(D2) It does not contain unit clauses, i. e. clauses with a single literal.

Otherwise, one can always fix it by replacing such literals and their negations by the corresponding truth values, and deleting them from Π .

(D3) It explicitly contains each of the *tertium non datur* clauses p ∨ p, with p ∈ Π+ . This special convention provides the trivial implications p ← p for any p ∈ Π .
2.2 We call valuation any mapping w whereby each literal p in Π is assigned a value wp in the interval of real numbers from 0 to 1. The number wp , sometimes denoted alternatively as w(p), will be seen as a degree of belief in the proposition p.

Generally speaking, the values of wp and wp need not add up to 1, but their sum can take any value from 0 to 2. Having wp + wp < 1 means a lack of information, whereas wp +wp > 1 means that some contradiction is present. A valuation that satisfies wp + wp = 1 for any p ∈ Π will be called balanced.

Later on, we will sometimes compare two valuations v and w. In that connection, v ≤ w will mean simply that vp ≤ wp for any p ∈ Π .

The truth assignments of classical propositional logic can be seen as balanced valuations that take only the values 0 and 1, where 1 means 'true' and 0 means 'false'. Since they are formally the same, in the sequel we will often identify truth assignments with balanced all-or-none beliefs.

We will also be interested in balanced valuations with values in {0, 1 2
, 1},
where the value 12 can be interpreted as 'undefined' or 'undecided'. We will refer to such valuations as partial truth assignments, a term that we are taking from satisfiability theory. They will be used mainly for specifying decisions, in which case the values 1, 0 and 12 can be interpreted as meaning respectively 'accepted', 'rejected' and 'undecided'.

A partial truth assignment u will be said to be definitely consistent with D, or with Φ(D), when, for each clause C ∈ D and every p ∈ C ,
the following implication holds: if uα = 0 for every α ∈ C\{p}, then up = 1.

When no undecidedness is present, a partial truth assignment being definitely consistent is equivalent to its being consistent as a truth assignment. When undecidedness is allowed, definite consistency requires every clause to contain at least one accepted literal, or alternatively, at least two undecided literals.