables with three sentences and more it is easy to make mistakes.
To simplify this and ensure that we are including the right number of possible truth-values we can extend the formula to \(2n^-i\). This formula tells us how many groups of T and F we should have in each column.
We can already see that there is a pattern at work by looking at the columns of the truth tables above. If we take the sentence \((P \lor Q) & R\) we can see that for each sentence:
- \(P\) consists in two sets of \({\textsf{T,T,T,T}}\) and \({\textsf{F,F,F,F}}\) with four elements per set
- \(Q\) consists in four sets of \({\textsf{T,T}}\) , \({\textsf{F,F}}\), \({\textsf{T,T}}\) , \({\textsf{F,F}}\) with two elements per set
- \(R\) consists in eight sets of \({\textsf{T}}\), \({\textsf{F}}\), \({\textsf{T}}\), \({\textsf{F}}\), \({\textsf{T}}\), \({\textsf{F}}\), \({\textsf{T}}\), \({\textsf{F}}\) with one element per set.
If we work through the formula we see that it returns 4, 2, 1:
$$\begin{equation} \begin{split} 2n^-1 = 3 -1 \\ = 2 \\ = 2 \cdot 2 \\ = 4 \end{split} \end{equation}$$
$$ \\begin{equation} \begin{split} 2n^-2 = 3 - 2 \\ = 1 \\ = 2 \cdot 1 \\ = 2 \end{split} \end{equation} $$
$$ \\begin{equation} \begin{split} 2n^-3 = 3 - 3 \\ = 0 \\ = 2 \cdot 0 \\ = 1 \end{split} \end{equation} $$
Truth-table concepts
Recursion
When we move to complex truth-tables with more than one connective we realise that truth-tables are recursive. The truth-tables for the truth-functional connectives provide all that we need to determine the truth-values of complex sentences:
The core truth-tables tell us how to determine the truth-value of a molecular sentence given the truth-values of its immediate sentential components. And if the immediate sentential components of a molecular sentence are also molecular, we can use the information in the characteristic truth-tables to determine how the truth-value of each immediate component depends n the truth-values of its components and so on.
Truth-value assignment
A truth-value assignment is an assignment of truth-values (either T or F) to the atomic sentences of SL.
When working on complex truth tables, we use the truth-assignment of atomic sentences to count as the values that we feed into the larger expressions at a higher level of the sentential abstraction.
Partial assignment
We talk about partial assignments of truth-values when we look at one specific row of the truth-table, independently of the others. The total set of partial assignments comprise all possible truth assignments for the given sentence.
Working through complex truth-tables
The truth-table below shows all truth-value assignments for the sentence \((\sim B \supset C) & (A \equiv B)\) :
A B C ( ~ B ⊃ C ) & ( A ≡ B )
T T T F T T T T T T T
T T F F T T F T T T T
T F T T F T T F T F F
T F F T F F F F T F F
F T T F T T T F F F T
F T F F T T F F F F T
F F T T F T T T F T F
F F F T F F F F F T F
As with algebra we work outwards from each set of brackets. The sequence for manually arriving at the above table would be roughly as follows:
- For each sentence letter, copy the truth value for it in each row.
- Identify the connectives in the atomic sentences and the main overall sentence.
- Work out the truth-values for the smallest connectives and sub-compound sentences. The first should always be negation and then the other atomic connectives.
- Feed-in the truth-values of the atomic sentences as values into the main connective, through a process of elimination you then reach the core truth-assignments: