Tools and Techniques in Modal Logic Marcus Kracht II ...

1.4. General Logic 17
Exercise 7. Show that for any class of algebras, HH(K) ⊆ H(K) as well as SS(K) ⊆
S(K) and PP(K) ⊆ P(K). Hint. For the last claim take a collection 〈I j : j ∈ J〉 of
pairwise disjoint index sets, and assume that there is an algebra Ai j for every i j ∈ I j,
j ∈ J. Then each B j := � k∈I j Ak is a product, and C := � j∈J B j is a general element
of PP(K). Now let K := � j∈J I j and put D := � k∈K Ak. Show that C � D.
1.4. General Logic
In our view, logic is the study of truth and consequence. In logic we study
(among other things) whether a statement ϕ follows from some set ∆ of other statements.
We usually write ∆ ⊢ ϕ if this is the case. We interpret this as follows: if all
χ ∈ ∆ are true then so is ϕ. Of course, we must specify what we mean by being true.
However, already on these assumptions there are some nontrivial things that can be
said about the relation ⊢. To write them down, we will — in accordance with our
notation in connection with modal logic — use lower case Greek letters for terms
of propositional logic, since these terms are thought of as formulae. We also will
henceforth not distinguish between L as a set of function symbols and the terms of
L, namely the set TmΩ(var); given this convention ⊢ ⊆ ℘(L) × L. Moreover, we
write Σ ⊢ Γ if for all ϕ ∈ Γ, Σ ⊢ ϕ. It is also customary to use Σ; ∆ for Σ ∪ ∆ and
Σ; ϕ instead of Σ ∪ {ϕ}. This notation saves brackets and is almost exclusively used
instead of the proper set notation.
(ext.) If ϕ ∈ Σ then Σ ⊢ ϕ.
(mon.) If Σ ⊆ ∆ then Σ ⊢ ϕ implies ∆ ⊢ ϕ.
(trs.) If Σ ⊢ Γ and Γ ⊢ ϕ then Σ ⊢ ϕ.
(Observe that (mon.) is derivable from (ext.) and (trs.).) For suppose that ϕ ∈ Σ.
Then if all χ ∈ Σ are true, then ϕ is true as well. Thus (ext.) holds. Furthermore, if
Σ ⊢ ϕ and Σ ⊆ ∆ and if all χ ∈ ∆ are true, then all terms of Σ are true and so ϕ is true
as well; this shows (mon.). The third rule is proved thus. If Σ ⊢ Γ and ∆ ⊢ ϕ and all
terms of Σ are true then all formulae of Γ are true by the first assumption and so ϕ is
true by the second.
In addition, there are two other postulates that do not follow directly from our
intuitions about truth–preservation.
(sub.) If Σ ⊢ ϕ and σ is a substitution then Σ σ ⊢ ϕ σ .
(cmp.) Σ ⊢ ϕ iff there exists a finite Σ0 ⊆ Σ such that Σ0 ⊢ ϕ.
The postulate (sub.) reflects our understanding of the notion of a variable. A variable
is seen here as a name of an arbitrary (concrete) proposition and thus we may plug in
all concrete things over which the variables range. Then the relation Σ ⊢ ϕ says that
for any concrete instances of the occurring variables, the concretization of Σ ⊢ ϕ is
valid. So, the rule p ∧ q ⊢ q ∧ p — being valid — should remain valid under all concretizations.
For example, we should have Aristotle was a philosopher and Socrates