I'm comfortable showing the validity of an argument in basic modal logic using tableaux but when asked how to explain why an argument is true using semantics I'm unsure what exactly I'm supposed to show, and how.
For example,
$\Box(p\land q) ⊢ \Box p\land\Box q$
I have begun in the following way:
The antecedent to this argument relies on:
The consequent to this argument relies on:
$\nu _{w}(\Box A)=1$ iff $\nu _{x}(A)=1$ at all worlds $x$ such that $wRx$, where A is a wff
Assuming this is true for wff $p$, as well as for $q$, the following interpretation should also hold true:
$\nu _{w}(\Box p\land \Box q)=1$, thanks to $\nu _{w}(A\land B)=1$ iff $\nu _{w}(A)=1$ and $\nu _{w}(B)=1$, where A and B are wffs
Thus, our consequent is semantically true. Using the following condition,
$\nu _{w}(A\supset B)=1$ iff $\nu _{w}(A)=0$ or $\nu _{w}(B)=1$,
it leads that the argument as a whole is also semantically true.
This feels like a longwinded and unstructured approach and I'm unsure that just assuming truth for the wffs p and q is correct. How would you approach this?
Thanks!
I am not sure what you are asking, as your semantic proof looks good to me. Since you ask how others would approach it, I will give a proof as well. If you can clarify the question, I will happily edit my answer to better suit your needs.
We want to show $\square (p \land q) \vdash \square p \land \square q$.
By completeness, it suffices to show $(\mathfrak{M}, x) \vDash \square p \land \square q$ whenever $(\mathfrak{M}, x) \vDash \square (p \land q)$.
(here $\mathfrak{M} = (W,R,\nu)$ is a model and $x \in W$)
Since $(\mathfrak{M}, x) \vDash \square(p \land q)$, we know every $y \in R(x)$ satisfies $(\mathfrak{M}, y) \vDash p \land q$.
But this happens exactly when $(\mathfrak{M},y) \vDash p$ and $(\mathfrak{M},y) \vDash q$.
Thus, every $y \in R(x)$ satisfies $(\mathfrak{M},y) \vDash p$, so $(\mathfrak{M},x) \vDash \square p$.
Similarly, every $y \in R(x)$ satisfies $(\mathfrak{M},y) \vDash q$, so $(\mathfrak{M},x) \vDash \square q$.
So $(\mathfrak{M},x) \vDash \square p \land \square q$, as desired.
The big idea here is to use what we know about the models in order to prove that certain formulas are valid. We can reason about $\mathfrak{M}$ using whatever tools we like from mathematics, rather than the (extremely limited) set of tools we have at our disposal when writing formal deductions in $K$ (with tableaus, or any of the other deduction systems at our disposal).
For this particular example, it would be just as easy to use a tableau approach, however other theorems in modal logic might not go as smoothly using tableaus, and it is important to have multiple tools at available.
I hope this helps! ^_^