Metatheory - University of Cambridge

2. Substitution 17
2.3 Duality
The final result of this chapter will use substitution to explore a very powerful
idea. To explain the idea, recall the following equivalences, known as the De
Morgan Laws:
(A ∧ B)
(A ∨ B)
⊨
⊨
⊨ ¬(¬A ∨ ¬B)
⊨ ¬(¬A ∧ ¬B)
These indicate that there is a tight ‘symmetry’ between conjunction and disjunction.
The notion of duality is a generalisation of this tight ‘symmetry’.
And in fact, it is worth introducing it with complete generality at the outset.
In TFL, our only primitive connectives are one-place (i.e. ‘¬’) and twoplace
(i.e. ‘∧’, ‘∨’, ‘→’ and ‘↔’). But nothing stops us from introducing three-,
four-, or five-place connectives; or, more generally, n-place connectives, for any
number n we like. (We shall, however, always assume that our connectives are
truth-functional.)
So, let ◁ be an n-place connective. The dual of ◁, which I shall generically
denote by underlining it, i.e. ◁, is defined to have a characteristic truth table
as follows:
◁(A 1 , . . . , A n )
⊨
⊨ ¬◁(¬A 1 , . . . , ¬A n )
This is the obvious generalisation of the De Morgan Laws to many-place connectives.
2 And it is immediate, on this definition, that ‘∧’ is the dual of ‘∨’,
and that ‘∨’ is the dual of ‘∧’. I encourage you also to check that ‘↑’ is the dual
of ‘↓’, and vice versa; and that ‘¬’ is its own dual.
Inspired by these examples, we might correctly conjecture the following:
Lemma 2.5. The dual of a dual is always the original connective; that is, ◁
is the same connective as ◁.
Proof. I start by noting:
◁(A 1 , . . . , A n )
◁(¬A 1 , . . . , ¬A n )
¬◁(¬A 1 , . . . , ¬A n )
¬◁(¬A 1 , . . . , ¬A n )
Next, observe that, by definition:
⊨
⊨
⊨
⊨
⊨ ¬◁(¬A 1 , . . . , ¬A n )
by definition
⊨ ¬◁(¬¬A 1 , . . . , ¬¬A n ) by Lemma 2.1
⊨ ¬¬◁(¬¬A 1 , . . . , ¬¬A n ) by the definition of ‘¬’
⊨ ◁(A 1 , . . . , A n ) by Lemma 2.2
◁(A 1 , . . . , A n )
⊨
⊨ ¬◁(¬A 1 , . . . , ¬A n )
Hence, by the transitivity of entailment:
◁(A 1 , . . . , A n )
⊨
⊨ ◁(A 1 , . . . , A n )
as required.
■
2 Hence some books talk of the ‘De Morgan dual’, rather than simply of the ‘dual’.