Logic, arguments, and writing

Basic concepts
A proposition is a sentence that is either true or false but is not both true and false. To each proposition, we assign a truth value, true or false, depending on whether the proposition is true or false. Note that not every sentence is a proposition. One example is "this statement is false".

Simple sentences have the form: subject-predicate, where the predicate describes the subject of the sentence. There are five standard ways to create new propositions from existing simple propositions: conjunction, disjunction, negation, conditional, biconditional, through the use of "and", "or", "not", "if-then", "if and only if" respectively. We call these items logical connectives.

Inclusive & Exclusive or
As a demonstration, we give the truth table for P or Q.

$\small \begin{array}{c|c|c} P&Q&P \vee Q \\ \hline T&T&T\\T&F&T\\F&T&T\\F&F&F\end{array}$

Note that in mathematics, "or" is used inclusively, contrary to the case in everyday discourse where "or" is used exclusively. When we say coffee or tea in real life, we mean exactly one of them but not both. In logic, when we say "A or B", it can exactly one of A and B or both.

Implication
P => Q - If P then Q
P: hypothesis of the implication / antecedent
Q: conclusion of the implication / consequent

For the proposition P => Q,
Converse: Q => P
Contrapositive: ~Q => ~P
Inverse: ~P => ~Q

Negation
~P - not-P / It is not the case that P.

Logical equivalence
Suppose A and B are propositions formed from a collection of propositions P, Q, R, S, ... using logical connectives.
The propositions A and B are logically equivalent if A and B have the same truth values for all possible truth values of the propositions P, Q, R, S, ...
≅ - logically equivalent

Let A be a proposition formed from propositions P, Q, R, ... using the logical connectives.
A is called a tautology if A is true for every assignment of truth values to P, Q, R, ...
A is called a contradiction if A is false for every assignment of truth values to P, Q, R, ...

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Relation to arguments and writing

Tautology
When a word repeats the meaning of another word in the same phrase it is called tautology.

Some common superfluities:
absolute certainty, actual facts (and its cousin, true facts), a downward plunge, advance warning, arid desert, attach together, circle round, burn down, connect/collaborate/couple/gather/join/link/meet/merge/unite together, each and every one, early beginnings, eat up, final completion, final upshot, forward planning, free gift, future prospects, grateful thanks, general consensus, have got (simply have is fine), hurry up, inside of, important essentials, in between, lend out, lonely isolation, more preferable, mutual cooperation, new beginner, new creation, mew innovation, original source, other alternative, outside of, over with, proceed onward, really excellent, reduce down, renew again, seldom ever, set a new world record, still continue, this day and age, totally finished, tiny little child, unexpected surprise, usual habit, whether or not, widow woman...

We should avoid these tautologies in our writing.

Double Negatives
~(~P) ≅ P
Example:
The bomb attack was not unexpected.
The bomb attack was expected.
Although they can be useful, they are also often confusing. If you want to get your message across, it's better to use the latter version.

Fallacy of affirming the consequent
Since $[(P \Rightarrow Q) \wedge Q] \Rightarrow P$ is not always true, $P$ is not a valid consequence of $P \Rightarrow Q$ and $Q$.
Example:
"If I went to the grocery store, then I bought milk."
"I bought milk. Therefore, I went to the grocery store." is NOT VALID.
I could have bought the milk at a gas station, from a street vendor, or from an illicit milk dealer.
One might be tempted to say that this reasoning is valid, because we might associate buying milk with going to a grocery store. The form of the argument, however, does not lead to that conclusion.
Moral of the story: The converse of a statement may not be true.

Fallacy of denying the antecedent
$\neg Q$ is not a valid consequence of $P \Rightarrow Q$ and $\neg P$. When $P$ is false and $Q$ is true, $[(P \Rightarrow Q) \wedge \neg P] \Rightarrow \neg Q$ is false and thus is not a tautology.
Example:
"If I went to the grocery store, then I bought milk."
"I didn't go to the grocery store. Therefore, I didn't buy milk." is WRONG.
I could have bought the milk elsewhere.

Question
It's only a matter of time for people to accept...
Such a statement is an unfalsifiable claim or tautology [?]