### 5.1.2 Semantics of the Propositional Calculus

**Semantics** specifies how to put symbols of the language into
correspondence with the world. Semantics can be used to
understand sentences of the language.
The semantics of propositional calculus is defined below.

An **interpretation**
consists of a function *π* that maps atoms to *{*true, false*}*.
If *π(a)=*true, we say atom *a* is **true** in the interpretation, or that the interpretation assigns *true* to *a*.
If *π(a)=*false, we say *a* is **false** in the interpretation.
Sometimes it is useful to think of *π* as the set of atoms that
map to true, and that the rest of the atoms map to false.

The interpretation maps each proposition to a **truth value**. Each proposition is either **true** in the
interpretation or **false** in the interpretation. An atomic
proposition *a* is true in the interpretation if *π(a)=*true;
otherwise, it is false in the interpretation. The truth value of a compound proposition is built
using the truth table of Figure 5.1.

Note that we only talk about the truth value in an interpretation. Propositions may have different truth values in different interpretations.

p | q | ¬p | p ∧q | p∨q | p ←q | p →q | p ↔q |

true | true | false | true | true | true | true | true |

true | false | false | false | true | true | false | false |

false | true | true | false | true | false | true | false |

false | false | true | false | false | true | true | true |

**Example 5.1:**Suppose there are three atoms:

*ai_is_fun*,

*happy*, and

*light_on*.

Suppose interpretation *I _{1}* assigns

*true*to

*ai_is_fun*,

*false*to

*happy*, and

*true*to

*light_on*. That is,

*I*is defined by the function

_{1}*π*defined by

_{1}*π*,

_{1}(ai_is_fun)=true*π*, and

_{1}(happy)=false*π*. Then

_{1}(light_on)=true*ai_is_fun*is true in*I*._{1}*¬ai_is_fun*is false in*I*._{1}*happy*is false in*I*._{1}*¬happy*is true in*I*._{1}*ai_is_fun∨happy*is true in*I*._{1}*ai_is_fun ←happy*is true in*I*._{1}*happy ←ai_is_fun*is false in*I*._{1}*ai_is_fun ←happy ∧light_on*is true in*I*._{1}

Suppose interpretation *I _{2}* assigns

*false*to

*ai_is_fun*,

*true*to

*happy*, and

*false*to

*light_on*:

*ai_is_fun*is false in*I*._{2}*¬ai_is_fun*is true in*I*._{2}*happy*is true in*I*._{2}*¬happy*is false in*I*._{2}*ai_is_fun∨happy*is true in*I*._{2}*ai_is_fun ←happy*is false in*I*._{2}*ai_is_fun ←light_on*is true in*I*._{2}*ai_is_fun ←happy ∧light_on*is true in*I*._{2}

A **knowledge base** is a set of propositions that the agent is given as being true. An
element of the knowledge base is an **axiom**.

A **model** of a set of propositions is an
interpretation in which all the propositions are
*true*.

If *KB* is a knowledge base and *g* is a proposition, *g* is a **logical consequence**
of *KB*, written as

KB g

if *g* is *true* in every model of *KB*.

That is,
no interpretation exists in which *KB* is *true* and *g* is
*false*. The definition of logical consequence places no constraints
on the truth value of *g* in an interpretation where *KB* is false.

If *KB g* we also say *g* **logically follows**
from *KB*, or *KB* **entails** *g*.

**Example 5.2:**Suppose

*KB*is the following knowledge base:

*sam_is_happy.*

*ai_is_fun.*

*worms_live_underground.*

*night_time.*

*bird_eats_apple.*

*apple_is_eaten ←bird_eats_apple.*

*switch_1_is_up ←sam_is_in_room ∧night_time.*

Given this knowledge base,

*KB bird_eats_apple.*

*KB apple_is_eaten.*

*KB* does not entail *switch_1_is_up* as there is a model of the
knowledge base where *switch_1_is_up* is false. Note that
*sam_is_in_room* must be false in that interpretation.