Formal Logic/Sentential Logic/Truth Tables

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Truth Tables

In the Formal Syntax, we earlier gave a formal semantics for sentential logic. A truth table is a device for using this form syntax in calculating the truth value of a larger formula given an interpretation (an assignment of truth values to sentence letters). Truth tables may also help clarify the material from the Formal Syntax.

Basic tables

Negation

We begin with the truth table for negation. It corresponds to clause (ii) of our definition for extended valuations.

$\mathrm {P} \,\!$ $\lnot \mathrm {P} \,\!$ T F F T

T and F represent True and False respectively. Each row represents an interpretation. The first column shows what truth value the interpretation assigns to the sentence letter $\mathrm {P} \,\!$. In the first row, the interpretation assigns $\mathrm {P} \,\!$ the value True. In the second row, the interpretation assigns $\mathrm {P} \,\!$ the value False.

The second column shows the value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\displaystyle \lnot \mathrm{P}\,\!} receives under a given row's interpretation. Under the interpretation of the first row, $\lnot \mathrm {P} \,\!$ has the value False. Under the interpretation of the second row, $\lnot \mathrm {P} \,\!$ has the value True.

We can put this more formally. The first row of the truth table above shows that when ${\mathfrak {v}}[\mathrm {P} ]\,\!$ = True, ${\overline {\mathfrak {v}}}[\lnot \mathrm {P} ]\,\!$ = False. The second row shows that when ${\mathfrak {v}}[\mathrm {P} ]\,\!$ = False, ${\overline {\mathfrak {v}}}[\lnot \mathrm {P} ]\,\!$ = True. We can also put things more simply: a negation has the opposite truth value than that which is negated.

Conjunction

The truth table for conjunction corresponds to clause (iii) of our definition for extended valuations.

$\mathrm {P} \,\!$ $\mathrm {Q} \,\!$ $\mathrm {P} \land \mathrm {Q} \,\!$ T T T T F F F T F F F F

Here we have two sentence letters and so four possible interpretations, each represented by a single row. The first two columns show what the four interpretations assign to $\mathrm {P} \,\!$ and $\mathrm {Q} \,\!$. The interpretation represented by the first row assigns both sentence letters the value True, and so on. The last column shows the value assigned to $\mathrm {P} \land \mathrm {Q} \,\!$. You can see that the conjunction is true when both conjuncts are true—and the conjunction is false otherwise, namely when at least one conjunct is false.

Disjunction

The truth table for disjunction corresponds to clause (iv) of our definition for extended valuations.

$\mathrm {P} \,\!$ $\mathrm {Q} \,\!$ $\mathrm {P} \lor \mathrm {Q} \,\!$ T T T T F T F T T F F F

Here we see that a disjunction is true when at least one of the disjuncts is true—and the disjunction is false otherwise, namely when both disjuncts are false.

Conditional

The truth table for conditional corresponds to clause (v) of our definition for extended valuations.

$\mathrm {P} \,\!$ $\mathrm {Q} \,\!$ $\mathrm {P} \rightarrow \mathrm {Q} \,\!$ T T T T F F F T T F F T

A conditional is true when either its antecedent is false or its consequent is true (or both). It is false otherwise, namely when the antecedent is true and the consequent is false.

Biconditional

The truth table for biconditional corresponds to clause (vi) of our definition for extended valuations.

$\mathrm {P} \,\!$ $\mathrm {Q} \,\!$ $\mathrm {P} \leftrightarrow \mathrm {Q} \,\!$ T T T T F F F T F F F T

A biconditional is true when both parts have the same truth value. It is false when the two parts have opposite truth values.

Example

We will use the same example sentence from Formal Semantics:

$\mathrm {P} \land \mathrm {Q} \rightarrow \lnot (\mathrm {Q} \lor \mathrm {R} )\ .\,\!$

We construct its truth table as follows:

$\mathrm {P} \,\!$ $\mathrm {Q} \,\!$ $\mathrm {R} \,\!$ $\mathrm {P} \land \mathrm {Q} \,\!$ $\mathrm {Q} \lor \ \mathrm {R} \,\!$ $\lnot (\mathrm {Q} \lor \mathrm {R} )\,\!$ $(\mathrm {P} \land \mathrm {Q} )\rightarrow \lnot (\mathrm {Q} \lor \mathrm {R} )\,\!$ T T T T T F F T T F T T F F T F T F T F T T F F F F T T F T T F T F T F T F F T F T F F T F T F T F F F F F T T

With three sentence letters, we need eight valuations (and so lines of the truth table) to cover all cases. The table builds the example sentence in parts. The $\mathrm {P} \land \mathrm {Q} \,\!$ column was based on the $\mathrm {P} \,\!$ and $\mathrm {Q} \,\!$ columns. The $\mathrm {Q} \lor \mathrm {R} \,\!$ column was based on the $\mathrm {Q} \,\!$ and $\mathrm {R} \,\!$ columns. This in turn was the basis for its negation in the next column. Finally, the last column was based on the $\mathrm {P} \land \mathrm {Q} \,\!$ and $\lnot (\mathrm {Q} \lor \mathrm {R} )\,\!$ columns.

We see from this truth table that the example sentence is false when both $\mathrm {P} \,\!$ and $\mathrm {Q} \,\!$ are true, and it is true otherwise.

This table can be written in a more compressed format as follows.

$\mathrm {P} \,\!$      $\mathrm {Q} \,\!$      $\mathrm {R} \,\!$      $\mathrm {P} \,\!$ (1)    $\land \,\!$      $\mathrm {Q} \,\!$ (4)    $\rightarrow \,\!$ (3)    $\lnot \,\!$      $(\mathrm {Q} \,\!$ (2)    $\lor \,\!$      $\mathrm {R} )\,\!$ T T T   T   F F   T   T T F   T   F F   T   T F T   F   T F   T   T F F   F   T T   F   F T T   F   T F   T   F T F   F   T F   T   F F T   F   T F   T   F F F   F   T T   F

The numbers above the connectives are not part of the truth table but rather show what order the columns were filled in.

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