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:

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