# Describing propositional formulae as "independent"

Suppose that the truth values of two (either atomic or compound) propositional formulae do not depend on each other.

Can we call the two formulae "independent", or is there an appropriate adjective?

For example, $$(p\land q)$$ and $$(p \lor q)$$ are "dependent" formulae because when the former is true, the latter cannot be false. Similarly, $$(p)$$ and $$(\lnot p)$$ are "dependent" formulae.

On the other hand, $$(p \land q)$$ and $$(r\to s)$$ are "independent" formulae.

I'd sacrifice brevity for clarity and just spell it out:

• “The two propositions' logical truth values are dependent/independent of each other.”

This dependency can be characterised like so:

• Two propositions have dependent logical truth values precisely when interpreting at least one of them as true (or as false) determines the truth value, under the same interpretation, of the other.

("Logical truth value", rather than "truth value", to indicate that the two propositions haven't been subjected to an interpretation, which determines their truth values and makes them necessarily dependent.)

Thus, $$(p∧q)$$ and $$(p∨q)$$ have dependent logical truth values, while $$((p → q) → r)$$ and $$(p → (q ↔ r))$$ have independent logical truth values. $$\begin{array}{cc|c@{}c@{}ccc@{}ccc@{}ccc@{}c@{}c} p&q&{}&(p&\land&q)&{}&(p&\land&q)\\\hline 1&1&&{}&\mathbf1&{}&\mathbf{}&{}&\mathbf1&{}\\ 1&0&&{}&\mathbf0&{}&\mathbf{}&{}&\mathbf0&{}\\ 0&1&&{}&\mathbf0&{}&\mathbf{}&{}&\mathbf0&{}\\ 0&0&&{}&\mathbf0&{}&\mathbf{}&{}&\mathbf0&{} \end{array}$$ $$\begin{array}{ccc|c@{}c@{}c@{}ccc@{}ccc@{}ccc@{}ccc@{}ccc@{}c@{}c@{}c} p&q&r&{}&((p\rightarrow q)&\rightarrow&r)&{}&(p&\rightarrow&(q\leftrightarrow r))\\\hline 1&1&1&&&\mathbf1&{}&&{}&\mathbf1\\ 1&1&0&&&\mathbf0&{}&&{}&\mathbf0\\ 1&0&1&&&\mathbf1&{}&&{}&\mathbf0\\ 1&0&0&&&\mathbf1&{}&&{}&\mathbf1\\ 0&1&1&&&\mathbf1&{}&&{}&\mathbf1\\ 0&1&0&&&\mathbf0&{}&&{}&\mathbf1\\ 0&0&1&&&\mathbf1&{}&&{}&\mathbf1\\ 0&0&0&&&\mathbf0&{}&&{}&\mathbf1 \end{array}$$ Trivially, each pair of non-contingent propositions has dependent logical truth values.

• I agree with your definition, I wait for others too in case they find out such a definition already given in the mathematical literature.
– Aria
Commented Jul 9, 2022 at 18:28
• I consider that boldfaced bullet point a description/elaboration rather than a definition, which might for example be "Two propositions are dependent precisely when...". The point, as corroborated by Mario's above comment, is that this definition (just created here) is nonstandard, and anyway not terribly crucial. Commented Jul 10, 2022 at 8:25