◀ ▲ ▶Branches / Logic / Definition: `$k$`-nary Connectives, Prime and Compound Propositions

The semantics of propositional logic involves so-called connectives, including the negation "$\neg$", the conjunction "$\wedge$", the disjunction "$\vee$", the inclusion "$\Rightarrow$", and the equivalence "$\Leftrightarrow$". These connectives are important concepts in mathematics on their own. Therefore, they deserve separate definitions. We will introduce them now step by step, starting with a formal definition of connectives.

Definition: $k$-nary Connectives, Prime and Compound Propositions

Let $L$ be a formal language with a syntax of propositional logic $PL0$. Let $L'\subset L$ be the subset of all strings of $L$ which are propositions, i.e. which have a semantics of $PL0$. A $k$-nary connective is an operation "$\circ$" used to connect $k$ ($k=1,2,\ldots$) propositions

• compatible with their syntax, i.e. if $p_1,\ldots,p_n$ are propositions, then $p_1\circ \ldots\circ p_n$ is also a proposition, and
• for which the semantics of $PL0$ is properly defined, i.e. if $[[p_1]]_I, [[p_2]]_I,\ldots, [[p_k]]_I$ are assigned truth values for some interpretation $I$, then also $[[p_1\circ\ldots\circ p_k]]_I$ is an assigned truth value in the same interpretation $I.$

Propositions being Boolean constants or a Boolean variables are called prime propositions.

Any proposition involing a connective $p:=p_1\circ\ldots\circ p_k$ is called a compound proposition.

Mentioned in:

Chapters: 1
Corollaries: 2
Definitions: 3 4
Examples: 5 6
Lemmas: 7 8 9
Proofs: 10 11 12 13 14

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References

Bibliography

1. Mendelson Elliott: "Theory and Problems of Boolean Algebra and Switching Circuits", McGraw-Hill Book Company, 1982