"First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.
First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables.
Rather than propositions such as "all men are mortal", in first-order logic one can have expressions in the form
"for all x, if x is a man, then x is mortal"
; where "for all x" is a quantifier, x is a variable, and "... is a man" and "... is mortal" are predicates.
This distinguishes it from propositional logic, which does not use quantifiers or relations;:
in this sense, propositional logic is the foundation of first-order logic."
"First-order logic is the standard for the formalization of mathematics into axioms"
"The first mention of the number line used for operation purposes is found in John Wallis's Treatise of Algebra (1685)
In his treatise, Wallis describes addition and subtraction on a number line in terms of moving forward and backward, under the metaphor of a person walking."
"The natural numbers are used for counting things, like "there are six coins on the table", in which case they are called cardinal numbers.
They are also used to put things in order, like "this is the third largest city in the country", which are called ordinal numbers.
Natural numbers are also used as labels, like jersey numbers on a sports team, where they serve as nominal numbers and do not have mathematical properties. "
'While propositional logic deals with simple declarative propositions, first-order logic additionally covers predicates and quantification.
A predicate evaluates to true or false for an entity or entities in the domain of discourse."
"There are two key parts of first-order logic.
The syntax determines which finite sequences of symbols are well-formed expressions in first-order logic, while
the semantics determines the meanings behind these expressions."
'Unlike natural languages, such as English, the language of first-order logic is completely formal, so that it can be mechanically determined whether a given expression is well formed.
There are two key types of well-formed expressions:
terms, which intuitively represent objects,
and formulas, which intuitively express statements that can be true or false.
The terms and formulas of first-order logic are strings of symbols,
where all the symbols together form the alphabet of the language."
https://en.wikipedia.org/wiki/First-order_logic#:~:text=First%2Dorder%20logic%2C%20also%20called,.%20is%20mortal%22%20are%20predicates.
As a demonstration of the principle, consider two contradictory statements—"All lemons are yellow" and "Not all lemons are yellow"—and suppose that both are true. If that is the case, anything can be proven, e.g., the assertion that "unicorns exist", by using the following argument:
We know that "Not all lemons are yellow", as it has been assumed to be true.
We know that "All lemons are yellow", as it has been assumed to be true.
Therefore, the two-part statement "All lemons are yellow or unicorns exist" must also be true,
since the first part of the statement ("All lemons are yellow") has already been assumed,
and the use of "or" means that if even one part of the statement is true,.
the statement as a whole must be true as well.
However, since we also know that "Not all lemons are yellow" (as this has been assumed), the first part is false, and
hence the second part must be true to ensure the two-part statement to be true, i.e., unicorns exist
(this inference is known as the
Disjunctive syllogism).
The procedure may be repeated to prove that unicorns do not exist
(hence proving an additional contradiction where unicorns do and do not exist), as well as any other well-formed formula.
Thus, there is an explosion of true statements.
https://en.wikipedia.org/wiki/Principle_of_explosion
"In classical logic, intuitionistic logic, and similar logical systems, the principle of explosion[a][b] is the law according to which any statement can be proven from a contradiction.
That is, from a contradiction, any proposition (including its negation) can be inferred; this is known as deductive explosion."
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"An example in English:
I will choose soup or I will choose salad.
I will not choose soup.
Therefore, I will choose salad."
P pop
https://en.wikipedia.org/wiki/Disjunctive_syllogism
"Disjunctive syllogism is closely related and similar to hypothetical syllogism, which is another rule of inference involving a syllogism. It is also related to the law of noncontradiction, one of the three traditional laws of thought."
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