@Zanna Indeed.

Btw have you noticed the close connection between truth tables and Smullyan's method of indexing?

no... I did not notice any connection

I liked that method as it was easy to understand

kind of like truth tables...

@Zanna One of the other ways to determine if a Boolean equation is valid (which Smullyan doesn't cover until the beginning of the second book) is to treat it as an assertion that two truth-functional schemata are logically equivalent. Every Boolean term, such as $A \cap B$, has a corresponding truth-functional schema in which set symbols become sentence letters and operations become truth-functional sentential connectives.

(So, for example, the Boolean term $A \cap B$, where $A$ and $B$ refer to sets, becomes the truth-functional schema $A \wedge B$, where $A$ and $B$ stand for arbitrary sentences.)

That is, pretend that:

(1) each symbol referring to a set (in Smullyan's notation, $A$, $B$, etc.) is instead a sentence letter,

(2) each operation on sets is instead the corresponding truth-functional sentential connective or combination of sentential connectives. At the risk of making something quite intuitive appear to be far more complicated than it really is, this is to say that you should read:

i. $A'$ as $\neg A$

ii. $A \cap B$ as $A \wedge B$

iii. $A \cup B$ as $A \vee B$

iv. $A \equiv B$ as $A \leftrightarrow B$

v. $A \rightarrow B$ as $A \rightarrow B$

vi. $A - B$ as $A \wedge \neg B$

[where the Boolean equations shown before "as" are in the notation Smullyan uses for Boolean equations, and the truth-functional schemata shown after "as" are in the notation we've been using for truth-functional schemata],

(3) the symbol for the empty set, often (and in Smullyan's usage) written $\varnothing$, is instead $\top$; and the symbol for the "universal set", which Smullyan writes as $I$, is instead $\bot$,

[recall that $\top$ (respectively, $\bot$) in a truth-functional schema, which some people write $T$ (respectively, $F$), is treated like a sentence letter, except that it may only be interpreted as true (respectively, false); alternatively, it may be viewed as a nullary (i.e., medadic, i.e., of arity zero) truth-functional sentential connective that always comes out true (respectively, false)], and

(4) $=$ is instead an assertion that the truth-functional schema written to its left is logically equivalent to the truth-functional schema written to its right.

If this does not seem immediately intuitive, then consider how you would use Venn diagrams to demonstrate that two truth-functional schemata are valid, and also how you would use Venn diagrams to demonstrate that a Boolean equation is valid.

Each cell (or, in the terminology Smullyan uses, basic region) of a Venn diagram represents a combination of yes-no answers. When the Venn diagram is used to visualize sets, the questions being answered are questions about membership. Calling the sets $A$, $B$, and so forth, if you draw an object $x$ in a particular cell, the questions you're giving yes-no answers to are "Is $x \in A$?", "Is $x \in B$?", and so forth.

When the Venn diagram is used to visualize sentences, it is as though each region is a container of possible worlds. Suppose you have three sentences, $P$, $Q$, and $R$, and you are only interested in those sentences and sentences formed as truth-functional compounds of them.

In this particular conceptualization of Venn diagrams, the region for $P$, which is not a cell, contains all the worlds that satisfy $P$ (i.e., in which $P$ comes out true), and likewise the regions for $Q$ and $R$, which are not cells, contain all the worlds that satisfy $Q$ and all the worlds that satisfy $R$, respectively.

The region for $P \wedge Q \wedge \neg R$, which is a cell, contains all the worlds that satisfy $P$ and satisfy $Q$ but do not satisfy $R$. Worlds in different cells can be distinguished by examining the truth of $P$, $Q$, and $R$, while worlds in the same cell cannot.

The technically precise analogue of thinking of the cells of a Venn diagram as containers for worlds is to think of them as representing sets of interpretations. This works for both meanings of "interpretation."

For truth-functional schemata with sentence letters $p$, $q$, and so forth, the rows of a truth table represent combinations of answers to yes-no questions "Is $p$ true?", "Is $q$ true?", and so forth -- and which combination agrees with reality will in general depend on the interpretation given to those sentence letters (i.e., on what sentences they are taken to stand for, and in particular, on the actual truth values of those sentences).

For boolean terms with sets $A$, $B$, and so forth, the individual indices ($1$, $2$, ...) in Smullyan's method of indexing represent combinations of answers to yes-no questions "Is $x \in A$?", "Is $x \in B$?", and so forth, and which combination agrees with reality will in general depends on what object $x$ you pick (and also on what sets $A$, $B$, and so froth are).

@EliahKagan I meant to say:

(Sorry about the extra ping; I just wanted that awful error of mine to be immediately apparent.)

pings are all good :)

@EliahKagan makes sense

I will re read the rest tomorrow if time permits

Cool. There is no hurry. :)