# Deductive systems which deduce the structure of logics

“Logics” are frequently defined as having both a “syntax” and a “semantics”.

For example, first-order logic is a deductive system or formal language with an alphabet (collection of symbols) and formation rules, which permit the creation of new expressions and/or judgments. (Perhaps a “judgment” is a well-formed expression which includes the equality symbol/relation.)

Thus, well-formed expressions follow from the formation rules. However, well-formed expressions are associated with a “semantics”, which in this case is one of two values (1 and 0, “true” or “false”).

Perhaps there are two formalizations which can be shown to be equivalent.

On the one hand, if we explicitly denote whether or not an expression is true or false, perhaps with an arrow to 0 or 1, we could define formation rules regarding how one can form new expressions, which include mention of the truth-value. For example:

``````If p -> 1 and q -> 1, you can form p && q -> 1.
``````

On the other hand, perhaps we could define formation rules which only form true expressions, if the starting expression is assumed to be. Thus,

``````If p && q, you can form !(p || q).
``````

In both cases, it seems like “logics” are designed with a distinction between rules of formation vs. rules of evaluation. If one used a particular sequence of formation rules to form a collection of expressions, it can have a variety of relationships to the truth-predicate, or evaluation function.

A choice of formation rules can lead to both p -> 1 and p -> 0, where p is an expression. This is called inconsistency.

Or, if the formation rules do not lead to an expression we intend to have, we call the system incomplete.

It seems like a logic is two deductive systems - the logic, and the model (the syntax and the semantics). To call a logic “incomplete” means we have some other way, than the formation rule, of determining when an expression would be true, and can show that the formation rules don’t form the expression.

A theory of pure equality is an example of a theory, in first order logic. A theory is the collection of all expressions formed from the formation rules. It has the property of decidability.

Zermelo-Fraenkel set theory is another theory of first-order logic. It begins with a collection of logical expressions, called “axioms”. Gödel showed it is incomplete.

Thus, theories are quotients of free algebras. From the collection of all expressions of first order logic, they add as axioms particular relations on expressions, which changes the overall structure of the collection of all expressions.

I want to understand deductive systems whose formation rules cause them to derive expressions which are interpreted as or have the effect of logical evaluation.

Is there a deductive system which generates expressions which we as humans feel embody the characteristic of a sort of “evaluation”? The truth-function emerges from the deductive system and is further applied to the expressions of the deductive system.

• Quite standard presentation, except for "rules of evaluation". What do you mean exactly? In classical propositional logic those are "defined" by truth tables for connectives. We define tautology: a formula that is evaluated as always TRUE and we track semantics with proof system, that means that we check that axioms are always TRUE (tautologies) and that rules of inference (like Modus Ponens) are truth preserving. Commented Nov 29, 2023 at 8:48
• With more detail, for each proof system we have a fundamental result called Soundness and Completeness Theorem that means that the system cannot prove anything that's wrong (soundness) and that can prove anything that's right (completeness) wrt to semantics for that "logic". Commented Nov 29, 2023 at 9:15
• Not clear to me what do you mena with "Is there a deductive system which generates expressions which we as humans feel embody the characteristic of a sort of “evaluation”? "... but in a sense yes. Truth table semantics emerged much later than basic "pattern" of inference codified by rules, like e.g. Modus Ponens and Contraposition. Commented Nov 29, 2023 at 9:24
• What You talk about is true. You are suspecting with You deductive intuition the presence of Absolute Knowledge. I wrote an article regarding. Sharing with You as is. Maybe it has what You are looking for.
– user71091
Commented Jan 20 at 6:26

I believe that what I was looking for is Tarski’s undefinability theorem, which I find quite devastating if I understand it correctly.

(I think), there is no deductive system which can emergently generate structures which are interpretable as “semantic” evaluations (if you enumerated all sentences of the language).

But I need to study it to fully understand it and understand why.

• Ha-ha-ha! :D Yes! Your answer here confirm that You feel with Your deductive intuition the presence of Absolute Knowledge! Why I think so? Tarski, has some example of solving kind of Squaring the Circle. Squaring the Circle is believed to be the hypostatis of Absolute Knowledge. Bibliography
– user71091
Commented Jan 20 at 6:30