March2017 ⊑Year2017

The language provides three term operators ⊓ (intersection), ⊔ (sum), and � (complement).

Since any pre-order can be expressed as a sub-relation of a partial order relation, and be extended to a partial order relation over its equivalence classes, the single sub-context relation together with the ⊓ operators allows the specification of arbitrarily many different partial order relations [24]. More accurately we may, for instance, want to say that the city of London is a *spatial* sub-context or a sub-region of England, and that March 2017 is a *temporal* sub-context or a sub-interval of the year 2017.

> London ⊓ Space⊑England March2017 ⊓ Time⊑Year2017

This and the following examples feature one simple spatial sub-context and one temporal sub-context relation. We can in the same manner however express, for instance, directional relations [25], temporal ordering relations (bi-directionally branching), and class hierarchies [9]. Ordering relations between thematic values, such as expressed by the comparative use of adjectives (Section 2) can also be added in the same way. The main purpose of the language is to facilitate expressing the common partial order core of all these theories, including the tractable fragments of these theories in a unified syntax.

A syntactic shorthand reflects – linguistically speaking – a topicalized adverbial position:

$$c: [a \sqsubseteq b] \stackrel{def}{\Leftrightarrow} c \sqcap a \sqsubseteq b$$

Space : ½ � London ⊑England

Time : ½ � March2017 ⊑Year2017

Spatially, London is a sub-context of England. Temporally, March 2017 is a subcontext of the year 2017. For entities such as cities or months, this may seem redundant. But contexts, such as a birthday party, which have both temporal and spatial extent can thus be located temporally within one context and spatially within another:

> Space : John<sup>0</sup> sBirthday⊑ London

*Towards a Fuzzy Context Logic DOI: http://dx.doi.org/10.5772/intechopen.95624*

> Time : John<sup>0</sup> sBirthday⊑ March2017

We can also reflect that speakers may choose to topicalize the other way around [26], as the last two sentences are logically equivalent to the following:

> John0 sBirthday : ½ � Space⊑ London John0 sBirthday : ½ � Time⊑ March2017

or, leveraging the propositional second layer,

John0 sBirthday : ð Þ ½ � Space⊑ London ∧ ½ � Time ⊑ March2017

where, for any propositional junctor ∘∈ ∧f g , ∨, ! :

*c* : *ϕ*<sup>1</sup> ∘ *ϕ*<sup>2</sup> ð Þ ⇔ *def c* : *ϕ*<sup>1</sup> ∘*c* : *ϕ*<sup>2</sup> and *c* : ¬*ϕ* ⇔ *def* ¬*c* : *ϕ*

Regarding John's birthday party: the location is in London, the time is in March 2017. Moreover, we can allow contexts to be stacked or combined, in order to express more complex contextualization:

> MarySays : John<sup>0</sup> sBirthday : ½ � Time ⊑ March2017 TomSays : John<sup>0</sup> sBirthday : ½ � Time⊑ August2017

Similarly to how we would express conflicting opinions in natural language, we can equivalently state:

John<sup>0</sup> sBirthday ⊓ Time : MarySays⊑ March2017 ∧ TomSays⊑ August2017 *d* : *c* : ½ �� *a*⊑*b d* : ½ �� *c* ⊓ *a*⊑ *b d* ⊓ *c* ⊓ *a*⊑*b*

Regarding John's birthday party and the time, Mary says in March 2017 and Tom says in August 2017. Context logic thus allows to reflect colloquial contextualizations well, but also to represent conflicting information.

#### **4.2 Context logic as a logical language**

Context logic thus employs two syntactic layers: the term layer with the term operators ⊓ , ⊔ , � and the propositional layer with the logic connectives ( ∧ , ∨, ¬, !). Context terms T *<sup>C</sup>* are defined over a set of variables **V** *<sup>C</sup>*: 1

Any context variable *v* ∈**V** *<sup>C</sup>* and the special symbols ⊤ and ⊥

are atomic context terms*:*

If *c* is a context term, then � *c* is a context term*:*

If *c* and *d* are context terms then *c* ⊓ *d* and *c* ⊔ *d* are context terms*:*

Context formulae L*<sup>C</sup>* are defined as follows:

If *c* and *d* are context terms then *c*⊑*d* is an atomic context formula*:*

<sup>1</sup> We leave out brackets as possible applying the following precedence: � , <sup>⊓</sup> , <sup>⊔</sup> , <sup>⊑</sup>, :, ¬, <sup>∧</sup> , <sup>∨</sup>, ! , \$. The scope of quantifiers is to be read as maximal.

If *ϕ* is a context formula, then¬*ϕ* is a context formula*:*

If *ϕ* and *ψ* are context formulae then *ϕ* ∧ *ψ*, *ϕ*∨*ψ* and *ϕ* ! *ψ* are context formulae*:*

We further define:

$$
\mathcal{L} = d \stackrel{def}{\Leftrightarrow} [\mathcal{c} \sqsubseteq d] \land [d \sqsubseteq \mathcal{c}] \tag{20}
$$

$$c \sqsubseteq d \stackrel{def}{\Leftrightarrow} [c \sqsubseteq d] \land \neg [d \sqsubseteq c] \tag{21}$$

Different variant semantics have been proposed [10, 11, 26]. The different approaches slightly differ in the resulting semantics, but all three employ a lattice structure for specifying the meanings of context terms, assigning a partial order to give a semantics to ⊑. Here, we give a semantics by mapping the language to a predicate logic with a single binary predicate *P*, describing a pre-order relation, to give the fundamental ⊑ its semantics. We use a function *τ*PL CL : L*<sup>C</sup>* � **V** *<sup>P</sup>* ! L*P*, where L*<sup>C</sup>* is the set of context logic formulae, **V***<sup>P</sup>* is a vocabulary of predicate logic variables, and L*<sup>P</sup>* is the set of predicate logic formulae. We also employ **V** *<sup>C</sup>*, the set of variables, as the set of constants for L*P*, and require **V***P*∩**V** *<sup>C</sup>* ¼ ∅:

$$
\pi\_{\rm CL}^{\rm PL}(\mathsf{T}\sqsubseteq\mathsf{T},m) = \mathsf{T} \tag{22}
$$

$$
\tau\_{\rm CL}^{\rm PL}(\mathsf{T} \sqsubseteq \bot, m) = \bot \tag{23}
$$

$$\pi\_{\rm CL}^{\rm PL}(\top \sqsubseteq v, m) = P(m, v), \text{for } v \in \mathcal{V}\_{\rm C} \tag{24}$$

$$
\tau\_{\rm CL}^{\rm PL}(\mathsf{T}\sqsubseteq\sim\mathcal{c},\mathsf{m}) = \tau\_{\rm CL}^{\rm PL}(\mathsf{c}\sqsubseteq\bot,\mathsf{m}) \text{ for } \mathsf{c}\in\mathcal{T}\_{\mathcal{C}}\tag{25}
$$

$$
\tau\_{\rm CL}^{\rm PL}(\top \sqsubseteq c \sqcap d, m) = \tau\_{\rm CL}^{\rm PL}(\top \sqsubseteq c, m) \land \tau\_{\rm CL}^{\rm PL}(\top \sqsubseteq d, m) \tag{26}
$$

*τ*PL CLð Þ¼ ⊤ ⊑*c* ⊔ *d*, *m*

$$\forall m', P(m', m) : \exists m'', P(m', m) : \tau\_{\text{CL}}^{\text{pl}}(\top \sqsubseteq c, m'') \lor \tau\_{\text{CL}}^{\text{pl}}(\top \sqsubseteq d, m'') \tag{27}$$
  $\text{where } m' \text{ and } m'' \text{ are new variables.}$ 

$$\begin{split} \tau\_{\rm CL}^{\rm PL}(c \sqsubseteq d, m) &= \forall m', P(m', m) : \tau\_{\rm CL}^{\rm PL}(\top \sqsubseteq c, m') \to \tau\_{\rm CL}^{\rm PL}(\top \sqsubseteq d, m') \\ \text{where } m' \text{ is a new variable.} \end{split} \tag{28}$$

$$
\tau\_{\rm CL}^{\rm PL}(\neg \phi, m) = \neg \tau\_{\rm CL}^{\rm PL}(\phi, m) \tag{29}
$$

$$
\tau\_{\rm CL}^{\rm PL}(\phi \wedge \nu, m) = \tau\_{\rm CL}^{\rm PL}(\phi, m) \wedge \tau\_{\rm CL}^{\rm PL}(\nu, m) \tag{30}
$$

$$
\tau\_{\rm CL}^{\rm PL}(\phi \vee \nu, m) = \tau\_{\rm CL}^{\rm PL}(\phi, m) \vee \tau\_{\rm CL}^{\rm PL}(\nu, m) \tag{31}
$$

$$
\tau\_{\rm CL}^{\rm PL}(\phi \to \psi, m) = \tau\_{\rm CL}^{\rm PL}(\phi, m) \to \tau\_{\rm CL}^{\rm PL}(\psi, m) \tag{32}
$$

We note that although we introduce new variables *m*<sup>0</sup> , *<sup>m</sup>*″ in (27) and (28), each new variable is only used together with the variable last introduced – *<sup>m</sup>*<sup>0</sup> with *<sup>m</sup>*, *<sup>m</sup>*″ with *m*<sup>0</sup> but not with *m* –, not with any other variables introduced before. This means, we can alternate between two variables and reuse *m* after *m*<sup>0</sup> , i.e., that **V***<sup>P</sup>* ¼ *m*, *m*<sup>0</sup> f g. We also note, that the context variables *v*∈**V** *<sup>C</sup>* are constants with respect to the predicate logic and that they only appear in the second position of *P* in (24). This property allows us to reformulate any binary expression *P m*ð Þ , *v* for *v*∈**V** *<sup>C</sup>* using a different monadic predicate *Pv* for each *v*∈ **V** *<sup>C</sup>*, and write *Pv*ð Þ *m* instead of *P m*ð Þ , *v* .

*Towards a Fuzzy Context Logic DOI: http://dx.doi.org/10.5772/intechopen.95624*

Consequently, the fragment of predicate logic required in application of *τ*PL CL alone is in the two-variable fragment known to be decidable. Moreover, the variables, such as *m* and *m*<sup>0</sup> , only occur together in the *atomic guard*, as *P m*<sup>0</sup> ð Þ , *m* , suggesting that the language as defined so far is in the so-called *guarded fragment* GF [27] defined as [cited after 28, p.1664f]:

Every atomic formula belongs to GF*:*

GF is closed under¬, ∧ , ∨, ! , \$ *:*

If *x*, *y* are tuples of variables, *α*(*x*, *y*) is an atomic formula, *ψ*(*x*,*y*) is in GF, and free (*ψ* ⊆ free (*α*)={*x*,*y*}, where free (*ϕ*) is the set of the free variables of *ϕ*, then the formulae

$$\begin{aligned} \exists \mathcal{y} &: a(\mathfrak{x}, \mathfrak{y}) \land \psi(\mathfrak{x}, \mathfrak{y})\\ \forall \mathcal{y} &: a(\mathfrak{x}, \mathfrak{y}) \to \psi(\mathfrak{x}, \mathfrak{y}) \\ &\text{belong to GF.} \end{aligned}$$

In order to obtain the reasoning capabilities, however, we would need to add pre-order axioms for *P*, so as to be able to specify ⊑ as a partial order relation:

$$\forall \mathbf{x}, \mathbf{y}, \mathbf{z}: P(\mathbf{x}, \mathbf{y}) \land P(\mathbf{y}, \mathbf{z}) \to P(\mathbf{x}, \mathbf{z}) \tag{33}$$

$$\forall \mathfrak{x}: P(\mathfrak{x}, \mathfrak{x}) \tag{34}$$

and we see that transitivity (13) cannot be axiomatized in the two-variable fragment, as it requires three variables. Fortunately, [28, 29] have shown that for GF<sup>2</sup> + PG – the guarded fragment limited to two variables and a single binary preorder that can only appear in the guard – is in 2-EXPTIME. Moreover, this result is a loose upper bound, since the language under inspection here can be expressed using the transitive binary relation *P* in only one direction – namely from wholes to parts –, using otherwise only the monadic predicates *Pv*, *v*∈**V** *<sup>C</sup>*, placing the translation of context logic with the axioms for <sup>⊑</sup> into the class MGF<sup>2</sup> <sup>þ</sup> *TG* , the two-variable monadic guarded fragment with one-way transitive guards, which is decidable and whose satisfiability problem is in EXPSPACE [28].

In addition to the pre-order axioms, we can also add a localized guarded variant of the so-called *weak supplementation principle* [30, Ch. 3] for ⊑ ensuring a minimal homogeneity constraint over *v*1, *v*<sup>2</sup> ∈ **V** *<sup>C</sup>*: 2

$$\begin{split} \forall \mathbf{x} : (\forall \mathbf{x'}, P(\mathbf{x'}, \mathbf{x}) : P(\mathbf{x'}, \nu\_1) \to P(\mathbf{x'}, \nu\_2)) \\ \qquad \land (\exists \mathbf{x'}, P(\mathbf{x'}, \mathbf{x}) : P(\mathbf{x'}, \nu\_2) \land \neg P(\mathbf{x'}, \nu\_1)) \\ \to (\exists \mathbf{x'}, P(\mathbf{x'}, \mathbf{x}) : P(\mathbf{x'}, \nu\_2) \land \neg \exists \mathbf{x'}', P(\mathbf{x'}, \mathbf{x'}) : P(\mathbf{x'}, \nu\_1)) . \end{split} \tag{35}$$

The principle says that, if for any *x* all its parts *x*<sup>0</sup> that are in *v*<sup>1</sup> are also in *v*2, but there is a part *x*<sup>0</sup> that is in *v*<sup>2</sup> but not in *v*<sup>1</sup> (paraphrasing: *v*<sup>1</sup> is a proper part of *v*2), then there is a part *x*<sup>0</sup> of *v*<sup>2</sup> that has no parts in *v*<sup>1</sup> (i.e.: *x*<sup>0</sup> does not overlap *v*1), i.e., is completely outside of *v*1. Axiom 35 ensures that the entities described by *v*1, *v*<sup>2</sup> ∈**V** *<sup>C</sup>* do not have, e.g., singular points that are not entities themselves in the domain under inspection. This axiom is required for proving several of the lattice laws. Note that we thus characterize a weak supplementation principle only for ⊑, that

<sup>2</sup> The interested reader may find a brief discussion on mereological and ontological properties in Section 4.5.

we, however, cannot formulate a weak supplementation principle for *P* without leaving the guarded fragment.

In order to do this, however, we have to employ *v*1, *v*<sup>2</sup> ∈**V** *<sup>C</sup>* as schema variables, i.e., we have to formally see this actually not as one axiom but ∣**V** <sup>2</sup> *<sup>C</sup>*∣ axioms. This means that for infinite **V** *<sup>C</sup>*, the axiomatization becomes infinite. For practical, finite knowledge bases, **V** *<sup>C</sup>* will be finite. If an infinite vocabulary **V** *<sup>C</sup>* is employed, a practical realization would be to use a unification mechanism suitable for the particular language **V** *<sup>C</sup>* employed.

Intuitively, the meaning of *a*⊑ *b* is that all parts of *a* are part of *b*. The reading thus corresponds to a universal quantification, and the properties expressed by contexts in this statement describe homogenous properties inherited from wholes to their parts. Correspondingly, ¬½ � *a*⊑ *b* expresses an existential quantification, stating that not all parts of *a* are parts of *b*, which means that there is a part of *a* that is not part of *b*, or that does not have property *b*. We can thus express heterogeneity.

The complement � *c*, is interpreted with respect to the pseudo-0-element ⊥: the atomic formula ⊤ ⊑ � *a* is interpreted as equivalent to *a*⊑ ⊥, meaning that no part is in *a*, implying universal quantification. There are thus two types of negation ¬ on the logical level and � on the context level. ⊥ is a pseudo-element, it disappears in the translation when applying (28). We do not need to assume that an empty element exists:

$$\begin{aligned} \tau\_{\rm CL}^{\rm PL}(\mathsf{c} \sqsubseteq \bot, m) &\equiv \forall m', P(m', m) : \tau\_{\rm CL}^{\rm PL}(\mathsf{T} \sqsubseteq \mathsf{c}, m') \to \tau\_{\rm CL}^{\rm PL}(\mathsf{T} \sqsubseteq \bot, m') \\\\ &\equiv \forall m', P(m', m) : \tau\_{\rm CL}^{\rm PL}(\mathsf{T} \sqsubseteq \mathsf{c}, m') \to \bot \\\\ &\equiv \forall m', P(m', m) : \neg \tau\_{\rm CL}^{\rm PL}(\mathsf{T} \sqsubseteq \mathsf{c}, m') \\\\ &\equiv \neg \exists m', P(m', m) : \tau\_{\rm CL}^{\rm PL}(\mathsf{T} \sqsubseteq \mathsf{c}, m') \end{aligned} \tag{36}$$

A crucial consequence of adopting weak supplementation (35) is (2). It says that if all parts *<sup>m</sup>*″ of a part *<sup>m</sup>*<sup>0</sup> have a part *<sup>m</sup>*‴ that is part of *<sup>a</sup>*, this is equivalent to *<sup>m</sup>*<sup>0</sup> being part of *a*:

$$P(\forall m'', P(m'', m') : \exists m''', P(m''', m'') : P(m''', a) \equiv P(m', a) \tag{37}$$

Proof (⫤): this holds immediately with the reflexivity (34) and transitivity (33) of *<sup>P</sup>*: if *P m*<sup>0</sup> ð Þ , *<sup>a</sup>* then all parts *<sup>m</sup>*″ of *<sup>m</sup>*<sup>0</sup> fulfill *P m*ð Þ ″, *<sup>a</sup>* by transitivity, and therefore there is a part *<sup>m</sup>*‴ of *<sup>m</sup>*″, namely *<sup>m</sup>*″ itself, by reflexivity, so that *P m*ð Þ ‴, *<sup>a</sup>* .

Proof (⊨): we prove the reverse direction by contradiction, applying (35). Assume <sup>∀</sup>*m*″, *P m*″, *<sup>m</sup>*<sup>0</sup> ð Þ : <sup>∃</sup>*m*‴, *P m*ð Þ ‴, *<sup>m</sup>*″ : *P m*ð Þ ‴, *<sup>a</sup>* and not *P m*<sup>0</sup> ð Þ , *<sup>a</sup>* , i.e., that there is an *m*<sup>00</sup> <sup>1</sup> that has *<sup>m</sup>*‴, *P m*ð Þ ‴, *<sup>a</sup>* but not *P m*<sup>00</sup> <sup>1</sup> , *<sup>a</sup>* . Then by (35) there has to be a part *m*<sup>00</sup> <sup>2</sup> of *<sup>m</sup>*<sup>0</sup> that does not have a part *<sup>m</sup>*‴ where *P m*ð Þ ‴, *<sup>a</sup>* . But this is prevented by the premise <sup>∀</sup>*m*″, *P m*″, *<sup>m</sup>*<sup>0</sup> ð Þ : <sup>∃</sup>*m*‴, *P m*ð Þ ‴, *<sup>m</sup>*″ : *P m*ð Þ ‴, *<sup>a</sup>* .

It can be shown (Section 4.4) that the definition of *τ*PL CL together with the two pre-order axioms and the local guarded variant of the weak supplementation principle is sufficient to characterize context terms as spanning a bounded lattice. We note that with a different axiomatization other types of lattice structures could be realized for different application domains.

#### **4.3 A fuzzy logic perspective on context logic**

This section shows context logic as specified above is a two-layered language with a generalized *t*-norm-based fuzzy logic at the term level and a classical

*Towards a Fuzzy Context Logic DOI: http://dx.doi.org/10.5772/intechopen.95624*

f g 0, 1 -based semantics at the formula level. From there it is a small step to also add a 0, 1 ½ �-based multivalued semantics to the formula level, so as to obtain a full twolayered fuzzy logic in Section 5.

To see that the context terms T *<sup>C</sup>* can be viewed as a generalized t-norm, we set the intersection ⊓ , the meet operation of the lattice, as the monoid operation and the term ⊤ as the identity element of the monoid. The monoid properties associativity and identity element are fulfilled by any lattice (see Section 4.4, (44) and (46)). For the generalized fuzzy logic semantics, the lattice meet-operation ⊓ will be shown to fulfill the properties of a *t*-norm, the join-operation ⊔ , those of the corresponding *s*-norm. Both are required to be commutative (1), associative (2), and support an identity element (4) and monotonicity (3) (for the full proofs see Section 4.4). We prove monotonicity for ⊓ (38) and ⊔ (39):

*τ*PL CLð½ � *a*⊑*c* ∧ ½ �! *b*⊑ *d* ½ � *a* ⊓ *b*⊑ *c* ⊓ *d* , *m*Þ � ∀*m*<sup>0</sup> , *P m*<sup>0</sup> ð Þ , *m* : *P m*<sup>0</sup> ð Þ! , *a P m*<sup>0</sup> ð Þ ð Þ ,*c* ∧ ∀*m*<sup>0</sup> , *P m*<sup>0</sup> ð Þ , *m* : *P m*<sup>0</sup> ð Þ! , *b P m*<sup>0</sup> ð Þ ð Þ , *d* ! ∀*m*<sup>0</sup> , *P m*<sup>0</sup> ð Þ , *m* : *P m*<sup>0</sup> ð Þ , *a* ∧ *P m*<sup>0</sup> ð Þ! , *b P m*<sup>0</sup> ð Þ ,*c* ∧ *P m*<sup>0</sup> ð Þ ð Þ , *d* (38) *τ*PL CLð½ � *a*⊑*c* ∧ ½ �! *b*⊑*d* ½ � *a* ⊔ *b*⊑*c* ⊔ *d* , *m*Þ � ∀*m*<sup>0</sup> , *P m*<sup>0</sup> ð Þ , *m* : *P m*<sup>0</sup> ð Þ! , *a P m*<sup>0</sup> ð Þ ð Þ ,*c* ∧ ∀*m*<sup>0</sup> , *P m*<sup>0</sup> ð Þ , *m* : *P m*<sup>0</sup> ð Þ! , *b P m*<sup>0</sup> ð Þ ð Þ , *d* ! ∀*m*<sup>0</sup> , *P m*<sup>0</sup> ð Þ , *m* : <sup>∀</sup>*m*″, *P m*″, *<sup>m</sup>*<sup>0</sup> ð Þ ð Þ : <sup>∃</sup>*m*‴, *P m*ð Þ ‴, *<sup>m</sup>*″ : *P m*ð Þ ‴, *<sup>a</sup>* <sup>∨</sup>*P m*ð Þ ‴, *<sup>b</sup>* ! <sup>∀</sup>*m*″, *P m*″, *<sup>m</sup>*<sup>0</sup> ð Þ ð Þ : <sup>∃</sup>*m*‴, *P m*ð Þ ‴, *<sup>m</sup>*″ : *P m*ð Þ ‴,*<sup>c</sup>* <sup>∨</sup>*P m*ð Þ ‴, *<sup>d</sup>* (39)

Proof (3): if every *m*<sup>0</sup> that is part of *a* is in *c* and every *m*<sup>0</sup> that is part of *b* is in *d*, then every *m*<sup>0</sup> that is part of *a* and *b* is also in both *c* and *d*. Proof (4): we see that it follows from this condition also that any *<sup>m</sup>*‴ that exists as part of any *<sup>m</sup>*″ in *<sup>a</sup>* or *<sup>b</sup>* must also be part of *<sup>c</sup>* or *<sup>d</sup>* in *<sup>m</sup>*″.

The generalized De Morgan law connects t-norms with s-norms (7). It follows for the translations of ⊓ and ⊔ directly from the De Morgan laws in predicate logic.

$$\begin{array}{lcl} \mathsf{P}^{\mathrm{PL}}\_{\mathrm{CL}}(a \sqcup b = \sim(\sim a \sqcap \sim b), m) \equiv \\\\ \forall m', P(m', m) : & (\forall m'', P(m'', m') : \exists m''', P(m''', m'') : \\\\ P(m''', a) \vee P(m''', b)) \\\\ \leftrightarrow \ \neg \exists m'', P(m'', m') : (\neg \exists m''', P(m''', m'') : P(m''', a)) \\\\ & \wedge (\neg \exists m''', P(m''', m'') : P(m''', b)) \end{array} \tag{40}$$

The residual can then be derived from its characterization:

$$r(a,b) = \sup \{ z | t(a,z) \prec b \} .$$

The operation ) with the definition

$$a \Rightarrow b \stackrel{def}{\Leftrightarrow} a \sqcup b \tag{41}$$

has the required property *t a*ð Þ , *z* ≼ *b* (with ⊓ the *t*-norm and ⊑, the lattice partial order ≼).<sup>3</sup>

*τ*PL CLð*a* ⊓ ð Þ � *a* ⊔ *b* ⊑*b; m*Þ � ∀*m*<sup>0</sup> , *P m*<sup>0</sup> ð Þ *; <sup>m</sup>* : <sup>ð</sup>∀*m*″, *P m*″*; <sup>m</sup>*<sup>0</sup> ð Þ : <sup>∃</sup>*m*‴, *P m*ð Þ ‴*; <sup>m</sup>*″ : <sup>¬</sup>∃*miv; P miv; <sup>m</sup>*‴ : *P miv; <sup>a</sup>* <sup>∨</sup>*P m*ð ÞÞ ‴*; <sup>b</sup>* <sup>∧</sup> *P m*<sup>0</sup> ð Þ! *; <sup>a</sup> P m*<sup>0</sup> ð Þ *; <sup>b</sup>* (42)

We prove that for any *m*<sup>0</sup> , *P m*<sup>0</sup> ð Þ , *m* :

$$\begin{aligned} \forall m'', P(m'', m'): &\quad \exists m''', P(m''', m''):\\ &\quad \left(\neg \exists m^{iv}, P(m^{iv}, m'''): P(m^{iv}, a) \lor P(m''', b)\right) \land P(m', a) \\ &\quad \forall P(m', b) \end{aligned}$$

and the term � *a* ⊔ *b* expresses the maximal element local to *m* with this property.

Proof: assume the antecedent is true, then because of transitivity of *P* (33) and the conjunct *P m*<sup>0</sup> ð Þ , *<sup>a</sup>* , there can be no *<sup>m</sup>*‴ part of *<sup>m</sup>*<sup>0</sup> for which all parts *<sup>m</sup>iv*, including *<sup>m</sup>*‴ itself fulfill ¬*P miv*, *<sup>a</sup>* . Therefore the second disjunct *P m*ð Þ ‴, *<sup>b</sup>* must be true. But if we know that for all *<sup>m</sup>*″ with *P m*″, *<sup>m</sup>*<sup>0</sup> ð Þ exists *<sup>m</sup>*‴, so that *P m*ð Þ ‴, *<sup>b</sup>* , we know by (2), a consequence of the localized guarded variant of the weak supplementation principle (35), that *P m*<sup>0</sup> ð Þ , *b* . To see that it is maximal, assume there is *m*<sup>0</sup> 1 outside of � *a* ⊔ *b* and *P m*<sup>0</sup> 1, *<sup>a</sup>* and *P m*<sup>0</sup> 1, *<sup>b</sup>* . To be outside of � *<sup>a</sup>* <sup>⊔</sup> *<sup>b</sup>*, there would have to be an *<sup>m</sup>*″, *P m*″, *<sup>m</sup>*<sup>0</sup> 1 so that for all *<sup>m</sup>*‴, *P m*ð Þ ‴, *<sup>m</sup>*″ there is *<sup>m</sup>iv*, so that *P miv*, *<sup>m</sup>*‴ and *P miv*, *<sup>a</sup>* and ¬*P m*ð Þ ‴, *<sup>b</sup>* , but this cannot be, because *P m*‴, *<sup>m</sup>*<sup>0</sup> 1 and by the assumption *P m*<sup>0</sup> 1, *<sup>b</sup>* , thus by transitivity (33) *P m*ð Þ ‴, *<sup>b</sup>* .

This result indicates that, at least with respect to the supplementation property expressed through (35), � *a* ⊔ *b* fulfills the characterization of a residual. We can also show continuity (54) and pre-linearity (55) (Section 4.4).

We are thus justified to say that context logic terms have a generalized *t*-norm semantics and we can give a *t*-norm-based semantics to context logic.

We obtain: a t-norm-based classical semantics for context logic is a structure ð Þ *I*, *i*, *a*, *LT*, ≼, 1*T*, *t*, *s* , where the terms are interpreted by *i* : T *<sup>C</sup>* ! *LT* together with the function *a* : **V** *<sup>C</sup>* ! *LT* assigning context terms and variables, respectively, to elements of a lattice *LT*, and the formulae, by the classical interpretation function *I* : L*<sup>C</sup>* ! f g 0, 1 :

<sup>3</sup> To understand the meaning of *<sup>a</sup>* ) *<sup>b</sup>*, we can translate

*τ*PL CLð Þ *Τ* ⊑ � *a* ⊔ *b*, *x* � ∀*m*<sup>0</sup> , *P m*<sup>0</sup> ð Þ , *<sup>m</sup>* : <sup>∃</sup>*m*″, *P m*″, *<sup>m</sup>*<sup>0</sup> ð Þ : ð Þ <sup>¬</sup>∃*m*‴, *P m*ð Þ ‴, *<sup>m</sup>*″ : *P m*ð Þ ‴, *<sup>a</sup>* <sup>∨</sup>*P m*ð Þ ″, *<sup>b</sup>* � ∀*m*<sup>0</sup> , *P m*<sup>0</sup> ð Þ , *<sup>m</sup>* : <sup>¬</sup>∀*m*″, *P m*″, *<sup>m</sup>*<sup>0</sup> ð Þ ð Þ : <sup>∃</sup>*m*‴, *P m*ð Þ ‴, *<sup>m</sup>*″ : *P m*ð Þ ‴, *<sup>a</sup>* ∨∃*m*″, *P m*″, *<sup>m</sup>*<sup>0</sup> ð Þ : *P m*ð Þ ″, *<sup>b</sup>* � ∀*m*<sup>0</sup> , *P m*<sup>0</sup> ð Þ , *<sup>m</sup>* : <sup>∀</sup>*m*″, *P m*″, *<sup>m</sup>*<sup>0</sup> ð Þ ð Þ : <sup>∃</sup>*m*‴, *P m*ð Þ ‴, *<sup>m</sup>*″ : *P m*ð Þ ‴, *<sup>a</sup>* ! <sup>∃</sup>*m*″, *P m*″, *<sup>m</sup>*<sup>0</sup> ð Þ : *P m*ð Þ ″, *<sup>b</sup>* � ∀*m*<sup>0</sup> , *P m*<sup>0</sup> ð Þ , *<sup>m</sup>* : *P m*<sup>0</sup> ð Þ! , *<sup>a</sup>* <sup>∃</sup>*m*″, *P m*″, *<sup>m</sup>*<sup>0</sup> ð Þ : *P m*ð Þ ″, *<sup>b</sup>* � ∀*m*<sup>0</sup> , *P m*<sup>0</sup> ð Þ , *m* : *P m*<sup>0</sup> ð Þ! , *a P m*<sup>0</sup> ð Þ , *b* ,

that is, for every part *m*<sup>0</sup> of *m* holds that: if *m*<sup>0</sup> is inside *a* it is inside *b*. Here the last two steps follow by (2) a consequence of (A9) .

*Towards a Fuzzy Context Logic DOI: http://dx.doi.org/10.5772/intechopen.95624*

$$i(v) = a(v), \text{for } v \in V\_C$$

$$i(\sim c) = \mathbf{1}\_T - i(c),$$

With 1*<sup>T</sup>* � *i c*ð Þthe pseudo-complement of the lattice

$$i(c \sqcap d) = t(i(c), i(d))$$

$$i(c \sqcup d) = s(i(c), i(d))$$

$$I(c \sqsubseteq d) = \mathbf{1} \text{ iff } i(c) \not\approx i(d)$$

$$I(\neg\phi) = \mathbf{1} - I(\phi)$$

$$I(\phi \wedge \psi) = \min\left(I(\phi), I(\psi)\right)$$

$$I(\phi \vee \psi) = \max\left(I(\phi), I(\psi)\right)$$

It only remains to show that the context term operators indeed support the lattice requirements.

#### **4.4 Proof: context logic with local, guarded weak supplementation characterizes a bounded lattice**

For the purpose of completeness, the proofs are listed here in detail. However, the results are part of basic, fundamental lattice theory and no novelty is claimed.

We prove that ⊓ and ⊔ fulfill the laws for a bounded lattice. We start by showing that ⊓ fulfills the laws of a semilattice: ⊓ is idempotent (7), associative (8), commutative (9), and has ⊤ as its neutral element (46).

$$
\mathfrak{a} \sqcap \mathfrak{a} = \mathfrak{a} \tag{43}
$$

$$a \sqcap (b \sqcap c) = (a \sqcap b) \sqcap c \tag{44}$$

$$a \sqcap b = b \sqcap a \tag{45}$$

$$
\mathfrak{a} \sqcap \mathsf{T} = \mathfrak{a} \tag{46}
$$

These properties hold, since ⊓ directly translates into ∧ :

$$
\tau\_{\rm CL}^{\rm PL}(\top \sqsubseteq c \sqcap d, m) = \tau\_{\rm CL}^{\rm PL}(\top \sqsubseteq c, m) \land \tau\_{\rm CL}^{\rm PL}(\top \sqsubseteq d, m)
$$

We show the translations:

$$\begin{split} \tau^{\rm PL}\_{\rm CL}(a \sqcap a = a, m) &\equiv \forall P(m', m) : P(m', a) \land P(m', a) \leftrightarrow P(m', a) \\ \tau^{\rm PL}\_{\rm CL}(a \sqcap (b \sqcap c) = (a \sqcap b) \sqcap c, m) &\equiv \forall P(m', m) : P(m', a) \land (P(m', b) \land P(m', c)) \\ &\leftrightarrow \left(P(m', a) \land P(m', b)\right) \land P(m', c) \\ \tau^{\rm PL}\_{\rm CL}(a \sqcap b = b \sqcap a, m) &\equiv \forall P(m', m) : P(m', a) \land P(m', b) \\ &\leftrightarrow P(m', b) \land P(m', a) \\ \tau^{\rm PL}\_{\rm CL}(a \sqcap \top = a, m) &\equiv \forall P(m', m) : P(m', a) \land \top \leftrightarrow P(m', a) \end{split}$$

We can see that all translations of properties are tautologies and follow directly from the properties of ∧ . The semantics of ⊔ requires a closer look. We first note that a basic requirement of extensionality holds:

$$\begin{aligned} \pi\_{\rm CL}^{\rm PL}(\mathsf{T}\sqsubseteq a, m) &\equiv P(m, a) \equiv \forall m', P(m', m) : P(m', a) \\ &\equiv \forall P(m', m) : \pi\_{\rm CL}^{\rm PL}(\mathsf{T}\sqsubseteq a, m') \end{aligned} \tag{47}$$

The property (47) holds because *P m*ð Þ , *a* entails *P m*<sup>0</sup> ð Þ , *a* for all *P m*<sup>0</sup> ð Þ , *m* because of transitivity of *P*. Also, for all *m*<sup>0</sup> , *P m*<sup>0</sup> ð Þ , *m* : *P m*<sup>0</sup> ð Þ , *a* entails *P m*ð Þ , *a* , since *P* is reflexive.

We can now prove the semilattice laws for ⊔ .

$$
\mathfrak{a} \sqcup \mathfrak{a} = \mathfrak{a} \tag{48}
$$

$$a \sqcup (b \sqcup c) = (a \sqcup b) \sqcup c \tag{49}$$

$$a \sqcup b = b \sqcup a \tag{50}$$

$$
\mathfrak{a} \sqcup \mathfrak{L} = \mathfrak{a} \tag{51}
$$

When we translate idempotency (48):

$$\begin{aligned} &\tau\_{\text{CL}}^{\text{PL}}(a\sqcup a=a,m) \\ \equiv&\forall m',P(m',m):(\forall m'',P(m'',m'):\exists m''',P(m''',m''):\\ &P(m''',a)\lor P(m''',a))\leftrightarrow P(m',a)\\ \equiv&\forall m',P(m',m):(\forall m'',P(m'',m'):\exists m''',P(m''',m''):\\ &P(m''',a))\leftrightarrow P(m',a) \end{aligned}$$

we see that the translation of ⊔ provides one direction of the proof. With (2), a consequence of weak supplementation, we obtain the other direction.

The other laws follow in a similar manner. We show associativity (49):

$$\begin{split} \pi\_{\text{CL}}^{\text{PL}}(a \sqcup (b \sqcup c) = (a \sqcup b) \sqcup c, m) &\equiv \forall m', P(m', m) : \\ \forall m'', P(m'', m') : \exists m''', P(m''', m'') : P(m''', a) \vee \forall m^{\text{iv}}, P(m^{\text{iv}}, m''') : \\ \exists m'', P(m^v, m^{\text{iv}}) : P(m^v, b) \vee P(m^v, c) \\ \leftarrow \forall m'', P(m'', m') : \exists m''', P(m''', m'') : (\forall m^{\text{iv}}, P(m^{\text{iv}}, m''') \\ \exists m^v, P(m^v, m^{\text{iv}}) : P(m^v, a) \vee P(m^v, b) \vee P(m''', c) \end{split}$$

By proving the following for any *m*<sup>0</sup> from which the above then follows directly via the associativity and commutativity of ∨:

$$\begin{aligned} &\forall m'', P(m'', m'): \exists m''', P(m''', m''): P(m''', a) \lor \forall m^{iv}, P(m^{iv}, m'''):\\ &\exists m'', P(m^v, m^{iv}): P(m^v, b) \lor P(m'', c)\\ &\equiv \quad \forall m'', P(m'', m'): \exists m''', P(m''', m''): P(m''', a) \lor P(m''', b) \lor P(m''', c) \end{aligned}$$

We prove in two steps.

Proof (⊨): assume we choose an arbitrary *<sup>m</sup>*″, *P m*″, *<sup>m</sup>*<sup>0</sup> ð Þ. The antecedent says that if there is an *<sup>m</sup>*‴, *P m*ð Þ ‴, *<sup>m</sup>*″ so that *P m*ð Þ ‴, *<sup>a</sup>* or there is *P m*ð Þ ‴, *<sup>m</sup>*″ so that for all its parts *miv*, we can find *<sup>m</sup><sup>v</sup>*, so that *P m<sup>v</sup>* ð Þ , *<sup>b</sup>* or *P m<sup>v</sup>* ð Þ ,*<sup>c</sup>* . If there is an *<sup>m</sup>*‴, *P m*ð Þ ‴, *<sup>a</sup>* , the consequent holds. If there is no such *<sup>m</sup>*‴, *P m*ð Þ ‴, *<sup>a</sup>* , there must be an *<sup>m</sup>*‴, so that all its parts *<sup>m</sup>iv* have a part *<sup>m</sup><sup>v</sup>* in *<sup>b</sup>* or *<sup>c</sup>*. Since each such *mv* is also a part of *<sup>m</sup>*″, we can conclude that for all *<sup>m</sup>*″, *P m*″, *<sup>m</sup>*<sup>0</sup> ð Þ there is an *<sup>m</sup>*‴ – namely, the *<sup>m</sup><sup>v</sup>* we identified –, so that *P m*ð Þ ‴, *<sup>b</sup>* <sup>∨</sup>*P m*ð Þ ‴,*<sup>c</sup>* .

Proof (⫤): assume we have for each *<sup>m</sup>*″, *P m*″, *<sup>m</sup>*<sup>0</sup> ð Þ: <sup>∃</sup>*m*‴, *P m*ð Þ ‴, *<sup>m</sup>*″ : *P m*ð Þ ‴, *<sup>a</sup>* <sup>∨</sup>*P m*ð Þ ‴, *<sup>b</sup>* <sup>∨</sup>*P m*ð Þ ‴,*<sup>c</sup>* and the consequent is false. In this case, there must be an *m*<sup>00</sup> <sup>1</sup> , so that *P m*ð Þ ‴, *<sup>a</sup>* must be false for all *<sup>m</sup>*‴, *P m*‴, *<sup>m</sup>*<sup>00</sup> 1 and that there is a part of any such *<sup>m</sup>*‴ so that all its subparts are neither in *<sup>b</sup>* nor in *<sup>c</sup>*. By the premise however, we know that *m*<sup>00</sup> <sup>1</sup> must have a part *<sup>m</sup>*″<sup>0</sup> <sup>1</sup> so that *P m*″<sup>0</sup> 1, *<sup>b</sup>* <sup>∨</sup>*P m*″<sup>0</sup> 1,*<sup>c</sup>* . But since *P* is transitive we know that for all parts *miv* <sup>1</sup> of *<sup>m</sup>*″<sup>0</sup> <sup>1</sup> holds either *P miv* <sup>1</sup> , *<sup>b</sup>* or

*Towards a Fuzzy Context Logic DOI: http://dx.doi.org/10.5772/intechopen.95624*

*P miv* ,*<sup>c</sup>* . By reflexivity we moreover know that each *<sup>m</sup>iv* has a part, namely itself, for which *P miv* , *<sup>b</sup>* or *P miv* ,*<sup>c</sup>* hold.

Applying this result twice via the associativity and commutativity of ∨, we can conclude (49) must hold:

$$\begin{split} &\forall m'', P(m'', m'): \exists m''', P(m''', m''): P(m''', a) \lor \forall m^{iv}, P(m^{iv}, m'''):\\ &\quad \exists m'', P(m^v, m^{iv}): P(m'', b) \lor P(m^v, c)\\ &\equiv \quad \forall m'', P(m'', m'): \exists m''', P(m''', m''): P(m''', a) \lor P(m''', b) \lor P(m''', c):\\ &\equiv \quad \forall m'', P(m'', m'): \exists m''', P(m''', m''): (\forall m^{iv}, P(m^{iv}, m''')\\ &\quad \exists m^v, P(m^v, m^{iv}): P(m^v a \lor P(m^v b) \lor P(m''', c)) \end{split}$$

Theorem 14 holds immediately given the definition of the translation for ⊔ and the commutativity of ∨:

$$\begin{aligned} \tau\_{\text{CL}}^{\text{PL}}(a \sqcup b = b \sqcup a) &\equiv \\ (\forall m', P(m', m) : \forall m'', P(m'', m') : \exists m'', P(m''', m'') : \\ P(m''', a) \vee P(m''', b)) \\ \leftarrow (\forall m', P(m', m) : \forall m'', P(m'', m') : \exists m''', P(m''', m'') : \\ P(m''', b) \vee P(m''', a)) \end{aligned}$$

Proving the neutral element property (51) requires (35).

$$\begin{array}{lcl} & \tau^{\mathsf{PL}}\_{\mathrm{CL}}(a \sqcup \bot = a, m) \\ \equiv & \forall m', P(m', m) : (\forall m'', P(m'', m') : \exists m''', P(m'', m'') : \\ & \tau^{\mathrm{PL}}\_{\mathrm{CL}}(\top \sqsubseteq a, m'') \vee \tau^{\mathrm{PL}}\_{\mathrm{CL}}(\top \sqsubseteq \bot, m'')) \\ & \leftrightarrow & \tau^{\mathrm{PL}}\_{\mathrm{CL}}(\top \sqsubseteq a, m') \\ \equiv & \forall m', P(m', m) : \quad (\forall m'', P(m'', m') : \exists m''', P(m''', m'') : P(m''', a) \vee \bot) \\ & \leftrightarrow & P(m', a) \\ \equiv & \forall m', P(m', m) : \quad (\forall m'', P(m'', m) : \exists m''', P(m''', m'') : P(m''', a)) \\ & \leftrightarrow & P(m', a) \end{array}$$

The proof follows immediately by (2).

In summary, we needed (35) for proving idempotency (48) and the neutral element (51). Associativity (49) and commutativity (50) were proven without using (35).

We have thus shown that ⊓ and ⊔ each create a semilattice structure over the *x*∈**V** *<sup>C</sup>*. When we prove the absorption laws, we see that the absorption law (52) can be proven without requiring (35), while the proof for the absorption law (53) uses it:

$$a \sqcap (a \sqcup b) = a \tag{52}$$

$$a \sqcup (a \sqcap b) = a \tag{53}$$

For (52):

$$\begin{aligned} &\pi\_{\text{CL}}^{\text{PL}}(a \sqcap (a \sqcup b) = a, m) \\ &\equiv \quad \forall m', P(m', m) : (P(m', a) \wedge \forall m'', P(m'', m') : \\ &\qquad (\exists m''', P(m''', m'') : P(m''', a)) \vee (\exists m''', P(m''', m'') : P(m''', b))) \\ &\leftrightarrow P(m', a) \end{aligned}$$

we show that for any *m*<sup>0</sup> :

$$\begin{aligned} P(m',a) \land \forall m'', P(m'',m') &: (\exists m''', P(m''',m'') : P(m''',a)) \\ &\qquad \lor (\exists m''', P(m''',m'') : P(m''',b)) \\ &\equiv P(m',a) \end{aligned}$$

Proof (⫤): this holds because of transitivity (33) and reflexivity (34) of *P*. If *P m*<sup>0</sup> ð Þ , *<sup>a</sup>* , is true, we know <sup>∀</sup>*m*″, *P m*″, *<sup>m</sup>*<sup>0</sup> ð Þ : ð Þ <sup>∃</sup>*m*‴, *P m*ð Þ ‴, *<sup>m</sup>*″ : *P m*ð Þ ‴, *<sup>a</sup>* is also true and thus also the disjunct. Therefore, the whole consequent must be true.

Proof (⊨): here we already know *P m*<sup>0</sup> ð Þ , *a* in the antecedent, so the consequent cannot be false.

We prove (53):

$$\begin{aligned} \tau^{\mathrm{PL}}\_{\mathrm{CL}}(a \sqcup (a \sqcap b) &= a, m) \\ \equiv & \forall m', P(m', m) : (\forall m'', P(m'', m') : (\exists m''', P(m''', m'') : P(m''', a))) \\ & \quad \vee (\exists m''', P(m''', m'') : P(m''', a) \land P(m''', b))) \\ \leftrightarrow P(m', a) \end{aligned}$$

by showing for any *m*<sup>0</sup> :

$$\begin{aligned} \forall m'', P(m'', m') &: (\exists m''', P(m''', m'') : P(m''', a)) \\ &\qquad \qquad \qquad \qquad \lor (\exists m''', P(m''', m'') : P(m''', a) \land P(m''', b)) \equiv P(m', a) \end{aligned}$$

Proof (⊨): <sup>∀</sup>*m*″, *P m*″, *<sup>m</sup>*<sup>0</sup> ð Þ : ð Þ <sup>∃</sup>*m*‴, *P m*ð Þ ‴, *<sup>m</sup>*″ : *P m*ð Þ ‴, *<sup>a</sup>* ∨ ∃<sup>ð</sup> *<sup>m</sup>*‴, *P m*ð Þ ‴, *<sup>m</sup>*″ : *P m*ð Þ ‴, *<sup>a</sup>* <sup>∧</sup> *P m*ð ÞÞ ‴, *<sup>b</sup>* is true iff <sup>∀</sup>*m*″, *P m*″, *<sup>m</sup>*<sup>0</sup> ð Þ : ð Þ <sup>∃</sup>*m*‴, *P m*ð Þ ‴, *<sup>m</sup>*″ : *P m*ð Þ ‴, *<sup>a</sup>* is true, and this entails *P m*<sup>0</sup> ð Þ , *a* by (2). Proof (⫤): this holds by transitivity and reflexivity.

The relation between the residual and the t-norm were covered by two additional axioms above: continuity (11) and pre-linearity (12):

$$
\pi \sqcap y \sqsubseteq z \text{ iff } \pi \sqsubseteq (y \Rightarrow z) \tag{54}
$$

$$(\mathfrak{x}\Rightarrow\mathfrak{y})\sqcup(\mathfrak{y}\Rightarrow\mathfrak{x})=\mathsf{T}\tag{55}$$

We prove continuity (54) by translation using *τ*PL CL.

$$\begin{split} &\tau\_{\mathrm{CL}}^{\mathrm{PL}}(\mathfrak{x}\sqcap\mathfrak{y}\sqsubseteq\mathfrak{z},m) \equiv \tau\_{\mathrm{CL}}^{\mathrm{PL}}(\mathfrak{x}\sqsubseteq(\mathfrak{y}\Rightarrow\mathfrak{z}),m) \text{ translates into } \mathtt{z} \\ &\forall m',P(m',m):P(m',\mathfrak{x})\wedge P(m',\mathfrak{y}) \to P(m',\mathfrak{z}) \\ \equiv &\forall m',P(m',\mathfrak{x}) \rightarrow \forall m'',P(m'',m'):\exists m''',P(m''',m''): \\ &\left(\neg \exists m''',P(m^{iv},m'''):P(m^{iv},\mathfrak{y})\right) \vee P(m''',\mathfrak{z}) \end{split}$$

Proof (⊨): assume the antecedent holds, and *P m*<sup>0</sup> ð Þ , *x* for some *m*<sup>0</sup> . Then, for <sup>∀</sup>*m*″, *P m*ð Þ ‴, *<sup>m</sup>*″ : … *P m*ð Þ ‴, *<sup>z</sup>* to be false, there must be an *<sup>m</sup>*<sup>00</sup> <sup>1</sup> , *P m*<sup>00</sup> <sup>1</sup> , *<sup>m</sup>*<sup>0</sup> , so that <sup>∀</sup>*m*‴, *P m*‴, *<sup>m</sup>*<sup>00</sup> 1 : <sup>∃</sup>*miv*, *P miv*, *<sup>m</sup>*‴ : *P miv*, *<sup>y</sup>* and <sup>∀</sup>*m*‴, *P m*‴, *<sup>m</sup>*<sup>00</sup> 1 : <sup>¬</sup>*P m*ð Þ ‴, *<sup>z</sup>* . However, if <sup>∀</sup>*m*‴, *P m*‴, *<sup>m</sup>*<sup>00</sup> 1 : <sup>∃</sup>*miv*, *P miv*, *<sup>m</sup>*‴ : *P miv*, *<sup>y</sup>* holds then by (2), *P m*<sup>00</sup> <sup>1</sup> , *<sup>y</sup>* and by transitivity also *P m*<sup>00</sup> <sup>1</sup> , *<sup>x</sup>* and by the assumption thus *P m*<sup>00</sup> <sup>1</sup> , *<sup>z</sup>* , which cannot hold since all parts *<sup>m</sup>*‴ of *<sup>m</sup>*<sup>00</sup> <sup>1</sup> including by reflexivity *m*<sup>00</sup> <sup>1</sup> itself fulfill <sup>¬</sup>*P m*ð Þ ‴, *<sup>z</sup>* .

Proof (⫤): assume the antecedent ∀*m*<sup>0</sup> , *P m*<sup>0</sup> ð Þ , *<sup>x</sup>* : … *P m*ð Þ ‴, *<sup>z</sup>* holds. For ∀*m*<sup>0</sup> , *P m*<sup>0</sup> ð Þ , *m* : *P m*<sup>0</sup> ð Þ , *x* ∧ *P m*<sup>0</sup> ð Þ! , *y P m*<sup>0</sup> ð Þ , *z* to be false, there must be *m*<sup>0</sup> 1, *P m*<sup>0</sup> 1, *<sup>m</sup>* , so that *P m*<sup>0</sup> 1, *<sup>x</sup>* and *P m*<sup>0</sup> 1, *<sup>y</sup>* must hold, but *P m*<sup>0</sup> 1, *<sup>z</sup>* must be false. But then we also know that <sup>∃</sup>*m*‴, *P m*ð Þ ‴, *<sup>m</sup>*″ : <sup>¬</sup>∃*miv*, *P miv*, *<sup>m</sup>*‴ : *P miv*, *<sup>y</sup>* cannot hold for any

*Towards a Fuzzy Context Logic DOI: http://dx.doi.org/10.5772/intechopen.95624*

*<sup>m</sup>*″, *P m*″, *<sup>m</sup>*<sup>0</sup> 1 . Thus, <sup>∃</sup>*m*‴, *P m*ð Þ ‴, *<sup>m</sup>*″ : *P m*ð Þ ‴, *<sup>z</sup>* must hold for all *<sup>m</sup>*″, and thus by (2) *P m*<sup>0</sup> 1, *<sup>z</sup>* .

We prove pre-linearity (55):

$$\begin{split} \tau^{\mathsf{PL}}\_{\mathrm{CL}}((\boldsymbol{\omega}\Rightarrow\boldsymbol{\mathsf{y}})\sqcup(\boldsymbol{\up}\Rightarrow\boldsymbol{\mathsf{x}}) &\equiv \\ \forall m^{\prime},P(m^{\prime},\boldsymbol{m}): [\exists\!m^{\prime},P(m^{\prime},\boldsymbol{m}^{\prime}): \forall m^{\prime\prime},P(m^{\prime\prime},\boldsymbol{m}^{\prime\prime}): \exists\!m^{\prime\prime},P(m^{\prime\prime},\boldsymbol{m}^{\prime\prime}): \\ &\Big{(}\cdot\exists\!m^{\prime\prime},P(m^{\prime\prime},\boldsymbol{m}^{\prime\prime}): P(m^{\prime\prime},\boldsymbol{\up}\}\vee P(m^{\dot{\boldsymbol{m}}},\boldsymbol{\up}\}] \\ \forall [\exists\!m^{\prime\prime},P(m^{\prime\prime},\boldsymbol{m}^{\prime\prime}): \forall m^{\prime\prime},P(m^{\prime\prime},\boldsymbol{m}^{\prime\prime}): \exists\!m^{\prime\prime},P(m^{\dot{\boldsymbol{m}}},\boldsymbol{m}^{\prime\prime}): \\ &\Big{(}\cdot\exists\!m^{\prime\prime},P(m^{\prime\prime},\boldsymbol{m}^{\dot{\boldsymbol{m}}}): P(m^{\prime},\boldsymbol{\up})) \lor P(m^{\dot{\boldsymbol{m}}},\boldsymbol{\up})] \end{split}$$

by showing for any*m*<sup>0</sup> , *P m*<sup>0</sup> ð Þ , *m* :

$$\begin{split} & \neg[\exists m'', P(m'', m') : \forall m''', P(m''', m'') : \exists m^{iv}, P(m^{iv}, m''') : \\ & \qquad \left( \neg \exists m'', P(m^v, m^{iv}) : P(m^v, \infty) \right) \lor P(m^{iv}, \jmath) ] \\ & \models [\exists m'', P(m'', m') : \forall m''', P(m''', m'') : \exists m^{iv}, P(m^{iv}, m''') : \\ & \qquad \left( \neg \exists m'', P(m^v, m^{iv}) : P(m^v, \jmath) \right) \lor P(m^{iv}, \infty)] \end{split}$$

Proof: we obtain for the antecedent:

$$\begin{aligned} \forall m'', P(m'', m') : \exists m''', P(m''', m'') : \forall m^{iv}, P(m^{iv}, m'') : \left(\exists m'', P(m^v, m^{iv}) : P(m^v, \infty)\right) \land \\ \neg P(m^{iv}, y) \end{aligned}$$

Since this holds for all *<sup>m</sup>*″, *P m*″, *<sup>m</sup>*<sup>0</sup> ð Þ it also holds for *<sup>m</sup>*<sup>0</sup> itself, i.e., it follows that:

$$\exists m''' , P(m''', m') : \forall m^{iv}, P(m^{iv}, m''') \; : \; \left( \exists m^v, P(m^v, m^{iv}) \; : \; P(m^v, \ge) \right) \land \neg P(m^{iv}, y) \; :$$

We rename the variables to better show the structure:

$$\begin{split} & \exists m'', P(m'', m') : \forall m''', P(m'', m'') : \neg P(m''', \jmath) \land \left( \exists m^{iv}, P(m^{iv}, m'') : P(m^{iv}, \varkappa) \right) \\ \equiv & \exists m'', P(m'', m') : \left( \forall m''', P(m'', m'') : \neg P(m''', \jmath) \right) \land \\ & \qquad \left( \forall m''', P(m''', m'') : \left( \exists m^{iv}, P(m^{iv}, m''') : P(m^{iv}, \varkappa) \right) \right) \end{split}$$

and by (2):

$$\equiv \exists m'', P(m'', m') : (\neg \exists m''', P(m''', m'') : P(m''', y)) \land P(m'', \varkappa).$$

We now know that *<sup>m</sup>*<sup>0</sup> has a part *<sup>m</sup>*″ that is in *<sup>x</sup>* and none of its parts is in *<sup>y</sup>*. With *P m*ð Þ ″, *<sup>x</sup>* , however we also know by transitivity of *<sup>P</sup>* that all parts *<sup>m</sup>*‴, *P m*ð Þ ‴, *<sup>m</sup>*″ fulfill *P m*ð Þ ‴, *<sup>x</sup>* , and thus by reflexivity of *<sup>P</sup>* that there is an *<sup>m</sup>iv*, *P miv*, *<sup>m</sup>*‴ , namely *miv* <sup>¼</sup> *<sup>m</sup>*‴, for each *<sup>m</sup>*‴, which fulfills *P miv*, *<sup>x</sup>* . Moreover, since all parts *<sup>m</sup><sup>v</sup>*, *P m<sup>v</sup>*, *<sup>m</sup>iv* are by transitivity also parts of *<sup>m</sup>*″, we know that ¬∃*m<sup>v</sup>*, *P m<sup>v</sup>*, *<sup>m</sup>iv* : *P miv*, *y* and thus:

$$\begin{aligned} \exists m'', P(m'', m') : \forall m''', P(m''', m'') : \exists m^{iv}, P(m^{iv}, m'') : \left( \neg \exists m'', P(m^v, m^{iv}) : P(m^v, y) \right) \land \\ P(m^{iv}, x), \end{aligned}$$

which entails the consequent.

#### **4.5 A note on mereological and ontological status**

The mereologically interested reader may notice that adding even the weakened variant of the weak supplementation principle is sufficient to collapse context logic term structures to a single level by (2). The reason for this is that the weak supplementation principle considerably strengthens the expressiveness of negation, which given the principle always ensures the existence of a fully negative individual. This is the case, although our system mereologically speaking is an MM system, i.e., supports M1-M4 [30] only, with M4 acting as an axiom schema.

We may note also, that we need not ensure product (M5) or sum (M6) to exists, nor do we need or posit a universal ⊤ or null object ⊥ to exist. The symbols ⊤, ⊥, ⊓ , ⊔ , � are, so to speak, "syntactic sugar" only. The assumed mereology thus is slightly weaker than MM and ontologically careful and minimalistic. For a deeper discussion, cf. [30, 31].

#### **4.6 Example: set-theoretical model**

To make the discussion more concrete, we briefly sketch a set-theoretical interpretation. An example of a suitable model is the set-theoretic lattice, assuming the set of all subsets of a base universe as the universe for the interpretation of the translation *τ*PL CLð Þ *ϕ*, *m* of a context formula *ϕ*, and mapping *t* to ∩ (17), *s* to ∪ (18), and the residual *r* according to (19). Note that, within this interpretation, the variables *m*, *m*<sup>0</sup> , etc., as well as the constants *a*, *b*,*c*, etc. of the translation *τ*PL CLð Þ *ϕ*, *m* range over sets, not elements, of the base universe. With a set-theoretical model ð Þ *<sup>I</sup>*, *<sup>i</sup>*, *<sup>a</sup>*, *<sup>U</sup>*, <sup>⊆</sup>, *<sup>B</sup>*, <sup>∩</sup>, <sup>∪</sup> , where *<sup>U</sup>* <sup>⊆</sup> <sup>2</sup>*<sup>B</sup>* for *<sup>B</sup>*, the base set, we get:

$$i(v) = a(v) \text{ for } v \in \mathcal{V}\_C$$

$$i(\sim c) = \mathbf{1}\_T - i(c) = B - i(c)$$

$$i(c \sqcap d) = t(i(c), i(d)) = i(c) \sqcap i(d)$$

$$i(c \sqcup d) = s(i(c), i(d)) = i(c) \cup i(d)$$

$$I(c \sqsubseteq d) = \mathbf{1} \text{ iff } i(c) \sqsubseteq i(d)$$

$$I(\neg \phi) = \mathbf{1} - I(\phi)$$

$$I(\phi \wedge \psi) = \min\left(I(\phi), I(\psi)\right)$$

$$I(\phi \vee \psi) = \max\left(I(\phi), I(\psi)\right)$$

We can show that, if the canonical interpretation ð Þ *I*, *i*, *a*, *U*, ⊆, *B*, ∩, ∪ is a model for formula *ϕ*, then there is a corresponding predicate logic model for *τ*PL CLð Þ *ϕ*, *m* with interpretations for *<sup>m</sup>*, *<sup>m</sup>*<sup>0</sup> from *<sup>U</sup>* <sup>⊆</sup>2*<sup>B</sup>* � f g <sup>∅</sup> , interpreting *<sup>P</sup>* as <sup>⊆</sup> and individuals *<sup>v</sup>*∈*VC* using an assignment function *<sup>a</sup>* : **<sup>V</sup>** *<sup>C</sup>* ! <sup>2</sup>*<sup>B</sup>*. While we allow the constants *v*∈*VC* to be empty, the variables used to describe their extension cannot.

The pre-order axioms for *P* obviously hold for ⊆. Also, the weak supplementation principle holds for ⊆:

$$\begin{array}{c} \forall \mathsf{x}: \ (\forall \mathsf{x'} \subseteq \mathsf{x}: \mathsf{x'} \subseteq a \to \mathsf{x'} \subseteq b) \\ \qquad \land (\exists \mathsf{x'} \subseteq \mathsf{x}: \mathsf{x'} \subseteq b \land \neg \mathsf{x'} \subseteq a) \\ \quad \rightarrow \left(\exists \mathsf{x'} \subseteq \mathsf{x}: \mathsf{x'} \subseteq b \land \neg \exists \mathsf{x'}' \subseteq \mathsf{x'}: \mathsf{x'} \subseteq a\right). \end{array}$$

Proof: assume a set *x*<sup>0</sup> ⊆ *x* supports that *x*<sup>0</sup> ⊆*a* implies *x*<sup>0</sup> ⊆*b*, and there is a set *x*0 <sup>1</sup> ⊆*x* supporting *x*<sup>0</sup> <sup>1</sup> ⊆*b* but not *x*<sup>0</sup> <sup>1</sup> ⊆*a*. We can then construct *x*<sup>0</sup> <sup>2</sup> ⊆*x* as *x*<sup>0</sup> <sup>2</sup> ¼ *x*<sup>0</sup> <sup>1</sup> � *a*, which supports that *x*<sup>0</sup> <sup>2</sup> ⊆*b* and none of its subsets *x*<sup>00</sup> <sup>2</sup> ⊆*x*<sup>0</sup> <sup>2</sup> supports *x*<sup>00</sup> <sup>2</sup> ⊆*a*.

We prove that ∩, ∪, � over non-empty sets *m*, *m*<sup>0</sup> , *<sup>m</sup>*″ fulfill the characteristic properties for translations for ⊓ , ⊔ , �, respectively:

$$m \subseteq a \cap b \quad \equiv m \subseteq a \land m \subseteq b \tag{56}$$

$$m \subseteq a \cup b \; \equiv \forall m' \subseteq m : \exists m'' \subseteq m' : m'' \subseteq a \lor m'' \subseteq b \tag{57}$$

$$\in m\subseteq \simeq a \quad \equiv \forall m' \subseteq m : \neg \exists m'' \subseteq m' : m'' \subseteq a \tag{58}$$

The case of (20) is immediately clear. For (21), we look at the definition of ∪ in terms of elements *P*∈*B*, which we call *points*:

$$\begin{aligned} \forall m \subseteq a \cup b &\equiv \forall P \in m : P \in a \lor P \in b \\ &\equiv \forall m' \subseteq m : \exists P \in m' : P \in a \lor P \in b \\ &\equiv \forall m' \subseteq m : \exists m'' \subseteq m' : m'' \subseteq a \lor m'' \subseteq b \end{aligned}$$

Proof (⊨): if the points in *m* are in *a* or in *b* in the first step, then, since the *m*<sup>0</sup> are non-empty, it follows that each *m*<sup>0</sup> ⊆ *m* has a point either in *a* or in *b*. In the second step, if there is a point *P* in each *m*<sup>0</sup> , that is in *<sup>a</sup>* or in *<sup>b</sup>*, then there is a set *<sup>m</sup>*″ <sup>⊆</sup> *<sup>m</sup>*<sup>0</sup> , namely the singleton containing *<sup>P</sup>*, for which *<sup>m</sup>*″<sup>⊆</sup> *<sup>a</sup>* or *<sup>m</sup>*″⊆*<sup>b</sup>* must hold.

Proof (⫤): assume that for every *m*<sup>0</sup> , there is a non-empty *<sup>m</sup>*″ <sup>⊆</sup> *<sup>m</sup>*<sup>0</sup> , with *<sup>m</sup>*″<sup>⊆</sup> *<sup>a</sup>* or *<sup>m</sup>*″<sup>⊆</sup> *<sup>b</sup>*, then, since *<sup>m</sup>*″ non-empty, it must have a point *<sup>P</sup>*<sup>∈</sup> *<sup>a</sup>* or *<sup>P</sup>*<sup>∈</sup> *<sup>b</sup>*. Since this holds for all non-empty sets *m*<sup>0</sup> , including all singleton sets, which have only one element, it must hold for all points *P*∈ *m*.

For (22), we similarly look at the definition of � in terms of elements of *m*, i.e., points:

$$\begin{aligned} m \subseteq \sim a &\equiv \forall P \in m : \neg P \in a \\ &\equiv \forall m' \subseteq m : \neg \exists P \in m' : P \in a \\ &\equiv \forall m' \subseteq m : \neg \exists m'' \subseteq m' : m' \subseteq a' \end{aligned}$$

Proof (⊨): the property carries over to all parts *m*<sup>0</sup> of *m* in the second step. The third step follows, because any set *<sup>m</sup>*″<sup>⊆</sup> *<sup>m</sup>*<sup>0</sup> must be non-empty, and if it contains a point, *<sup>m</sup>*″<sup>⊆</sup> *<sup>a</sup>* cannot be true, since no point in *<sup>m</sup>*<sup>0</sup> is in *<sup>a</sup>* and <sup>⊆</sup> is transitive.

Proof (⫤): as in the proof for ⊔ , we can argue over singleton sets. If for all sets *m*0 , no subset *<sup>m</sup>*″ is subset of *<sup>a</sup>*, then this also holds for the singletons, and thus no set *m*<sup>0</sup> has a point *P* in *a*, but this again holds also for singleton sets *m*<sup>0</sup> ⊆ *m*, and thus all points of *m* are outside *a*.

We have thus seen that the set-thoretical standard model is a concrete example of a structure for interpreting context terms and formulae.
