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.

















