4.2 Inconsistent LP Models of Arithmetic
4.2.1 LP Language and Semantics
The logic LP is a strong-Kleene semantics proposed originally in Priest(1979) (and hinted before by Ansejo(1966)) in order to model contradictions (and logical para- doxes, in particular) without explosion. The languageLP is the language of first-order
logic, including function symbols and identity (with terms and formulae defined in- ductively in the usual way).
Definition(LP-Structure) For a signatureL, anLP-L-structureMcon- sists of:
1. a non-empty set of elementsM called the domain ofM; 2. for each constant symbolc in the signature, an objectcM∈M. 3. for each n-ary function symbol f in the signature, a function fM :
Mn →M;
4. for each n-ary relation symbol R in the signature, an ordered tuple RM=hR+M, R−M
i ⊆Mn×Mn
A denotation function defined for anLP-structure is very much like the classical first- order structure with the exception of the relation symbols. Intuitively, R+M is the extension of the relation in the structure, that is, the set of objects or tuples true ofR, andR−M its anti-extension. Clearly, in a classical structure we may associate a relation with its anti-extension by R−M = Mn\R+M; i.e. classically, the anti- extension of a relation is the complement of its extension. InLP this does not need to hold: though we stipulateR+M∪R−M
=Mn (and so we have excluded middle),
it is not assumed R+M∩R−M
6
= ∅. This is why we have to explicitly define the anti-extension of a relation.
Definition(LP-model) For a theoryT in the signature of L, theLP-L- structure Mis a model ofT iff, for every formulaϕ∈T,M |=LP ϕ.
Since we are working with a non-classical semantics, logical consequence |=LP does
not behave classically. For convenience, we first define a valuation v, relative to a structure M’s interpretation function, as a function taking formulas to truth values such thatv(ϕ)∈
℘
({1,0})−∅, where{1}and{0,1}are designated values.1 Now, ifϕ is atomic andt1..,tn are terms, then 1[0]∈v(R(t1, ..., tn)) iffhtM1 , ...tMn i ∈R+[−]M
.2
The other cases are as follows: • 1[0]∈v(¬ϕ) iff 0[1]∈v(ϕ);
• 1[0]∈v(ϕ∧ψ) iff 1[0]∈v(ϕ) and[or] 1[0]∈v(ψ);
• 1[0] ∈v(∀xϕ) iff 1[0]∈ v(ϕ(x/d))) for all[some]d∈D, with v(ϕ(x/d))) being the valuation that results in assigning to the variable xthe elementd.3
Disjunction, implication and existential quantification have their normal definitions through the dual of the other logical connectives, such that:
1Intuitively, ‘v(ϕ) ={1}’ can be read as ‘ϕ isonly true, ‘v(ϕ) ={0}’ as ‘ϕ isonly false’, and
‘v(ϕ) ={1,0}’ as ‘ϕisbothtrue and false’. Despite intuitive this neednot be the case. As Barrio & Da R´e(2018, 161-162) persuasively argue ‘There is no intrinsically dialetheic value forLP. [...] [T]he intermediate value might be interpreted in a non-dialetheic fashion. In other words, to admit three-valued semantics does not compel us to interpret the intermediate semantic value, this pure value, in any particular way.’
2This is to be read 1 ∈ v(R(t1, ..., tn)) iff htM
1 , ...t M n i ∈ R+ M , and 0 ∈ v(R(t1, ..., tn)) iff htM 1 , ...tMn i ∈R− M
. Throughout we will adopt this convention where ‘x[y] iff z[w]’ is to be read as the two clauses: ‘x iff z’ and ‘y iff w’.
3For simplicity, we take the names of the elements of the domain as being the elements themselves,
• v(ϕ∨ψ) =v(¬(¬ϕ∧ ¬ψ)); • v(ϕ→ψ) =v(¬ϕ∨ψ); • v(∃xϕ) =v(¬∀x¬ϕ).
We then say that a model satisfies a formula if it comes out at least true under the model (though it may be both true and false):
Definition(LP-Satisfaction) Given anLP-structureM and formula ϕ, we say that ϕ is satisfiable in M and write M |=LP ϕ iff the valuation
function under the model’s interpretation is such that 1∈v(ϕ).
It is easy to see that every classical model is isomorphic to anLP-model in which all atoms (and, therefore, all formulas) take either the value {1} or{0}. Hence,
Definition(Classical/ConsistentLP-Model) AnLP-model Mis aclas- sical or consistent LP-model iff, for every atomicϕ, v(ϕ)∈ {{1},{0}}. Definition(LP-Validity) Given a formulaϕ, we say that ϕisvalid and write |=LP ϕiff 1∈v(ϕ) for every modelMand associated valuation v.
Theorem For arbitraryϕ,|=LP ϕiff|=Lϕ4
Proof. We first show a small Lemma:
Lemma Letv be an arbitraryLP-valuation. For arbitrary n- ary relationR, we define
v∗(R(t1, ..., tn)) = (
v(R(t1, ..., tn)) iffv(R(t1, ..., tn))∈ {{1},{0}}
{1} otherwise
Then, for arbitraryϕ,v∗(ϕ)⊆v(ϕ).
Proof. The proof is by induction on the complexity of ϕ.
(⇒)Suppose|=LP ϕ. Thenϕis (at least) true under everyLP-valuation
v. Since every classical valuation (that is, without formulas both true and false) is an LP-valuation, it follows that ϕ is true under every classical valuation. Hence,|=Lϕ.
(⇐) By contraposition. Suppose 6|=LP ϕ. There is an LP-valuation v
under which ϕis only false. This means that 16∈v(ϕ).
By the above Lemma, there is a valuation v∗ such that v∗(ϕ) ⊆ v(ϕ). From this latter fact and from 1 6∈v(ϕ), it follows 16∈ v∗(ϕ). Since by constructionv∗ is a classical valuation that falsifiesϕ, we have6|=Lϕ.
Now, for semantic consequence:
Definition(LP-Consequence) Given a formulaϕand set of formulas Γ, we say that ϕ is a (semantic) consequence of Γ and write Γ |=LP ϕ iff
for every model Msuch that M |=LP γ for all γ ∈Γ it is the case that
M |=LP ϕ.
Theorem If Γ|=LP ϕ, then Γ|=L ϕ. The converse of the implication is
false.
Proof. If Γ |=LP ϕ then all valuations are truth preserving. Hence, all
classical valuations are truth-preserving. Therefore, Γ|=L ϕ.
To see why the converse does not hold it suffices to give a counterexample. Consider an LP-valuation v with v(α) = {1,0} and v(β) = {0}. Then, v(α→β) ={1,0}, from where it follows: α, α→β6|=LP β.
We end this section with some notable classical validities that do not hold inLP. It is easy to show that the there areLP-models where the following fail (Priest, 1979,§ III.14.)5:
ϕ, ϕ→ψ|=ψ ϕ→ψ,¬ψ|=¬ϕ ϕ→ψ, ψ→χ|=ϕ→χ