160 lines
5.5 KiB
Markdown
160 lines
5.5 KiB
Markdown
---
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categories:
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- Mathematics
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- Logic
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tags: []
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---
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# Truth-tables
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We are already familiar with truth-tables from the previous entry on the
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_truth-functional connectives_ and the relationship between sentences,
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connectives and the overall truth-value of a sentence. Here we will look in
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further depth at how to build truth-tables and on their mathematical relation to
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binary truth-values. We will also look at examples of complex truth-tables for
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large compound expressions and the systematic steps we follow to derive the
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truth conditions of compound sentences from their simple constituents.
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## Formulae for constructing truth-tables
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For any truth-table, the number of rows it will contain is equal to $2n$ where:
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- $n$ stands for the number of sentences
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- $2$ is the total number of possible truth values that the sentence may have:
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true or false.
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When we count the number of sentences, we mean atomic sentences. And we only
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count each sentence once. Hence for a compound sentence of the form
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$(\sim B \supset C) & (A \equiv B)$, $B$ occurs twice but there are only three
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sentences: $A$, $B$, and $C$.
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Thus for the sentence $P & Q$ ,we have two sentences so $n$ is 2 which equals 4
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rows (2 x 2):
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```
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P Q P & Q
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T T T
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T F F
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F T F
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F F F
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```
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For the sentence $(P \lor Q) & R$ we have three sentences so $n$ is 3 which
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equals 8 rows (2 x 2 x 2):
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```
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P Q R ( P ∨ Q ) & R
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T T T T
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T T F F
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T F T T
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T F F F
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F T T T
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F T F F
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F F T F
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F F F F
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```
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For the single sentence $P$ we have one sentence so $n$ is 1 which equals 2 rows
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(2 x 1):
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```
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P P
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T T
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F F
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```
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This tells us how many rows the truth-table should have but it doesn't tell us
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what each row should consist in. In other words: how many Ts and Fs it should
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contain. This is fine with simple truth-tables since we can just alternate each
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value but for truth-tables with three sentences and more it is easy to make
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mistakes.
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To simplify this and ensure that we are including the right number of possible
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truth-values we can extend the formula to $2n^-i$. This formula tells us how
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many groups of T and F we should have in each column.
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We can already see that there is a pattern at work by looking at the columns of
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the truth tables above. If we take the sentence $(P \lor Q) & R$ we can see that
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for each sentence:
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- $P$ consists in two sets of ${\textsf{T,T,T,T}}$ and ${\textsf{F,F,F,F}}$ with
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**four** elements per set
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- $Q$ consists in four sets of ${\textsf{T,T}}$ , ${\textsf{F,F}}$,
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${\textsf{T,T}}$ , ${\textsf{F,F}}$ with **two** elements per set
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- $R$ consists in eight sets of ${\textsf{T}}$, ${\textsf{F}}$, ${\textsf{T}}$,
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${\textsf{F}}$, ${\textsf{T}}$, ${\textsf{F}}$, ${\textsf{T}}$, ${\textsf{F}}$
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with **one** element per set.
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If we work through the formula we see that it returns 4, 2, 1:
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$$\begin{equation} \begin{split} 2n^-1 = 3 -1 \\ = 2 \\ = 2 \cdot 2 \\ = 4 \end{split} \end{equation}$$
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$$
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\\begin{equation} \begin{split} 2n^-2 = 3 - 2 \\ = 1 \\ = 2 \cdot 1 \\ = 2 \end{split} \end{equation}
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$$
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$$
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\\begin{equation} \begin{split} 2n^-3 = 3 - 3 \\ = 0 \\ = 2 \cdot 0 \\ = 1 \end{split} \end{equation}
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$$
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## Truth-table concepts
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### Recursion
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When we move to complex truth-tables with more than one connective we realise
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that truth-tables are recursive. The truth-tables for the truth-functional
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connectives provide all that we need to determine the truth-values of complex
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sentences:
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> The core truth-tables tell us how to determine the truth-value of a molecular
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> sentence given the truth-values of its
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> [immediate sentential components](Syntax%20of%20sentential%20logic.md). And if
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> the immediate sentential components of a molecular sentence are also
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> molecular, we can use the information in the characteristic truth-tables to
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> determine how the truth-value of each immediate component depends n the
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> truth-values of _its_ components and so on.
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### Truth-value assignment
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> A truth-value assignment is an assignment of truth-values (either T or F) to
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> the atomic sentences of SL.
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When working on complex truth tables, we use the truth-assignment of atomic
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sentences to count as the values that we feed into the larger expressions at a
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higher level of the sentential abstraction.
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### Partial assignment
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We talk about partial assignments of truth-values when we look at one specific
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row of the truth-table, independently of the others. The total set of partial
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assignments comprise all possible truth assignments for the given sentence.
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## Working through complex truth-tables
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The truth-table below shows all truth-value assignments for the sentence
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$(\sim B \supset C) & (A \equiv B)$ :
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```
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A B C ( ~ B ⊃ C ) & ( A ≡ B )
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T T T F T T T T T T T
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T T F F T T F T T T T
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T F T T F T T F T F F
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T F F T F F F F T F F
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F T T F T T T F F F T
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F T F F T T F F F F T
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F F T T F T T T F T F
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F F F T F F F F F T F
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```
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As with algebra we work outwards from each set of brackets. The sequence for
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manually arriving at the above table would be roughly as follows:
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1. For each sentence letter, copy the truth value for it in each row.
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1. Identify the connectives in the atomic sentences and the main overall
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sentence.
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1. Work out the truth-values for the smallest connectives and sub-compound
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sentences. The first should always be negation and then the other atomic
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connectives.
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1. Feed-in the truth-values of the atomic sentences as values into the main
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connective, through a process of elimination you then reach the core
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truth-assignments:
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