Also known as 'primitive notions' and 'basic concepts'; that doubting requires these concepts would be important later.

1.1 Objects, things

• Unanalyzable

• Motivations:

  • 'thing', 'object', 'individual object'

1.2 Set

• 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...}'

1.2.1 Set membership

• An object in a non-empty set is its element/member.

• Symbols: 'x is in A' ≝ 'x ∈ A'

1.2.2 Empty set

• A set with no member

• Symbols: '∅'

1.2.3 Subset and superset

• 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'

1.3 State of affairs (SoA)

1.3.1 Maximal state of affairs (MSoA)

• 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'

1.3.2 Local state of affairs (LSoA)

• Def: any subset of a MSoA

• AKA: a situation, a circumstance, a proposition

1.4 Negation

• 'not'

• Symbols: ¬

• irreducible

• Motivation: natural languages

1.5 Actual world, obtainment, and truth

1.5.1 Actuality and obtainment

• 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

1.5.2 Possible world

• Def: a MSoA not containing both LSoAs A and B when B stands for A not obtaining

1.5.3 Truth

• 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

1.5.4 Falsity

• 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'

1.6 Properties

• 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

1.6.1 Truth values

• Properties of LSoA

• 'True' and 'false' [corresponding to truth (1.5.3) and falsity (1.5.4)]

1.6.2 Sortals

• Properties of objects

• Allowing objects to be counted

• Examples: being a particle, being a wave

• Contrasted with: characterizing properties (redness, stillness, momentum, charge, etc.)

1.6.3 Relations

• 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

1.6.4 Quantities

• 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.

1.7 Universal quantification

• All, every

• Irreducible

• Motivation: natural languages

1.8 Existence

• An instantiation of property(s) in a world

• Irreducible

1.9 Implication

• 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 (→)

1.10 Tuple

• 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,...,a_n⟩ is identical to ⟨b₁, b₂,...,b_n⟩ only when a₁ is identical to b₁, a₂ is identical to b₂,..., and a_n is identical to b_n

Implications

• 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.