Every positive formula is satisifiable

Question

We say that a propositional logical formula is positive if it does not include the negation connective ¬ anywhere in it (but it may still use ∧, ∨, ↔, →, and propositions). Show that all positive formulas are satisfiable formulas.

Any tips/ideas how I can show that?

Answer 1

Note: This procedure assumes you are referring to a propositional language $$L_V=\left<V,\rightarrow,\bot \right>$$ where $V$ is the set of proposition symbols of $L_V$. $L_\Sigma^*$ denotes the set of all strings of $L_V$.

Hint:

We are going to define the set of positive formulas of $L_V$, call it $POS_{L_V}$:

Definition: The set POS of positive well-formed formulas of $L_V$ is the set satisfying

1. $\alpha \in POS$ if $\alpha \in V$
2. $(\alpha \rightarrow \beta) \in PROP$ if $\alpha, \beta \in POS$
3. No string that is not obtained by (1) or (2) belongs to POS.

Now it is easier to put the statement we want to prove:

$$\phi \in POS \Rightarrow M \vDash \phi$$

For some interpretation $M$.

So is it easier to proceed now? What about a induction on the structure of POS?