Also known as 'primitive notions' and 'basic concepts'; that doubting requires these concepts would be important later.
Unanalyzable
Motivations:
'thing', 'object', 'individual object'
a set of distinct object(s), an empty set
a set is an object with at least one or no object within it, otherwise unanalyzable, irreducible
Symbols: if 'A' denotes a set, 'A = {object1, object2, object3...}'
An object in a non-empty set is its element/member.
Symbols: 'x is in A' ≝ 'x ∈ A'
A set with no member
Symbols: '∅'
A is a subset of B and B is a superset of A only when every element of A is also in B
Symbols:
Def. 1: 'A is a subset of B' ≝ 'A ⊆ B'
Def. 2: 'A is a superset of B' ≝ 'A ⊇ B'
Def: The way things are; the set of everything except itself
Also known as (AKA): a world [the universe, and every object outside it (if there is any)], including the writer of this text]
Motivations:
Possible worlds semantics of alethic modal logic
En: 'this is the way it is,' 'as things are now,' 'why is it like this?,'
Fr: 'en l'état actuel des choses,' 'dan l'ordre des choses,' 'l'état de la situation,' 'comment vont les choses'
De: 'die Dinge'
Ar: 'لماذا هو هكذا؟'
Th: 'ทำไมมันเป็นอย่างนี้'
Def: any subset of a MSoA
AKA: a situation, a circumstance, a proposition
'not'
Symbols: ¬
irreducible
Motivation: natural languages
The actual world ≝ the MSoA (1.3.1) that obtains
AKA: the real world, reality
Obtainment is unanalyzable.
Motivations: Abstractionist possible worlds semantics of alethic modal logic and those same motivations for 1.3.1
Def: a MSoA not containing both LSoAs A and B when B stands for A not obtaining
With P standing in for any LSoA:
Def. 3: 'P is true' ≝ 'P is a subset of the actual world (1.4.1)'
Def. 4: 'P is possibly true' ≝ 'P is a subset of at least one possible world'
Def. 5: 'P is necessarily true' ≝ 'P is a subset of every possible world'
Def. 6: Fact [or truth (2nd sense)] ≝ an LSoA that obtains
Def. 7: 'P is false' ≝ 'P is not a subset of the actual world (1.4.1)'
Def. 8: 'P is possibly false' ≝ 'it is not the case that P is a subset of every possible world'
Def. 9: 'P is necessarily false' ≝ 'P is not a subset of any possible world'
AKA: predicates, attributes, qualities, features, characteristics
An object in a world instantiates/exemplifies a property
Irreducible
Motivations:
Cross-cultural categorization based on resemblances and translatability between those words from different languages
Truth values (true and false)
Recognition of new instances (discovery)
Usage in mathematics across cultures
Usage in science across cultures
Properties of LSoA
'True' and 'false' [corresponding to truth (1.5.3) and falsity (1.5.4)]
Properties of objects
Allowing objects to be counted
Examples: being a particle, being a wave
Contrasted with: characterizing properties (redness, stillness, momentum, charge, etc.)
n-ary properties, where n > 1
AKA: polyadic properties
The two or more objects involved in the instantiation are the relation's arguments.
Motivations:
transitive verbs ('x loves y'), valency of verbs
prepositions ('between x and y')
relational nouns ('x is the mother of y')
functions (in mathematics)
arity
AKA: numbers
Example: natural numbers, fractions, etc.
Motivations: mathematics across cultures, its usage in daily lives, its usage in science, the use of science in technology.
All, every
Irreducible
Motivation: natural languages
An instantiation of property(s) in a world
Irreducible
AKA: logical implication
Motivation: Invoked in situations such as when two distinct SoAs obtaining implies that at least one SoA obtains
Irreducible
Used as a basis for logical implication (⊨) and the material conditional (→)
Def: a finite ordered list of objects
n-tuple ≝ an ordered list of n objects, where n ≥ 1
Symbols: where A denotes a tuple, 'A = ⟨object1, object2,..., objectn⟩
⟨a1, a2,...,an⟩ is identical to ⟨b1, b2,...,bn⟩ only when a1 is identical to b1, a2 is identical to b2,..., and an is identical to bn
Propositions are now local states of affairs (LSoA), and different sentences from (or not from) within the same language can refer to the same LSoA.
Combined correspondence and deflationary theory of truth:
Correspondence: a proposition (LSoA) being true means that it is identical to a LSoA that obtains / that is a subset of the actual world.
Deflationary: A proposition is now simply a state of affairs and the object doing the referring is a sentence in a language.
Relativism about truth is false by this definition.
Coherence theory of truth is false by this definition.
Social constructionism about reality is false by this definition.
The semantics of intuitionistic logic doesn't define 'truth' the same way as it is defined here.