# On the Idea of a “Validity Calculus” Analogous to Probability Calculus

The question might be judged as belonging to logic-fiction rather than to serious logic ...

Some well formed formulas are true in all possible cases/ interpretations , some are true in no possible case/ interpretation, some are contingent ( true in some cases, false in other ones).

Now, amongst contingent formulas, some are true in more cases than others are.

For example, (A --> B) is true in 3 cases out of 4, while (A&B) is true in only 1 case out of 4.

So, one could say that some formilas are closer to validity than others. This suggests an analogy with the theory of probability, in which some some propositions are considered as being closer to certainty than others.

Is it possible to think of rules concerning, so to say, the " degrees" of validity of propositional calculus formulas.

Is it possible to take a theoretical advantage of the fact that all formulas are not at the same distance from validity.

• It sounds what you suggest is the number of $1$'s in the truth table of the formula, or equivalently, the number of terms in disjunctive normal form. Nevertheless, assigning each propositional variable a random variable will measure exactly this.. – Berci Oct 30 at 22:06
• Avoiding numerical interpretations, this is exactly what the Lindenbaum algebra $L$ (of the empty theory, in propositional logic) captures: $L$ is the partial order of propositional sentences, ordered by $$\varphi\ge\psi\quad\iff\quad\vdash\varphi\rightarrow\psi,$$ modulo the corresponding equivalence relation $\dashv\vdash$. The Lindenbaum algebra has the advantage of capturing a finer level of detail than just the number of satisfying valuations, as well as being more broadly applicable (e.g. to things like first-order logic where there's no long an obvious notion of "random valuation"). – Noah Schweber Oct 31 at 3:24

Expanding my comment into an answer:

While there is a definite connection with probability in the propositional case per Berci's comment - treating each propositional atom as an independent coin-flip - this connection doesn't generalize well outside that context. For example, in first-order logic there is no obvious probability distribution on the "set" (or rather, any sufficiently large set) of structures. There is however an approach to "comparative truth" which applies to basically all logics and is fundamentally algebraic as opposed to probabilistic, namely the Lindenbaum algebra: the partial order given by equivalence classes of sentences modulo equivalence over $$T$$, with the ordering given by $$T$$-provability. The partial ordering corresponds exactly to "at least as true as (assuming $$T$$)."

Typically this partial order comes with additional structure due to the interaction between the syntax and the deduction relation. For example, in the case of classical propositional logic we get the structure of a Boolean algebra, in the case of intuitionistic propositional logic we get the structure of a Heyting algebra, classical first-order logic gives a cylindrical algebra, and so forth. Usually the phrase "Lindenbaum algebra" refers to the partial order equipped with such natural structure; arguably I should call the bare partial order the "Lindenbaum order."

It's worth noting however that there are serious algebraic issues here, not directly relevant to your question but foundational to the meaning of the Lindenbaum algebra.

The starting point is the observation that syntax may play poorly with the Lindenbaum algebra. As noted above, some syntactic structure translated to structure in the Lindenbaum algebra, but this is not true in general: for example, in reasonable propositional modal logics we can have $$T\vdash \varphi\leftrightarrow\psi$$ but $$T\not\vdash (\Box \varphi)\leftrightarrow(\Box\psi)$$. So while the set of sentences can be thought of as a free algebra in an appropriate sense, this doesn't interact well with deductions.

The interplay between the deduction relation and the syntactic algebra is quite nuanced; see Blok and Pigozzi's book or the first bit of Czelakowski's book for a good description of the situation. This all belongs to the general subject of algebraic logic.

• NoahSchweber.- Thanks for this detailed and accurate answer. – Eleonore Saint James Nov 1 at 11:26