1 Sets, Relations and Arguments

A set is a binary relation iff

it contains only ordered pairs.

A binary relation R is reflexive on a set S iff

for all elements d of S the pair <d, d> is an element of R

A binary relation R is symmetric on a set S iff

for all elements d, e of S: if <d, e> ∈ R then <e, d> ∈ R

A binary relation R is asymmetric on a set S iff

for no elements d, e of S: <d, e> ∈ R and <e, d> ∈ R

A binary relation R is antisymmetric on a set S iff

for no two distinct elements d, e of S: <d, e> ∈ R and <e, d> ∈ R

A binary relation R is transitive on a set S iff

for all elements d, e, f of S: if <d, e> ∈ R and <e, f> ∈ R, then <d, f> ∈ R

A binary relation R is an equivalence relation on S iff

R is reflexive on S, symmetric on S and transitive on S

A binary relation R is symmetric/asymmetric/antisymmetric/transitive iff

it is symmetric/asymmetric/antisymmetric/transitive on all sets

A binary relation R is a function iff

for all d, e, f: if <d, e> ∈ R and <d, f> ∈ R then e = f

The domain of a function R is

the set {d : there is an e such that <d, e> ∈ R}

The range of a function R is

the set {e : there is a d such that <d, e> ∈ R}

R is a function into the set M iff

all elements of the range of the function are in M

If d is the domain of a function R one writes R(d) for

the unique object e such that <d, e> is in R.

An n-place relation is

a set containing only n-tuples

An n-place relation is called

a relation of arity n

An argument consists of

a set of declarative sentences (the premises) and a declarative sentence (the conclusion) marked as the concluded sentence

An argument is logically valid iff

there is no interpretation under which the premises are all true and the conclusion false

A set of sentences is logically consistent iff

there is at least one interpretation under which all sentences of the set are true

A sentence is logically true iff

it is true under any interpretation

A sentence is a contradiction iff

it is false under any interpretation

Sentences are logically equivalent iff

they are true under exactly the same interpretations