theory

Definition (theory)

A theory is a set $T$ of sentences such that for any sentence $\phi$, if $T \models \phi$, then $\phi \in T$.

(a theory is closed under logical consequence)

Notes

• A theory is decidable if the problem of belonging to $T$ is decidable
  • exists a decision procedure for reasoning about $T$
• A theory $T$ is complete if for any sentence $\phi$ we have $\phi \in T$ or $\neg \phi \in T$

References

1. M. Avanzini, Lecture Notes, Topic: “weak monadic second-order logic (WMSO).” M1-AL, Centre Inria d’Université Côte d’Azur, 2021. https://www-sop.inria.fr/members/Martin.Avanzini/teaching/2021/AL/slides/w2.pdf
2. https://en.wikipedia.org/wiki/Theory_(mathematical_logic)