breaking down posts into first-order logic

you know this

[logic-of-posts]

The premise of this blog is self-evident. If you want to refer to me for some reason then use it/its pronouns.

I have nothing else to say so I may as well write a primer to concepts on this blog.

Propositions

A proposition is a statement that can either be true or false. Here are some propositions and their truth values:

"This is a blog on Tumblr." (true)

"1+1=2" (true)

"Gravity is a repelling force." (false)

"5<2" (false)

Propositions are represented as propositional variables. For instance, the proposition "It is raining" might be represented as r. I will be using p, q, and r as example variables henceforth, but keep in mind that each does hypothetically correspond to a proposition.

Connectives

There are three fundamental logical connectives:

Conjunction, ∧, "and"

Disjunction, ∨, "or"

Negation, ¬, "not"

The statement p∧q is only true if both p AND q are true. If either, or both, are false, then it is false as well.

The statement p∨q is true if at least one p OR q is true. The only way that this statement can be false is if neither p nor q is true.

The statement ¬p is true if p is false. If p is true, then ¬p is false.

I would love to demonstrate this with truth tables but Tumblr doesn't support tables :(

Derived Connectives

There are three other connectives worth mentioning, but they are technically just combinations of the three connectives mentioned before. I will give them as part of a logical formula because it is easy to say what they are equivalent to

Conditional, p→q ≡ ¬p∨q

Biconditional, p↔q ≡ (p→q)∧(q→p)

Exclusive or, p⊕q ≡ (p∨q)∧¬(p∧q)

For whatever reason, XOR is probably the rarest connective? You don't really see it a lot. But it is a valid connective, and you might see it here.

That symbol with the three lines (≡) represents equivalence, which I will discuss slightly later.

Equivalence

The "slightly later" is NOW!

Firstly: a propositional formula is a tautology if it is always true and a contradiction if it is always false. The most simple tautology is p∨¬p, and the most simple contradiction is p∧¬p. Anything that is not a tautology or contradiction is a contingency.

If the logical formula p↔q is a tautology, then p≡q. There are a whole host of equivalence laws that can be used to simplify logical formulas, which I don't feel like listing out, so go read this Wikipedia article and come back, I guess.

I use my shitty plagiarized hodgepodge program to find the logically equivalent formula that has the smallest possible amount of connectives. It uses truth tables, which means it's globally minimal, but takes a really long time :(

First-Order Logic

First-order logic is like propositional logic, except it also includes predicates and quantifiers.

Open Sentences

Just as a heads up, this whole section will be more mathy than the previous, because I'm most familiar with these concepts in a math way. But, as you'll see, they don't have to be applied in a math way at all.

That being said. Look at these two propositions:

1>2

3>2

The first one is false, and the second one is true. But is this true or false?

x>2

Well, it can't be said because x is undefined. But how would one define it? Well, that's done via

Predicates

So. Let me define "x>2" as P(x). And I'll gloss over this right now but let's say the universe of discourse of x is integers. But now I have done that:

P(1) is false (because 1>2 is false)

P(2) is false (because 2>2 is false)

P(3) is true (because 3>2 is true)

P(4) is true (because 4>2 is true)

And so on.

I will say that the the vast majority of people will, when discussing a predicate, use x, y, and z as placeholder variables. I will do that in this very post. But everywhere else, I will use Greek letters. Because they are fun.

Quantifiers

There are 3 main quantifiers, though the last one is quite rare.

Universal quantifier, "∀"

Existential quantifier, "∃"

Unique existential quantifier, "∃!"

Using these, we can turn predicate statements back into nice easy propositions:

∀x(P(x)) "For all x, the statement x>2 is true." (false)

∃x(P(x)) "There exists at least one x to where x>2 is true." (true)

∃!x(P(x)) "There exists exactly one x such that x>2 is true." (false)

Again, the placeholder variables are usually x, y, and z. I will probably stick by that as well but I might use different letters if I want.

Now to discuss the universe of discourse, which I will represent with 𝔻 for convenience. Earlier, I said it was integers for x, but it can be any set. It could be all real numbers. It could be all natural numbers that are a power of 2. It could be all kittens in the world. As long as it is a set. It represents all possible variables that those quantifiers can check for.

But let's say the universe of discourse of, say, k, is simply {1, 2, 3}. And let's say that Q(k) is just some predicate. I really don't know or care what it is. These equivalences follow:

∀k(Q(k)) ≡ Q(1)∧Q(2)∧Q(3)

∃k(Q(k)) ≡ Q(1)∨Q(2)∨Q(3)

∃!k(Q(k)) is more complicated unfortunately...

Theoretically, this means that any of those equivalence laws from before can be applied. But since the universe of discourse is incredibly large or infinite in the majority of cases, these equivalences do not mean much. But the most important thing is probably De Morgan's laws:

¬∀x(P(x)) ≡ ∃x¬(P(x))

¬∃x(P(x)) ≡ ∀x¬(P(x))

Unfortunately ∃! has nothing like this.

Advanced Quantifiers

I don't know know how to define this generally, so I will do some symbolization and leave you to do some inductive reasoning. I hope I don't mess up symbolization here because that would be awkward.

So! I shall define a few things:

The universe of discourse of all variables is all types of living creatures

E(x,y) = "x eats y"

C(x) = "x eats animals"

H(x) = "x eats plants"

L(x) = "x photosynthesizes"

A(x) = "x is an animal"

P(x) = "x is a plant"

Now. Here are a couple of symbolization examples.

"All plants photosynthesize." ≡ ∀x(P(x)→L(x))

"There exists at least one animal that eats plants." ≡ ∃x(A(x)∧H(x))

"Everything eats plants, animals, or photosynthesize." ≡ ∀x(C(x)∨H(x)∨L(x))

"No animals photosynthesize." ≡ ¬∃x(A(x)∧L(x))

Now. Time for nested quantifiers:

"All animals eat something."≡ ∀x(A(x)→(∃y(E(x,y))))

"There exists at least one thing that eats everything." ≡ ∃x(∀y(E(x,y)))

"There exists at least one animal that eats all plants." ≡ ∃x(A(x)∧(∀y(P(y)→E(x,y))))

Again. Please tell me if I fucked up the symbolization here.

[bunnywand-z]

"the best laid plans" "i'm hatching a plan" so a plan is a type of egg ? 🥚

[logic-of-posts]

H(α) = "α can hatch."

L(α) = "α is laid."

E(α) = "α is an egg."

𝔻 of x = "all things"

∀x((H(x)∧L(x))→E(x))

Minimized equivalent:

∀x((H(x)∧L(x))→E(x))

[contemptible-scoundrel]

dude he's a pro and he's dating a noob, it's a problematic skill gap, it's not a healthy pwnage dynamic

[logic-of-posts]

𝔻 of x = "All people."

𝔻 of y = "All people."

P(α) = "α is a pro."

N(α) = "α is a noob."

D(α,β) = "α is dating β."

S(α,β) = "α and β have a problematic skill gap."

∀x(∀y((P(x)∧N(y)∧D(x,y))→S(x,y)))

Minimized equivalent:

∀x(∀y((P(x)∧N(y)∧D(x,y))→S(x,y)))

[hungwy]

The dream doesn't let him taste or smell very much so he was really excited when you showed up!

[logic-of-posts]

t = "Mr. 985 can taste much."

s = "Mr. 985 can smell much."

e = "Mr. 985 is excited that you showed up."

¬(t∨s)→e

Minimized equivalent:

s∨t∨e

[derorii]

But he can't taste or smell much, right?

[logic-of-posts]

Not out of the question that I fucked up symbolization but conditionals are horribly unintuitive outside of basic "if p then q" structure

[akashicrecord]

*freaking the fuck out because I have nowhere to lay my eggs*

[logic-of-posts]

p = "There is a place for Tumblr user akashicrecord to lay eggs."

f = "Tumblr user akashicrecord will freak out."

¬p→f

Minimized equivalent:

f∨p

[footlongdingledong]

we all got that one homie who was added during the limited time collaboration event

[logic-of-posts]

𝔻 of x = "All people."

𝔻 of y = "All people."

H(α,β) = "α is a homie of β."

C(α) = "α was added during the limited-time collaboration event."

∀x(∃!y(H(y,x)∧C(y)))

Minimized equivalent:

∀x(∃!y(H(y,x)∧C(y)))

[hungwy]

The dream doesn't let him taste or smell very much so he was really excited when you showed up!

[logic-of-posts]

t = "Mr. 985 can taste much."

s = "Mr. 985 can smell much."

e = "Mr. 985 is excited that you showed up."

¬(t∨s)→e

Minimized equivalent:

s∨t∨e

[crabussy]

was about to write a post like "eating a decadent easter egg in bed fuck yes" and thought nah twelve thousand strangers don't want to hear about that. and then I unwrapped the foil and it was unexpectedly white chocolate and I started making delighted chimp sounds and I feel like THAT is worth sharing with twelve thousand strangers <- dude.... no...?

[logic-of-posts]

d = "Eating a decadent easter egg in bed."

w = "This experience is worth sharing online."

u = "The easter egg is unexpectedly white chocolate."

c = "Making delighted chimp sounds."

¬(d→w)∧((u∧c)→w)

Minimized equivalent:

¬((d→(c∧u))∨w)

[penultimate-step]

Funny thing about those posts about people who make strange assumptions from seeing limited amount of media, the "guy who has only seen boss baby" and "when does he steal the bread," those kinds of posts, I did actually personally have a moment like that as a kid.

Some of the games that were most formative to me when I was just starting out in playing videogames as a little kid of around seven years old: Mario 64, Kirby Super Star, and Super Smash Bros. All of these games had bosses which take the form of hands coming to attack the player character. So from this I devised the (very reasonable) theory that it was intrinsic to gaming that every game had you fight some guy's disembodied hand. "When do I fight the hand?"

[logic-of-posts]

𝔻 of x = "All things."

V(α) = "α is a video game."

H(α) = "α contains a hand-themed boss."

¬(∀x(V(x)→H(x)))

Minimized equivalent:

∃x(¬(V(x)→H(x)))

[eroscestlavie]

Nothing has it for me. Not even Poob.

[logic-of-posts]

𝔻 of x = "All things."

H(α) = "α has it for Tumblr user eroscestlavie."

¬(∃x(H(x)))

Minimized equivalent:

∀x(¬H(x))

[markovchaingang]

new report finds that over 30% of uranium in nuclear power plants is lost via engineers having a little bite when they think nobody's watching

[logic-of-posts]

𝔻 of x = "All things."

U(α) = "α is uranium."

E(α) = "α is consumed by engineers when they think nobody is watching."

¬(∀x(U(x)→E(x)))∧(∃x(U(x)∧E(x)))

Minimized equivalent:

(∃x(¬(U(x)→E(x))))∧(∃x(U(x)∧E(x)))

[hymnsofheresy]

i just woke up from a dream where i was being interrogated by a bunch of people asking me if “furbies are kosher” firstly…. im not jewish. secondly……..what the fuck

[hymnsofheresy]

please stop sending me asks pertaining to the kosher status of furbies. i really do not know. this was just a manifestation of my subconscious. im assuming that they are not kosher because furbies aren’t even food. but who knows! ask a rabbi, if you must. 

[natashatroyka-deactivated201902]

Jew here! Furbies are actually worse than unkosher–they are not permissible as food, even for gentiles. This is because the Torah teaches that it is forbidden for any human to eat the meat of an animal that is still alive, and the Furby cannot die.

[hymnsofheresy]

hi this is the most ominous description of a furby i have ever heard

[logic-of-posts]

𝔻 of x = "All things."

𝔻 of y = "All pieces of meat."

A(α) = "α is alive."

F(α,β) = "α comes from β."

P(α) = "α is permissable to eat."

∀x(∀y((A(x)∧F(y,x))→¬P(y)))

Minimized equivalent:

∀x(∀y((A(x)∧F(y,x))→¬P(y)))

[pukicho]

I think all items should cost 8 dollars. That way evrrything costs 8 dollard

[logic-of-posts]

c = "All items costs 8$."

c→c

Minimized equivalent:

⊤

[stupid-twink-girl]

[logic-of-posts]

f = "You are fucked up in the head."

p = "Your father was present."

k = "The therapist knows what's wrong with you."

(f∧p)→¬k

Minimized equivalent:

(f∧p)→¬k

[dragonsapphic]

[dragonsapphic]

I didn’t nail a jar of peanut butter to my ceiling for 50 notes.

[logic-of-posts]

p = "Tumblr user dragonsapphic nailed peanut butter to her ceiling."

n = "Tumblr user dragonsapphic was expecting to get only 50 notes."

¬(p→n)

Minimized equivalent:

¬(p→n)

[archangel-fucking-judas]

I love when my cat meows at me and I can see her little teeths cause she looks like this

[archangel-fucking-judas]

My cat is being oh so meanies to me but I can't take her seriously because everytime she opens her mouth to yell at me she becomes a stapler remover

[logic-of-posts]

m = "Tumblr user archangel-fucking-judas' cat meows."

r = "Tumblr user archangel-fucking-judas' cat looks like a staple remover."

s = "Tumblr user archangel-fucking-judas can take their cat seriously."

(m→r)→¬s

Minimized equivalent:

(m→r)→¬s

[logic-of-posts]

wait so what connectives do you use?

Here are the connectives I use:

and

or

not

conditional

biconditional

xor

I don't have nand or nor because I think they're kind of stupid, tbh. a↓b≡¬(a∨b) and a↑b≡¬(a∧b). Admittedly a⊕b≡¬(a↔b) though but I still have it because I think it is a useful thing to take special note of I guess.