Comma or no comma?

What can be an unfathomable point of the English syntax – use of commas with relative clauses – is actually quite straightforward if you apply a bit of logic. Relative clauses are something of a speciality of predicate logic, in fact. The problem touches on the question of anaphora – more about which another time.

There is a correlation between the scope of the quantifier and the use of commas, which looks like this:

narrow scope of quantifier – comma required
wide scope of quantifier – no comma

Some examples:

(1) Merchants flatter politicians, who are corrupt.

(2) Merchants flatter any politician who is corrupt.

(3) A merchant will flatter any politician who can help him.

(1.1) (x)[Mx ⊃ (∃y)(Py • Fxy)] • (y)(Py ⊃ Cy)

(2.1) (x){Mx ⊃ (y)[(Py • Cy) ⊃ Fxy)]}

(3.1) (x){Mx ⊃ (y)[(Py • Hyx) ⊃ Fxy)]}


Sentence (1.1) – narrow scope of quantifier over ‘merchant’ – says that each merchant flatters some politician and that all politicians are corrupt. The main connective is a conjunction, which means that if there is at least one politician who is not corrupt, the sentence is false.

Sentence (2.1) – wide scope of quantifier over ‘merchant’ – says that each merchant flatters corrupt politicians. The main connective is a conditional, which means that the fact that there may be some politicians who are not corrupt and whom merchants flatter does not falsify the sentence.

Sentence (3.1) – wide scope of quantifier over ‘merchant’ – leaves us no option really, because if the pronoun ‘him’ is to be a bound variable, the scope of quantification over ‘a merchant’ must extend to ‘him’.

The same applies to commas around relative clauses modifying the subject of the sentence:

(4) Merchants, who buy off politicians, are corrupt.

(5) Merchants who buy off politicians are corrupt.

(4.1) (x)(Mx ⊃ Cx) • (x)[Mx ⊃ (∃y)(Py • Bxy)]

(5.1) (x){[Mx • (∃y)(Py • Bxy)] ⊃ Cx}