foundations of computational agents
The third edition of Artificial Intelligence: foundations of computational agents, Cambridge University Press, 2023 is now available (including full text).
A proposition is a sentence, written in a language, that has a truth value (i.e., it is true or false) in a world. A proposition is built from atomic propositions using logical connectives.
An atomic proposition, or just an atom, is a symbol. We use the convention that propositions consist of letters, digits and the underscore (_) and start with a lower-case letter.
For example, ai_is_fun, ${\text{lit\_l}}_{1}$, live_outside, mimsy and sunny are all atoms.
Propositions can be built from simpler propositions using logical connectives. A proposition or logical formula is either
an atomic proposition or
a compound proposition of the form
$\mathrm{\neg}p$ | “not $p$” | negation of $p$ |
---|---|---|
$p\wedge q$ | “$p$ and $q$” | conjunction of $p$ and $q$ |
$p\vee q$ | “$p$ or $q$” | disjunction of $p$ and $q$ |
$p\to q$ | “$p$ implies $q$” | implication of $q$ from $p$ |
$p\leftarrow q$ | “$p$ if $q$” | implication of $p$ from $q$ |
$p\leftrightarrow q$ | “$p$ if and only if $q$” | equivalence of $p$ and $q$ |
where $p$ and $q$ are propositions.
The operators $\mathrm{\neg}$, $\wedge $, $\vee $, $\to $, $\leftarrow $ and $\leftrightarrow $ are logical connectives.
Parentheses can be used to make logical formulae unambiguous. When parentheses are omitted, the precedence of the operators is in the order they are given above. Thus, a compound proposition can be disambiguated by adding parentheses to the subexpressions in the order the operations are defined above. For example,
$$\mathrm{\neg}a\vee b\wedge c\to d\wedge \mathrm{\neg}e\vee f$$ |
is an abbreviation for
$$((\mathrm{\neg}a)\vee (b\wedge c))\to ((d\wedge (\mathrm{\neg}e))\vee f).$$ |