What is a logic, anyway?

A famous example from Frege

First a BNF to define our WFFs:
- birds ::= 'doves' | 'crows' | . . .
- stuff ::= 'tar' | 'snow' | . . .
- color ::= 'black' | 'white' | . . .
- spred ::= 'is' | 'is not'
- ppred ::= 'are' | 'are not'
- lcon ::= 'and' | 'or'
- ssent ::= stuff ' ' spred ' ' color
- psent ::= birds ' ' ppred ' ' color
- sent ::= ssent | psent
- wff ::= sent | '(' wff ') ' lcon ' (' wff ')'
Some examples of WFFs are then:
- crows are black
- snow is white
- (doves are black) or (doves are not black)

Our models are very simple
- a set of individuals (the domain)
- a set of named disjoint subsets of that set (the colourings)
For example, one possible model would be

domain

{a, b, c, d, e, f, g, h}

colourings

{black:{a, b, c},
white:{d, e, f, g}}

Finally the constructive definition of validity.
- A mapping from WFFs to conditions on a model
- We use double square-brackets 〚〛to denote this mapping
We treat our birds and stuff as variables
- they have an arbitrary association with individuals in a model
- We denote the variable assignment function with the Greek letter iota (ι)
The mapping rules then are as follows, using X and Y is meta-variables over parts of WFFs:
- 〚(X) or (Y)〛 == (〚X〛) ∨ (〚Y〛)
- 〚(X) and (Y)〛 == (〚X〛) ∧ (〚Y〛)
- 〚X is Y〛 == ι(X) ∈ 〚Y〛
- 〚X are Y〛 == ι(X) ∈ 〚Y〛
- 〚X is not Y〛 == ι(X) ∉ 〚Y〛
- 〚X are not Y〛 == ι(X) ∉ 〚Y〛
- 〚white〛 == the set named 'white' in the model
- 〚black〛 == the set named 'black' in the model

Given the specific model above
and a variable assignment which maps 'doves' to 'f'
the interpretation of the last example WFF above, namely
(doves are black) or (doves are not black)
is
- ( f ∈ {a, b, c} ) ∨ ( f ∉ {a, b, c} )
which is evidently true
- it should be clear that indeed it's a tautology
All of which may serve to explain why Frege was not being childish when he, famously, said
- 〚Snow is white〛 is true just in case 〚snow〛 is 〚white〛
  [punctuation added :-]