An introduction to non-classical logic: from if to is

Zotero

If every proof-theoretically valid inference is semantically valid (so that ⊢ entails |=) the proof-theory is said to be sound. If every semantically valid inference is proof-theoretically valid (so that |= entails ⊢) the proof-theory is said to be complete. (38)to-process ❧ Semantic validity: $\models$ Proof-theoretic/Syntactic Validity: $⊢$ If $⊢$ entails $\models$, the system is sound, since the syntax corresponds to our semantics.

If $\models$ entails $⊢$, the system is complete, since our semantics are all syntactically valid.

An interpretation of the language is a function, ν, which assigns to each propositional parameter either 1 (true), or 0 (false). (39)to-process

A is a logical truth (tautology) (|= A) iff it is a semantic consequence of the empty set of premises (φ |= A), that is, every interpretation makes A true. (39)to-process ❧ $\Sigma \models A$ iff there is no interpretation ($v$) where all $\Sigma$ are true and $A$ false.

Tautologies then are a special case, where $\models$ is shorthand for $\phi \models A$.

ρ (rho), reflexivity: for all w, wRw. σ (sigma), symmetry: for all w1, w2, if w1Rw2, then w2Rw1. τ (tau), transitivity: for all w1, w2, w3, if w1Rw2 and w2Rw3, then w1Rw3. η (eta), extendability: for all w1, there is a w2 such that w1Rw2. (70)to-process

Historically, the systems Kρ, Kη, Kρσ, Kρτ and Kρστ are known as T, D, B, S4 and S5, respectively (71)to-process

Every normal modal logic, L, is an extension of K, in the sense that if “|= K A then” |= L A. (71)to-process