Tag: lattice

Atomless: The Calculus of Systems of (Almost) Any Logic 

Bas van FraassenLeave a comment

A logic L is a closure operator on a set of sentences S (of a syntax SYNT). An L-theory is a subset of S closed under L, and an L-theorem is a sentence that belongs to all L-theories.  (A logic may not have any theorems.)  A sentence A is an L-consequence of set of sentences X exactly if A is in L(X).

The L-theories, being the L-closed sets, form a complete lattice CL, which Tarski called the calculus of systems of that logic.  This lattice is bounded above and below, respectively by S and L(Λ), the L-closure of the empty set.  The latter contains only the set of theorems of L, if any, and is therefore also said to be trivial.  

In a lattice, an atom is an element that is not the bottom but has no element between it and the bottom.  Precisely:

Definition.  An L-theory X is an atom of the calculus of systems of L if and only if X is not trivial but has no non-trivial proper sub-theory.

So far, so general.  I wish to remark that in almost all cases, and certainly for all familiar logics, this lattice is not atomic, in fact has no atoms at all.

Definition.  Logic L is disjunction minimal exactly if:

  1. S is a set of finite strings generated from an infinite set of propositional variables
  2. The syntax SYNT includes a binary connective v such that
    1. if A, B are sentences then A v B is an L-consequence of A
    1. if A is not an L-theorem, and q is a propositional variable that does not occur in A, then A v q is not an L-theorem, and A is not an L-consequence of A v q.

Result.   If logic L is at least  disjunction minimal then its calculus of systems CL has no atoms.

Argument.

Suppose X is a non-trivial L-theory, and A is a member of X that is not an L-theorem.  Since A is finite there is a propositional variable q which does not occur in A.  Then A v q is a member of X, and is not an L-theorem, and A is not an L-consequence of A v q.

Therefore L({A v q}) is a non-trivial proper sub-theory of X, for it is part of X, includes A v q which is not an L-theorem, and it does not include member A of X.  Hence X has a proper non-trivial sub-theory.

So, if L is at least disjunction minimal, and X is any non-trivial L-theory, then X is not an atom of calculus CL. Hence CL has no atoms.

Remark. Almost all, but not all logics in the literature are at least disjunction minimal. An exception is the Weak Kleene logic WK3 where A does not entail A v B. See Jc Beall (2016) “Off-Topic: a new interpretation of Weak Kleene logic”. Australasian Journal of Logic (13: 6), Article 1.