Chapter 9 Characteristic truth tables

Any sentence of TFL is composed of sentence letters, possibly combined using sentential connectives. The truth value of the compound sentence depends only on the truth value of the sentence letters that comprise it. In order to know the truth value of ‘ $(D\wedge E)$ ’, for instance, you only need to know the truth value of ‘ $D$ ’ and the truth value of ‘ $E$ ’.

We introduced five connectives in chapter 5. So we just need to explain how they map between truth values. For convenience, we abbreviate ‘True’ with ‘T’ and ‘False’ with ‘F’. (But, to be clear, the two truth values are True and False; the truth values are not letters!)

Negation

For any sentence $\mathscr{A}$ : If $\mathscr{A}$ is true, then $\neg$ $\mathscr{A}$ is false; and if $\neg$ $\mathscr{A}$ is true, then $\mathscr{A}$ is false. We can summarize this in the characteristic truth table for negation:

$\mathscr{A}$	$\neg$ $\mathscr{A}$
T	F
F	T

Conjunction

For any sentences $\mathscr{A}$ and $\mathscr{B}$ , $\mathscr{A}$ $\wedge$ $\mathscr{B}$ is true if and only if both $\mathscr{A}$ and $\mathscr{B}$ are true. We can summarize this in the characteristic truth table for conjunction:

$\mathscr{A}$	$\mathscr{B}$	$\mathscr{A}\wedge\mathscr{B}$
T	T	T
T	F	F
F	T	F
F	F	F

Note that the truth value for $\mathscr{A}\wedge\mathscr{B}$ is always the same as the truth value for $\mathscr{B}\wedge\mathscr{A}$ . Connectives that have this property are called commutative.

Disjunction

Recall that ‘ $\vee$ ’ always represents inclusive or. So, for any sentences $\mathscr{A}$ and $\mathscr{B}$ , $\mathscr{A}\vee\mathscr{B}$ is true if and only if either $\mathscr{A}$ or $\mathscr{B}$ is true. We can summarize this in the characteristic truth table for disjunction:

$\mathscr{A}$	$\mathscr{B}$	$\mathscr{A}\vee\mathscr{B}$
T	T	T
T	F	T
F	T	T
F	F	F

Like conjunction, disjunction is commutative.

Conditional

We’re just going to come clean and admit it: Conditionals are a mess in TFL. Exactly how much of a mess they are is philosophically contentious. We’ll discuss a few of the subtleties in sections 10.3 and 13. For now, we are going to stipulate the following: $\mathscr{A}\rightarrow\mathscr{B}$ is false if and only if $\mathscr{A}$ is true and $\mathscr{B}$ is false. We can summarize this with a characteristic truth table for the conditional.

$\mathscr{A}$	$\mathscr{B}$	$\mathscr{A}\rightarrow\mathscr{B}$
T	T	T
T	F	F
F	T	T
F	F	T

The conditional is not commutative. You cannot swap the antecedent and consequent without changing the meaning of the sentence; $\mathscr{A}\rightarrow\mathscr{B}$ and $\mathscr{B}\rightarrow\mathscr{A}$ have different truth tables.

Biconditional

Since a biconditional is to be the same as the conjunction of the conditionals running in both directions, we will want the truth table for the biconditional to be:

$\mathscr{A}$	$\mathscr{B}$	$\mathscr{A}\leftrightarrow\mathscr{B}$
T	T	T
T	F	F
F	T	F
F	F	T

Unsurprisingly, the biconditional is commutative.