@Zanna In what way?

Do you mean there are a number of things, relating to what I've said about logic and set theory, that you wish to try out but have not yet tried? Or something else?

I mean I am still rereading from a long time ago

Ah.

That you're rereading most of what I've written here about logic and set theory makes me feel like I should have written more carefully and with an actual... plan.

it's fine :)

What part are you rereading currently?

(If you wish to say.)

this is the message I last anchored myself at

@Zanna There is more that I can say, and should perhaps have said at the time, about the idea of empty conjunctions and disjunctions.

Relatedly, there is more that I can say, and probably should have said at the time, on the topic of logical equivalence (such as in De Morgan's laws, and in the preceding topic of expressing "→" in other ways, specifically by "¬" and "∨" and by "¬" and "∧").

I'm not sure if you want me to say more about any of that now or not, though.

if you feel like saying anything about it I will appreciate reading it, though I might not be attending to it for a while

I have said "logically equivalent" a number of times, without defining that phrase. Are you familiar with it?

Also, are you familiar with truth tables?

When I said that, I probably thought I was defining or adequately characterizing logical equivalence. I don't believe I was, though.

@EliahKagan no, I'm not familiar with it

@EliahKagan I have done some things with those

@Zanna Had the claims I made using that phrase made sense in context?

yes, it seemed clear what you were showing about those examples

@Zanna Cool! Do you know how to write the truth tables for each of the commonly used truth-functional sentential connectives, "¬", "∧", "∨", "→", and "↔"?

Also, and I should perhaps have asked this first: what kinds of things have you done with truth tables?

I don't remember how to do it or exactly what it was I did, but I remember it being easy and helpful

@EliahKagan no but I feel like I could get it with some hints hahaha

@Zanna Recall that truth-functional sentential connectives are called "truth-functional" because the truth value (i.e., truth or falsity) of the compound sentence formed with them is determined entirely by the truth values of the simpler sentences being composed.

so if thing 1 is true/false and thing 2 is true/false, what's the outcome for the whole sentence when connected with all these things?

Right. We usually use sentence letters like p and q for thing 1 and thing 2 respectively.

So a truth table is, as you probably know, a table that has one row for each combination of truth values of each component sentence.

I can picture it

For example, "¬" is a unary truth-functional connective. It attaches to a single sentence to produce a compound sentence whose truth value is the opposite of that of the sentence it attaches to. For a single sentence, there are two truth values. So a truth table for "¬" needs, and has, only two rows:

p   ¬p
T   ⊥
⊥   T

To represent truth values, I've used "T" and "⊥", but some people use "T" and "F" and some people use "t" and "f".

ok :)

so if p is true, not p is false

and vice versa

@EliahKagan The advantage of "⊥" over "F" or (to a lesser extent) "f" comes when it is used in a context where the structure of some or all sentences is also shown. That is "F" for false could be confused with a predicate, as in Fx.

@Zanna Yes. And equipped with a truth table that shows the effect of one or more connectives, one can easily extend it with columns for combinations of those connectives. For example:

p   ¬p   ¬¬p
T   ⊥    T
⊥   T    ⊥

Even if you did not know what the symbol "¬" meant at the start (since there are four possible unary truth-functional sentential connectives, it's just that "¬" is the only one that's actually useful), the first two columns of the truth table would tell you, effectively defining it, and would also directly facilitate writing the third column.

nice

As you know, "∧" is a binary truth-functional connective. It attaches to two sentences to produce a compound sentence that is true just when both of those sentences are true. For two sentences (and without knowing anything about the structure of those sentences, so as sometimes to be able to know that the truth value of one depends on the truth value of the other), there are four possible combinations of truth values. So a truth table for "∧" needs, and has, four rows:

p   q   p ∧ q
T   T     T
T   ⊥     ⊥
⊥   T     ⊥
⊥   ⊥     ⊥

If you're on a mobile device, this might not be easy to show, but if you're at a computer, or have paper on hand, then I suggest making a truth table for "∨". Otherwise, you could describe specifically how the truth table for "∨" differs from the truth table for "∧".

p   q     p v q
T   T       T
T   ⊥       T
⊥   T       T
⊥   ⊥       ⊥

oh no

yesss

Sry, was afk.

@Zanna Yes.

How about "→"?

no problem!

I have to dash but will come back when possible :/

Okay.