Basic Modal Logic question #1

Question

Its about two weeks I have started Cresswell's "A New Introduction To modal Logic". Now I've got a few questions on the text and I would deeply thank you if you help me clarify on them. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

The first among them will be as follows: To prove $K$-Validity of the system $K$ (i.e. that every theorem of the system $K$ is $K$-Valid; or in other words, is valid in every frame $(W,R)$ ), the book is using induction on the set of theorems. So: (a) It first tries to prove it for all valid formulas of Propositional Calculus (i.e. Tautologies of PC), plus the axiom $K$, are $K$-Valid; And (b) it shows that the Transformation rules of the system $K$, preserve $K$-validity.

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Now in part (a), to prove that every tautology of PC is $K$-Valid, the author argues in the following way:

In any model, a PC wff is evaluated in any world without reference to any other world. Therefore, since a valid wff has the value $1$ for every value-assignment to the variables, it has the value $1$ in every world in every model, i.e. it is valid on every frame.

But the author has defined validity of a formula in world, recursively, such that its base is such that if $V$ is a Value-assignmet, then "For any atomic formula $a$ , and any world $w$ ,either $V(a,w)=1$ or $V(a,w)=0$" [And then he continues to define valuation of other formulas, recursively; that we don't care here] So,the way the book shows is that: Value-assignmet are functions of form $V: PROP\times W \longrightarrow \{0,1\}$ that as we saw, are not dependent to any kind of valuation of propositions in PC.

So why the author argues as above, while there is no connection between valuations of formulas in PC and valuations of them (in worlds) in the system $K$?

Answer 1

The recursive definition of valuation of other formulas is the key. In particular, the valuation of formulas constructed using non-modal connectives only uses the valuation of the component formulas in the current world.

More precisely, for every modal valuation $V$ and every world $w$, you can define a PC-valuation $V_w$ by $V_w(a)=V(a,w)$, and then you can show by induction on the complexity of $\phi$ that if $\phi$ is a PC-formula, then $V(\phi,w)=V_w(\phi)$ (or, $w\models_V\phi\iff\models_{V_w}\phi$, or whatever notation Hughes and Cresswell use for valuation of compound formulas). If $\phi$ is a PC-valid, then $V_w(\phi)=1$, thus $V(\phi,w)=1$. Since $V$ and $w$ were arbitrary, $\phi$ is $K$-valid.

Answer 2

Well, i think the unclarity is more a matter of philosophy than math. You said we can define the valuation of formulas in PC by combining it with the world conception.And Then it results that a formula in PC is valid iff it is valid in every world. Now the ambiguity arises here for me: When i am in the context of PC, i understand truth of an atomic proposition independent from anything else[and then other formulas, recursively on them, by their definition of truth]. No tool in mathematical logic is to check the truth of an atomic proposition; It's a matter of philosophy and theories of truth; and for example if we accept the Correspondence approach to truth, it's somehow a matter of corresponding to the real world(take intuitively the "world" as what we are living in- or the state of affairs). Now the question is that when we define the truth of PC formulas by the modal concept 'world'(Or as you said, a member of the arbitrarily chosen set W in a model) and we say that for example v(a,w)=1 and v(a,w')=0 for an atomic formula a , what is the meaning of such assertions? What is the meaning of truth in a world here? I don't understand it neither mathematically nor non-mathematically!

Tell me if i am still unclear with the question.