@EliahKagan that one we did earlier

p   q   p -> q
T   T     T
T   ⊥     ⊥
⊥   T     T
⊥   ⊥     T

@EliahKagan I think that one should come out like this

p   q   p <-> q
T   T     T
T   ⊥     ⊥
⊥   T     ⊥
⊥   ⊥     T

is that right?

sorry I was away for like a week

that was not intentional

things occurred

@Zanna Yes. Those truth tables are both correct.

@Zanna No problem. I hope they were good or neutral things.

@EliahKagan oh, good :)

you have made me think about the good things and feel grateful for them, like managing to write a letter to one of my friends and talking with another friend about a book I'm enjoying :)

@Zanna So as I mentioned, in addition to providing a way to define an arbitrary truth-functional sentential connective, truth tables provide a way to determine the truth conditions produced by attaching sentences with truth-functional sentential connectives.

By "truth conditions" I mean the effect, of each combination of truth values for the sentences that are connected, on the truth value of the compound. Or to put it another way, which combinations of truth values make the compound true and which make it false.

@Zanna I have done this?

In light of this, I recommend making a truth table that verifies De Morgan's laws.

(You could do this with one truth table or two.)

@EliahKagan yes, you hoping they were good prompted me to think about whether they were

Ah. :)

p   q   ¬(p ∧ q)  ¬p ∨ ¬q     ¬(p ∨ q)  ¬p ∧ ¬q
T   T     ⊥          ⊥           ⊥          ⊥
T   ⊥     T          T           ⊥          ⊥
⊥   T     T          T           ⊥          ⊥
⊥   ⊥     T          T           T          T

aligning stuff is hard

idk if that is right, or if it is even what I'm supposed to be doing XD

That truth table is correct.

that's good! The alignment could be better...

It is not really complete, because it has a column ¬(p ∧ q) but no column for p ∧ q, and it has a column for ¬(p ∨ q) but no column for p ∨ q.

hey I didn't know there was a thing called vimtutor

ooh

In practice such an omission is typically fine, because the absent information is easily verified and well-known. However, this is true also of De Morgan's laws themselves, so relative to the goal of demonstrating De Morgan's laws, it would benefit from the inclusion of the missing columns.

@Zanna Are you writing them in the textarea that appears in the SE chat interface?

If so, that explains why alignment is hard to achieve. This interface does not use a monospaced font for the textarea itself, even though it renders in a monospaced font when it is formatted as code.

I recommend making them in a text editor and then pasting them in.

How did you learn vim?

(I am not suggesting vimtutor is the best or only way.)

@Zanna Can you say exactly what it is about that truth table that verifies the correctness of De Morgan's laws?

(This is not a trick question. I am disregarding the arguably important, arguably unimportant missing columns. I am saying, that truth table does verify De Morgan's laws... but why?)

@EliahKagan To be complete, it should also have columns for ¬p and ¬q.

That is, given any compound that has a column in a truth table, to fully demonstrate the correctness of the truth values in that column, the truth table should have columns for each compound strictly contained within that compound.

After all, it is on the truth values of those smaller compounds (or individual sentence letters, once you get down to p, q, r, etc.) that the truth value of the larger compound depends.

One way to look at it is through expression trees. For example, for the third and fourth columns of that table:

(generated from dreampuf.github.io - bit.ly/36uTqbP)

The leaves are sentence letters, and the internal nodes are truth-functional sentential connectives.

In a full truth table for a truth-functional compound, a column corresponds to that compound, which is associated with the full expression tree, and to the left of it there are columns associated with each subtree.

@EliahKagan I don't think I have learned vim, that is I'm using it at a verrrryyy low level

but I have played Vim Adventures and read/done the Vim parts in The Linux Command Line

@Zanna Out of curiosity, what editor do you use?

@Zanna Well, vimtutor is mostly just about showing and practicing the commands for navigation and editing.

@user3140225 I use Vim :)

← 8 messages moved from Raiders of the Lost Downboat

@Zanna 0h! I thought that you'd be an expert on the editor of your choice and, since you mentioned that you use vim at a low level, I am surprised that you actually use vim!

well, I know how to use vim better than all the other editors :D

Hahaha! That almost sounds like a joke!

@EliahKagan I should try it :)

@EliahKagan That is, what information in the truth table demonstrates that if you have a compound sentence of the form ¬(p ∧ q) then you can also express it in the form ¬p ∨ ¬q, and vice versa, and that if you have a compound sentence of the form ¬(p ∨ q) then you can also express it in the form ¬p ∧ ¬q, and vice versa?

well, they get the same results

@Zanna Indeed.

In particular, their columns in the truth table are identical.

yes...

This may sound obvious but I think it was worth mentioning. :)

As we've been discussing them, sequences of symbols like "¬(p ∧ q)" are not actually sentences. This is because, although we have given meaning to the connectives, we have not given meaning to the sentence letters.

A sequence of symbols like "¬(p ∧ q)" is said to be a schema. (The plural of "schema" is "schemata".) They are forms that sentences can take. The kind of formal logic that studies the truth relationships between compound sentences based on the meaning of truth-functional sentential connectives is called truth-functional logic or propositional logic.

It is a kind of logic on its own, but it is also a fragment of first-order logic (which is what we've been doing, and which is the kind of logic that modern set theory is built on -- to propositional logic, FOL adds things like names, predicates, variables of quantification, quantifiers, function symbols, definite descriptions, and a built-in notion of identity through a = predicate).

Schemata are sometimes called propositional formulas, and sentence letters are sometimes called propositional variables.

The notion of an interpretation of a schema is important, but there are actually two things that can mean.

In the strictest sense, an interpretation of a truth-functional schema is an assignment of actual sentences to its sentence letters.

However, sometimes what one means by an interpretation is merely the ascription of truth values to its sentence letters.

Two truth-functional schemata are said to be equivalent when they are true under all the same interpretations.

As a simple example, p is equivalent to p, but it is not equivalent to q.

There are interpretation of p and q where they have the same truth value. But they do not have the same truth value under all interpretations. In contrast, p has the same truth value as itself under all interpretations.

Likewise, ¬(p ∨ q) has the same truth value as ¬p ∧ ¬q under all interpretations.

Another phrase for equivalent, in this context, is logically equivalent. For truth-functional schemata, there is no ambiguity in speaking of equivalence--that is, there is nothing that it can mean to say truth-functional schemata are equivalent other than that they are logically equivalent--but in other contexts it is possible for confusion to arise.

¬(p ∧ q) ↔ (¬p ∨ ¬q)

And with a column for:

¬(p ∨ q) ↔ (¬p ∧ ¬q)

If you want to do more, you could also add the missing columns for ¬p, ¬q, p ∧ q, and p ∨ q. If you want to do less, you could just describe what the columns for ¬(p ∧ q) ↔ (¬p ∨ ¬q) and ¬(p ∨ q) ↔ (¬p ∧ ¬q) look like, rather than making another truth table. :)