A.1. Definitions and notation for basic concepts

Most deductive properties or relation concerns a set or some specific number of assumptions and a set or some specific number of alternatives. When there is only one alternative, it is a conclusion. This is shown in the following table, where cells are labeled in boldface by the concept expressed as a noun, with the verbal or adjectival form shown in italics.

alternatives
set
Σ		 two
ψ1, ψ2		 one
(concl.) ψ		 none
set
Γ		 relative
exhaustiveness
Γ renders Σ
exhaustive		 entailment

Γ entails ψ		 inconsistency

Γ is
inconsistent
two
φ1, φ2		 mutual
exclusiveness
φ1 and φ2 are
mutually
exclusive
one
φ		 implication
φ implies ψ		 absurdity
φ is absurd
none exhaustiveness

Σ is exhaustive		 (joint)
exhaustiveness
ψ1 and ψ2 are
(jointly)
exhaustive		 tautologousness
ψ is
tautologous
(or is a
tautology)

Not appearing in the table are two relations that each abbreviate conjunctions of two claims drawn from the ones above.

conjunctive relation component relations
(logical) equivalence
φ and ψ are
(logically) equivalent		φ implies ψψ implies φ
contradictoriness
φ and ψ are
contradictory		φ and ψ are
mutually
exclusive		φ and ψ are
jointly
exhaustive

There are also two alternative ways of applying the concept of inconsistency:

alternative statements (for assumptions Γ and φ)
exclusion
Γ excludes φ		 relative inconsistency
φ is inconsistent with Γ		 inconsistency of the union
Γ with φ added is inconsistent

Note that in this case all sentences involved count as assumptions.

All concepts appearing in the first table can be defined in the same way, as saying that their assumptions cannot be separated from their alternatives. This idea can be stated more specifically in two ways:

Negative definition: there is no possible world in which the assumptions (if any) are all true while the alternatives (if any) are all false.

Positive definition: in each possible world in which the assumptions (if any) are all true, at least one alternative is true.

When there are no assumptions or no alternatives, the corresponding clause may be dropped from the negative form. The same is true for the clause regarding assumptions in the positive form; and, if there are no alternatives, that definition can be restated as: in each possible world, the assumptions are not all true (i.e., at least one is false).

The following table gives an explicit definition for each of these concepts and also indicates compact notation for the concept.

concept negative definition positive definition
φ is a tautology
⊨ φ 		There is no possible world in which φ is false. φ is true in every possible world.
φ is absurd
φ ⊨		 There is no possible world in which φ is true. φ is false in every possible world.
φ implies ψ
φ ⊨ ψ 		There is no possible world in which φ is true and ψ is false. ψ is true in every possible world in which φ is true.
φ and ψ are
mutually exclusive
φ ▵ ψ 		There is no possible world in which φ and ψ are both true. In each possible world, at least one of φ and ψ is false.
φ and ψ are
(jointly) exhaustive
φ ▿ ψ 		There is no possible world in which φ and ψ are both false. In each possible world, at least one of φ and ψ is true.
φ and ψ are
(logically) equivalent
φ ≃ ψ 		There is no possible world in which φ and ψ have different truth values. In each possible world, φ and ψ have the same truth value as each other.
φ and ψ are
contradictory
φ ⋈ ψ 		There is no possible world in which φ and ψ have the same truth value. In each possible world, φ and ψ have opposite truth values.
Γ is inconsistent
Γ ⊨  		There is no possible world in which all members of Γ are true. In each possible world, at least one member of Γ is false.
Γ is exhaustive
⊨ Γ 		There is no possible world in which all members of Γ are false. In each possible world, at least one member of Γ is true.
Γ entails φ
Γ ⊨ φ 		There is no possible world in which φ is false while all members of Γ are true. φ is true in every possible world in which all members of Γ are true.
Γ excludes φ
Γ, φ ⊨  		There is no possible world in which φ is true while all members of Γ are true. φ is false in every possible world in which all members of Γ are true.
Γ renders Σ
exhaustive
Γ ⊨ Σ 		There is no possible world in which all members of Γ are true while all members of Σ are false. In each possible world in which all members of Γ are true, at least one member of Σ is true

All these concepts can be expressed in terms of relative exhaustiveness and also in terms of entailment. To express them in terms of relative exhaustiveness, simply list the assumptions (if any) to the left of ⊨ and list the alternatives (if any) to its right. The expression in terms of entailment for the concepts in the first table is shown below.

alternatives
Σ ψ1, ψ2 ψ none

Γ		 Γ, Σ⋈ ⊨ ⊥ Γ ⊨ ψ Γ ⊨⊥
φ1, φ2 φ1, φ2 ⊨ ⊥
φ φ ⊨ ψ φ ⊨ ⊥
none Σ⋈ ⊨ ⊥ ψ1⋈ ⊨ ψ2 ⊨ ψ

Here θ⋈ is any sentence contradictory to θ (such as its negation); and Σ⋈ is any result of replacing each member of Σ by a sentence that is contradictory to it. The joint exhaustiveness of ψ1 and ψ2 may also be expressed by ψ2⋈ ⊨ ψ1 and by ψ1⋈, ψ2⋈ ⊨ ⊥. The general rule is that alternatives can be dropped if their contradictories are made assumptions (and vice versa) and that ⊥ may used as a conclusion if there are no alternatives already.

Glen Helman 09 Sep 2012