Basics of Propositions and Logic

Background and Goal of this Article

Goal: Understand the basics of propositions and logic

Why I wrote this article

When you start learning university mathematics, notation using logical symbols becomes frequently used.

At first, I tried to understand it somehow by the literal meaning of the Japanese, but
when logical formulas became complex, I felt a limit because I didn't know how to interpret them to understand them strictly, so I want to firmly understand propositions and logic here.

First, starting with the general definitions and procedures for propositions,
the purpose of this article is to be able to read logical formulas.

Personally, the generally handled procedures for propositions didn't click with the world handled in algebra, so
I will redefine propositions and logical symbols in the form of operations by mappings, but since the volume of text became too large, I will leave it for the next article.

My misunderstandings before learning

The following is a famous axiom of union, but even though I knew I should read it while being careful about dependencies from the left, I tried to interpret it in Japanese in a strange way, and it was meaningless to me. (Many people write it in easy-to-understand, rough Japanese, but conversely, I didn't understand why it could be interpreted that way... It might just be me.)

$$
　\forall X\exists Y\forall v( \exists w( v\in w\land w\in X) \leftrightarrow v\in Y)
$$

For example, the evaluation by $${\leftrightarrow}$$ in the parentheses,
for it to be true for any $${v}$$, wouldn't it be false unless $${Y}$$ contains any $${v}$$? Is that always true as an axiom? I had such misunderstandings.

General procedures for propositions

Propositions and logical symbols

A proposition is generally defined as "a statement for which either $${true: 1}$$ or $${false: 0}$$ is determined."
And as for logical symbols, there are $${\neg,\land,\lor,\rightarrow,\forall,\exist}$$, etc. As I will state in the next article, it seems that $${=}$$ is also often treated as one of the logical symbols (of course, its type is different from other logical symbols, but it means a symbol handled in a logical formula).

Of course,
　negation$${\neg}$$...$${\neg P}$$ (not P)
　logical conjunction$${\land}$$...$${P\land Q}$$ (P and Q)
　logical disjunction$${\lor}$$...$${P\lor Q}$$ (P or Q)
　implication$${\rightarrow}$$...$${P\rightarrow Q}$$ (if P then Q)
　universal quantifier$${\forall}$$...$${\forall x(P(x))}$$ (P for any x)
　existential quantifier$${\exist}$$...$${\exist x(P(x))}$$ (there exists some appropriate x such that P)
are the ones.

For the top four, the propositions by the bottom two quantifiers treat the proposition $${P}$$ regarding the variable $${x}$$ as a function like $${P(x)}$$.
Propositional logic covers the top four, but a logical system that can include variables like the bottom two is called predicate logic and is distinguished. (Since things other than numbers also become variables, $${x}$$ is also called a term.)

Since predicate logic is an extension of propositional logic, basically predicate logic is assumed, but in this article, I often write it like propositional logic by treating the propositional function $${P(x)}$$ as $${P}$$.
I would also like to touch on how predicate logic and propositional logic are essentially different in the next article.

Note that in the above, for each operator, the result of acting on propositions $${P, Q}$$ is written to the right of the ellipsis.
The result of this action is also a proposition, and the top four are called "compound propositions", and the bottom two are called "universal propositions" and "existential propositions" respectively.
Without fear of being misunderstood, I think they are all the same in that they are created by combining one or more propositions and logical symbols. (Universal propositions and existential propositions should also be reproducible by connecting them with logical conjunctions and logical disjunctions if you consider the case where the number of elements in the domain of $${x}$$ is finite, as they are ordinary propositions if you pick out one $${x_1}$$ from the domain of $${x}$$.)

Conversely, when you decompose a compound proposition into its constituent element propositions, "atomic propositions", the smallest unit of propositions, are found.
When an atomic proposition is written in the form of a function and has a variable $${x}$$, it seems to be distinguished as an "atomic formula", and those that connect atomic formulas with logical symbols are called "compound formulas". Atomic propositions can also be said to be constants where atomic formulas have no variables, and their compound formulas can be said to be compound propositions.
Therefore, from now on, I will basically use the terms atomic formula and compound formula, including atomic propositions.

Logical operations by implication

The logical symbols listed above can further evaluate truth or falsehood from a set of truth/falsehood results of propositions.
I think the result that doesn't match everyday intuition is "implication", so I will confirm only that below.
A table that shows the results of logical operations when propositions are treated as variables and only their truth or falsehood is focused on, as shown below, is called a truth table.
Note that in propositional logic, the determination of the truth or falsehood of propositions $${P}$$ and $${Q}$$ is independent.

$$
\begin{array}{|c|c|c|c|c|c|} \hline
P & \neg P & Q & P\rightarrow Q & P\land(P\rightarrow Q) & (P\land(P\rightarrow Q))\rightarrow Q \\\hline
 1 & 0 & 1 & 1 & 1 & 1\\\
1 & 0 & 0 & 0 & 0 & 1\\\
0 & 1 & 1 & 1 & 0 & 1\\\
0 & 1 & 0 & 1 & 0 & 1\\\hline
\end{array}
$$

Looking at the truth table, $${P\rightarrow Q}$$ becomes true when $${P}$$ is false or $${Q}$$ is true.
It doesn't intuitively fit that $${P\rightarrow Q}$$ becomes true if $${P}$$ is false, but in everyday life, when saying if $${P}$$ then $${Q}$$, it is often assumed that $${P}$$ is true.
Therefore, as seen in $${P\land(P\rightarrow Q)}$$, when we consider $${P}$$ to be true, if $${P}$$ then $${Q}$$ is true, then $${Q}$$ should also be true, which might fit well with the everyday meaning.

And even if we claim that if $${P}$$ then $${Q}$$ is true in a world where $${P}$$ is false ($${\neg P}$$ is true), it does not mention the truth or falsehood of Q. (Since $${P}$$ and $${Q}$$ are independent.)

Trying to resolve the misunderstanding before learning

My misunderstanding about the axiom of union can be resolved by the discussion so far.

$$
　\forall X\exists Y\forall v( \exists w( v\in w\land w\in X) \leftrightarrow v\in Y)
$$

First, since it is an axiom, it is a premise that this proposition is true.
I thought, "Wouldn't it be false unless $${Y}$$ contains any $${v}$$ for it to be true for any $${v}$$," but actually, it is true even if $${Y}$$ does not contain a certain $${v}$$.

For example, when considering $${X=\{\{1,2\},\{2,3\}\}}$$, the premise is that a set $${Y=\{1,2,3\}}$$ exists as stated by this proposition, but
at this time, first, considering $${v=1}$$, $${1\in \{1,2\}\in X}$$, and the chosen $${Y}$$ also becomes $${1\in Y}$$.
Then, both $${\exists w( v\in w\land w\in X) \rightarrow v\in Y}$$ and $${\exists w( v\in w\land w\in X) \leftarrow v\in Y}$$ are evaluated as true.

Next, considering $${v=4}$$, $${\exists w( v\in w\land w\in X)}$$ is false, but in the evaluation of implication, $${\exists w( v\in w\land w\in X) \rightarrow v\in Y}$$ becomes true.
And for the chosen $${Y}$$, $${v\in Y}$$ is false, but in the evaluation of implication, $${\exists w( v\in w\land w\in X) \leftarrow v\in Y}$$ becomes true.
In other words, it is an axiom that stands on the premise that there exists a $${Y}$$ for which the logical evaluation is always true in this way.

By the way, if you choose $${Y=\{1,2,3,4\}}$$, then when $${v=4}$$, $${v\in Y}$$ is true and $${\exists w( v\in w\land w\in X)}$$ is false, so $${\exists w( v\in w\land w\in X) \leftarrow v\in Y}$$ becomes false.

Further Questions

While I have become able to read logical expressions to some extent with my current understanding, there are some points that personally bother me.
It is said that these things called propositions are 'assertions for which either true: 1 or false: 0 is determined,' but is the syntax 'A is B' itself called a proposition, or is it called a proposition including the function to evaluate its truth value?

In the case of atomic propositions (atomic formulas), it seems to be dealing with syntax, and in the case of compound propositions (compound formulas), it feels like it could be interpreted as either the combination via logical symbols acting on the syntax itself or acting on the evaluation function.

My goal is to understand propositions and logic through mappings to deepen my understanding of this area, but I will leave that for the next article.