## 1. Fuzzy connectives based on t-norms



The standard set of truth-values for fuzzy logics is the real unit
interval \([0,1]\) with its natural ordering \(\leq\), ranging from
total falsity (represented by \(0\)) to total truth (represented by
\(1\)) through a continuum of intermediate degrees of truth (see
Sections
 5
 and
 7
 for alternative interpretations of truth constants and alternative
sets of truth-values). A fundamental assumption of (mainstream)
mathematical fuzzy logic is that connectives are to be interpreted
truth-functionally over the set of degrees of truth. Such
truth functions are assumed, in the standard setting, to behave
classically on the extremal values \(0\) and \(1\). A very natural
behavior of conjunction, disjunction, and negation is achieved by
imposing \(x \land y = \min\{x,y\}\), \(x \lor y = \max\{x,y\}\), and
\(\neg x = 1 - x\) for each \(x,y \in [0,1]\).



These three truth functions yield the original semantics of fuzzy
logic proposed by Joseph Goguen (1975), later studied formally by,
e.g., Gehrke, Walker, & Walker (1997). Many researchers still
refer to it as "the fuzzy logic". However, it has soon
become apparent that this semantic framework is too poor to (1)
encode/model numerous interesting aspects of reasoning and (2) to
support a fully fledged theory of mathematical logic. 



Another, non-idempotent, conjunction \(\&\) is typically added to
account for the intuition that applying a partially true hypothesis
twice might lead to a different degree of truth than using it only
once. Such a conjunction is usually interpreted by a binary operation
on \([0,1]\), which is not necessarily idempotent, but still
associative, commutative, non-decreasing in both arguments and has
\(1\) as neutral element. These operations are called *t-norms*
(*triangular norms*) and their mathematical properties have
been thoroughly studied (see Klement, Mesiar, & Pap 2000).
Prominent examples of t-norms are the already mentioned function
\(\min\), the standard product \(\cdot\) of real numbers, and the
Lukasiewicz t-norm: \(x \*\_{L} y=\max\{x+y-1,0\}\). These
three t-norms are actually continuous functions and any other
continuous t-norm can be described as an *ordinal sum* of these
three basic ones (Ling 1965; Mostert & Shields 1957).



Interestingly, each left-continuous t-norm determines a suitable
choice for implication. Indeed, it is known that a t-norm \(\ast\) is
left-continuous if, and only if, there is a unique binary operation
\(\Rightarrow\_\ast\) on \([0,1]\) satisfying the so-called
*residuation condition:* 

\[
x \ast y \leq z\ \text{ if and only if }\ x \leq y \Rightarrow\_\ast z.
\] 

 The function
\(\Rightarrow\_\ast\) is known as *residuum of \(\ast\)*, and it
can be shown that \(x \Rightarrow\_\ast y = \max\{z \mid x \ast z \leq
y\}\). The residua of the three mentioned prominent t-norms are: \(x
\Rightarrow\_\min y = 1\) if \(x\leq y\) and \(y\) otherwise; \(x
\Rightarrow\_\cdot y = 1\) if \(x\leq y\) and \(\frac{y}{x}\)
otherwise; and \(x \Rightarrow\_{L} y = \min\{1, 1 - x+y\}\).



In the general t-norm setting, the negation is interpreted using the
residuum. Namely, for a given left-continuous t-norm \(\ast\) we set
\(\neg\_\ast x = x\Rightarrow\_\ast 0\). For the Lukasiewicz t-norm
we obtain the negation \(\neg\_{L} x = 1 - x\) (known as
Lukasiewicz negation or the standard involutive negation) whereas for the
minimum and the product t-norms we obtain the same negation (known as
Godel negation): \(\neg\_\mathrm{G} 0 = 1\) and \(0\) otherwise.



## 2. MTL: A fundamental fuzzy logic



The weakest logic with connectives interpreted by truth functions of
the type described above is MTL (*Monoidal T-norm based Logic*,
Esteva & Godo 2001). It is a logic with the primitive binary
connectives \(\mathbin{\&}, \to, \wedge\) and a truth-constant
\(\overline{0}\), and derivable connectives defined as: 

\[
\begin{align}
\varphi \lor \psi &= ((\varphi \to \psi) \to \psi) \land ((\psi \to \varphi) \to \varphi),\\
\neg \varphi &= \varphi \to \overline{0}, \\
\varphi \leftrightarrow \psi &= (\varphi \to \psi) \land (\psi \to \varphi),\\
\overline{1} &= \neg \overline{0}.
\end{align}
\]

MTL is defined as a consequence relation by means of the semantics
given by all left-continuous t-norms. Namely, given a particular
left-continuous t-norm \(\ast\) and a mapping from propositional
variables to \([0,1]\), we define the \(\ast\)-evaluation of all
formulas by interpreting \(\&\) as \(\ast\), the implication
\(\to\) as its residuum \(\Rightarrow\), and \(\land\) and
\(\overline{0}\) as \(\min\) and \(0\), respectively.



A formula \(\varphi\) is a consequence of a set of formulas \(\Gamma\)
in MTL, denoted \(\Gamma \models\_\mathrm{MTL} \varphi\), if for each
left-continuous t-norm \(\ast\) and each \(\ast\)-evaluation \(e\)
such that \(e(\psi) = 1\) for each \(\psi \in \Gamma\) we have
\(e(\varphi) = 1\); in words, if the premises are totally true, so is
the conclusion. A formula \(\varphi\) that always evaluates to \(1\)
(i.e. \(\emptyset\models\_\mathrm{MTL}\varphi\)) is called valid or a
tautology of MTL. Note that the formula \(\varphi \mathbin{\&}
\psi \to \varphi \land \psi\) is a tautology in MTL but the converse
implication \(\varphi \land \psi \to \varphi \mathbin{\&} \psi\)
is not, i.e., the conjunction \(\&\) is stronger than
\(\land\).



MTL can also be presented by a Hilbert-style proof system with the
following axioms:

 

\[
\begin{align}
 (\varphi \to \psi) & \to ((\psi \to \chi) \to (\varphi \to \chi)) \\
 \varphi \mathbin{\&} \psi & \to \varphi \\
 \varphi\mathbin{\&}\psi & \to\psi\mathbin{\&}\varphi \\
 \varphi \land \psi & \to \varphi \\
 \varphi \land \psi & \to \psi \land \varphi \\
 (\chi \to \varphi) & \to ((\chi\to \psi) \to (\chi \to \varphi\wedge \psi)) \\
 (\varphi\mathbin{\&}\psi \to \chi) & \to (\varphi\to(\psi \to \chi)) \\
 (\varphi\to(\psi \to \chi)) & \to (\varphi\mathbin{\&}\psi \to \chi) \\
 ((\varphi \to \psi) \to \chi) & \to (((\psi \to \varphi)\to \chi)\to \chi) \\
 \overline{0}& \to\varphi \\
\end{align}
\]


and *modus ponens* as the only inference rule: from \(\varphi\)
and \(\varphi \to \psi\), infer \(\psi\). We write \(\Gamma
\vdash\_\mathrm{MTL} \varphi\) whenever there is a proof of \(\varphi\)
from \(\Gamma\) in this system, i.e. there is a finite sequence of
formulas which ends with \(\varphi\) and its members are instances of
the axioms, elements of \(\Gamma\), or follow from previous ones by
the inference rule (*modus ponens*). This Hilbert-style system
is a strongly complete (finitary) axiomatization of the logic MTL,
i.e. for each set of premises \(\Gamma\) we have: \(\Gamma
\models\_\mathrm{MTL} \varphi\) iff \(\Gamma \vdash\_\mathrm{MTL}
\varphi\). The validity problem of \(\mathrm{MTL}\) is known to be
decidable, however its computational complexity has not yet been
determined.


## 3. Lukasiewicz logic




Lukasiewicz logic can be defined by adding the axiom (known as Wajsberg axiom)\[((\varphi \to \psi)
\to \psi) \to ((\psi \to \varphi) \to \varphi)\] to the Hilbert-style proof system for MTL. It corresponds to the finitary version of

 the consequence relation defined with respect to
evaluations based on the Lukasiewicz t-norm (in symbols: for each
*finite* set of formulas \(\Gamma\) and each formula
\(\varphi\), we have \(\Gamma \models\_{L} \varphi\) iff \(\Gamma
\vdash\_{L}
 \varphi\)).[2]




This logic was an early example of a many-valued logic, introduced by
Jan Lukasiewicz and Alfred Tarski (1930), well before the
inception of the theory of fuzzy sets, by means of an equivalent
axiomatic system (with *modus ponens* as the only inference
rule):

 

\[
\begin{align}
\varphi &\to (\psi \to \varphi)\\
(\varphi \to \psi) &\to ((\psi \to \chi) \to (\varphi \to \chi))\\
((\varphi \to \psi) \to \psi) &\to ((\psi \to \varphi) \to \varphi)\\
(\neg \psi \to \neg \varphi) &\to (\varphi \to \psi)\\
((\varphi \to \psi) &\to (\psi \to \varphi)) \to (\psi \to \varphi). 
\end{align}
\]


Lukasiewicz logic is the only t-norm based fuzzy logic where all
connectives are interpreted by continuous functions, including the
implication given by the function \(x \Rightarrow\_{L}
y=\min\{1,1-x+y\}\). McNaughton's theorem (1951) states that
real-valued functions over [0,1] that interpret formulas of
Lukasiewicz logic are exactly the continuous piecewise linear
functions with integer coefficients. In terms of computational
complexity, the validity problem for this logic is asymptotically not
worse than in classical logic: it remains coNP-complete.


## 4. Godel-Dummett logic



Godel-Dummett logic, also known as Dummett's LC or
simply as Godel logic, is another early example of a many-valued
logic with truth-values in \([0,1]\). It was introduced by Michael
Dummett (1959) as the extension of
 intuitionistic logic
 by the axiom 

\[(\varphi \to \psi) \lor (\psi \to \varphi).\]

 This formula enforces a linear order in the
underlying (Kripke-style as well as algebraic) semantics. It also
appears in the context of Kurt Godel's observation that it
is impossible to characterize intuitionistic logic by finite truth
tables (Godel 1932). In the fuzzy logic setting,
Godel-Dummett logic can alternatively be obtained as an
axiomatic extension of MTL by adding the axiom \(\varphi \to \varphi
\mathbin{\&} \varphi\), which amounts to requiring the idempotence
of \(\&\), and hence the interpretation of both conjunctions
coincides. Semantically, Godel-Dummett logic can also be
defined as the consequence relation given by the minimum t-norm. It is
distinguished as the only t-norm based logic where the validity of a
formula in a given evaluation does not depend on the specific values
assigned to the propositional variables, but only on the relative
order of these values. In this sense, Godel-Dummett logic
can be seen as a logic of comparative truth. Like for Lukasiewicz
logic, the computational complexity of testing validity remains
coNP-complete.


## 5. Other notable fuzzy logics



Besides MTL (the logic of all left-continuous t-norms) and
Lukasiewicz and Godel-Dummett logics (each induced by
a particular t-norm), one can consider logics *induced* by sets
of t-norms or, in general, arbitrary axiomatic extensions of MTL. In
particular, the logic of all *continuous* t-norms
(Hajek's basic fuzzy logic BL) is obtained by adding the
axiom 

\[\varphi\mathbin{\&}(\varphi\to{{\psi}}) \to
\psi\mathbin{\&}(\psi\to\varphi)\]

 to those of MTL. Actually, for any set of continuous
t-norms there is a finite axiomatization of the corresponding logic
(Esteva, Godo, & Montagna 2003; Hanikova 2014); in most
cases the axiomatization captures semantic consequence from
*finite* sets of premises. In particular, the logic of the
third prominent continuous t-norm (algebraic product), known as
Product logic, is the extension of BL by the axiom 

\[\neg\varphi \vee ((\varphi\to\varphi\mathbin{\&}{{\psi}})\to{{\psi}}).\]

 On the
other hand, not all axiomatic extensions of MTL can be given a t-norm
based semantics. For example, classical logic can be axiomatized as
MTL \(+\) \(\varphi\vee\neg \varphi\), but the axiom of excluded
middle is not a tautology under any t-norm based interpretation.



There are also reasons to consider weaker fuzzy logics. For example,
it can be argued that the assumptions forcing the interpretation of
the conjunction & to be a t-norm are too strong. In particular,
the assumption that \(1\) is its neutral element enforces a definition
of tautology as a formula always evaluated to \(1\) and the
consequence relation as preservation of the value \(1\) - that
is, \(1\) is the only *designated value* in the
 semantics.[3]
 A natural way to introduce logics with more than one designated
truth-degree is to assume that the neutral element for the
interpretation of & is a number \(t <1\). It can be shown that
in this situation the designated truth-degrees are exactly those
greater than or equal to \(t\). Such interpretations of conjunctions
are called *uninorms*. The logic of left-continuous uninorms
was axiomatized in Metcalfe & Montagna (2007).



Analogously, one may argue against commutativity or even against
associativity of &. Axiomatizations of resulting logics are
described in the literature (see Cintula, Horcik, &
Noguera 2013; Jenei & Montagna 2003); an exception is the logic of
non-commutative uninorms for which no natural axiomatic system is
known.



Finally, taking into account that fuzzy logics, unlike classical
logic, are typically not functionally complete, one can increase their
expressive power by adding new connectives. The most commonly
considered connectives are: truth-constants \(\bar r\) for each
rational number \(r\in (0,1)\); unary connectives \(\sim\) and
\(\triangle\) interpreted as \({\sim}x = 1-x\) and \(\triangle x = 1\)
if \(x=1\) and \(0\) otherwise; a binary connective \(\odot\)
interpreted as the usual algebraic product, etc. (Baaz 1996; Esteva,
Gispert, Godo, & Noguera 2007; Esteva, Godo, & Montagna 2001;
Esteva, Godo, Hajek, & Navara 2000).



A thorough overview of all the kinds of propositional fuzzy logics
mentioned in this section (and a general theory thereof) can be found
in the *Handbook of Mathematical Fuzzy Logic* (3 volumes,
Cintula et al. 2011a,b, 2015).


## 6. Predicate logics



The propositional logic MTL can be given a first-order counterpart
MTL\(\forall\) in a predicate language \(\mathcal{P\!L}\) (defined as
in the classical case) in the following way. The semantics is given by
structures in which predicate symbols are interpreted as functions
mapping tuples of domain elements into truth-values. More precisely,
given a left-continuous t-norm \(\ast\), a \(\ast\)-structure
\({\mathbf M}\) consists of a non-empty domain of elements \(M\), a
function \(f\_{\mathbf M}\colon M^n\to M\) for each \(n\)-ary function
symbol \(f\in \mathcal{P\!L}\), and a function \(P\_{\mathbf M}\colon
M^n\to [0,1]\) for each \(n\)-ary predicate symbol \(P\in
\mathcal{P\!L}\). Fixing a valuation \({\mathrm v}\) of object
variables in \(M\), one defines values of terms
(\(\|f(t\_1,\dots,t\_n)\|^{\mathbf M}\_{\mathrm v} = f\_{\mathbf
M}(\|t\_1\|^{\mathbf M}\_{\mathrm v},\dots,\|t\_n\|^{\mathbf M}\_{\mathrm
v})\)) and truth-values of atomic formulas
(\(\|P(t\_1,\dots,t\_n)\|^{\mathbf M}\_{\mathrm v} = P\_{\mathbf
M}(\|t\_1\|^{\mathbf M}\_{\mathrm v},\dots,\|t\_n\|^{\mathbf M}\_{\mathrm
v})\)). Truth-values of a universally/existentially quantified formula
are computed as infimum/supremum of truth-values of instances of the
formula where the quantified variable runs over all elements of the
domain \(M\). Formally: 

\[
\begin{align}
\|\forall x\, \varphi\|^{\mathbf M}\_{\mathrm v} & = \inf\{\|\varphi\|^{\mathbf M}\_{{\mathrm v}[x{:}a]} \mid a\in M\}\\
 \|\exists x\,\varphi\|^{\mathbf M}\_{\mathrm v} & = \sup\{\|\varphi\|^{\mathbf M}\_{{\mathrm v}[x{:}a]}\mid a\in M\},\\
 \end{align}
\]

 where \({\mathrm v}[x{:}a]\) is
the valuation sending \(x\) to \(a\) and keeping values of other
variables unchanged. The values of other formulas are computed by
interpreting, as in propositional semantics, \(\&\) as \(\ast\),
the implication \(\to\) as its residuum \(\Rightarrow\_\ast\), and
\(\land\) and \(\overline{0}\) as \(\min\) and \(0\),
respectively.



The first-order logic MTL\(\forall\) is then defined as the
consequence relation given by the preservation of total truth (value
\(1\)) in \(\ast\)-structures (allowing \(\ast\) to run over all
left-continuous t-norms). More precisely, we say that a first-order
formula \(\varphi\) is a consequence of a set of formulas \(\Gamma\)
(in symbols: \(\Gamma \models\_{\mathrm{MTL}\forall} \varphi\)) if, for
each left-continuous t-norm \(\ast\) and each \(\ast\)-structure
\({\mathbf M}\), we have \(\|\varphi\|^{\mathbf M}\_{\mathrm v} = 1\)
for each valuation v, whenever \(\|\psi\|^{\mathbf M}\_{\mathrm v} =
1\) for each valuation v and each \(\psi \in \Gamma\).



MTL\(\forall\) can be given a strongly complete Hilbert-style proof
system with the following axioms:


* (P) The first-order
instances of the axioms of MTL
* \((\forall1 )\) 
\(\forall x\,\varphi(x)\to\varphi(t)\), where the term \(t\) is
substitutable for \(x\) in \(\varphi\)
* \((\exists1)\) 
\(\varphi(t)\to \exists x\, \varphi(x)\), where the term \(t\) is
substitutable for \(x\) in \(\varphi\)
* \((\forall2)\) 
\(\forall x\,(\chi\to\varphi)\to(\chi\to \forall x\,\varphi)\), where
\(x\) is not free in \(\chi\)
* \((\exists2)\) 
\(\forall x\,(\varphi\to\chi)\to(\exists x\,\varphi\to\chi)\), where
\(x\) is not free in \(\chi\)
* \((\forall3)\) 
\(\forall x\,(\chi\vee\varphi)\to\chi\vee \forall x\,\varphi\), where
\(x\) is not free in \(\chi\).



The deduction rules of MTL\(\forall\) are *modus ponens* plus
the rule of *generalization:* from \(\varphi\) infer \(\forall
x\,\varphi\).



There are two ways of introducing first-order counterparts for other
propositional t-norm based fuzzy logics. On the one hand, the
axiomatization of MTL\(\forall\) can be extended by adding the
first-order instances of propositional axioms like those seen in the
previous sections. In this manner one obtains syntactic presentations
of first-order variants of, e.g., Lukasiewicz logic,
Godel-Dummett logic, Product logic, or Hajek's
basic fuzzy logic BL. On the other hand, the semantic definition of
MTL\(\forall\) can be easily modified by defining the consequence
relations given by the corresponding (sets of) t-norms. Then, the
natural question is whether, in each case, these two routes lead to
the same logic (as it happened for MTL). For soundness there is no
problem, as the axiomatic systems are easily checked to be sound with
respect to their corresponding classes of structures. As for
completeness, there is no general answer and the problem has to be
considered case by case.



For Godel-Dummett logic the axiomatic system is strongly
complete with respect to its semantics. However, the set of
tautologies of the semantics for Lukasiewicz logic is not
recursively axiomatizable as shown by Bruno Scarpellini (1962). Emil
Ragaz (1981) proved that it is actually \(\Sigma\_2\)-complete in the
sense of arithmetical hierarchy. The situation is even worse in the
case of Product logic and Hajek's basic fuzzy logic BL,
where the sets of first-order tautologies of all structures given by
continuous t-norms are as complex as true arithmetics (Montagna 2001).
Completeness can be achieved either by adding a suitable infinitary
inference rule to the Hilbert-style proof system, as done by Louise
Hay (1963) for Lukasiewicz logic, or by generalizing the set of
truth-values (see next section).



First-order counterparts of weaker fuzzy logics can be studied in
analogous, syntactic and semantic, ways; see the survey presentation
in Cintula, Horcik, & Noguera 2014. 


## 7. Algebraic semantics



One of the main tools in the study of fuzzy logic is that of
 algebraic semantics.
 Roughly speaking, the idea is to replace the real unit interval with
an arbitrary set and to interpret the connectives as operations of
corresponding arities on that set.



An MTL-algebra, introduced by Francesc Esteva and Lluis Godo
(2001), is a tuple \({\mathbf A} = \langle A, \&, \to, \wedge,
\vee, \overline{0}, \overline{1} \rangle\) where


* \(\langle A, \wedge, \vee, \overline{0}, \overline{1} \rangle\) is
a bounded lattice
* \(\langle A, \&, \overline{1} \rangle\) is a commutative
monoid
* \((x\to y)\vee(y\to x) = \overline{1}\)
* \(x\mathbin{\&} y \leq z\) iff \(x \leq y\to z\) (where
\(\leq\) is the lattice order induced by \(\wedge\) or \(\vee\)).



If the lattice order is total, then \({\mathbf A}\) is called an
MTL-chain. It is worth noting that MTL-algebras are a subvariety of
residuated lattices which provide the algebraic semantics for
 substructural logics;
 the first pointer to the tight connection between these two families
of logics. 



MTL-algebras are a generalization of the t-norm based semantics
explained above and provide a sound and complete semantics for
 MTL.[4]




MTL-chains are the basic building blocks of the whole class of
algebras, in the sense that each MTL-algebra can be decomposed as a
*subdirect product* of chains. This implies that the logic is
also complete with respect to the semantics of MTL-chains, which is
then used as the first step in the proof of its completeness with
respect to the t-norm based semantics (Jenei & Montagna 2002).



Algebraic semantics is a universal tool for propositional logics. In
particular, for any arbitrary fuzzy logic studied in the literature
(even those not supporting a t-norm based semantics such as
finite-valued fuzzy logics or the logic of non-commutative uninorms)
one can find a corresponding class of algebras which can be decomposed
as subdirect products of chains. This fact has led Libor Behounek
and Petr Cintula (2006) to propose a definition of fuzzy logics as
logics that are complete with respect to totally ordered algebraic
structures.



The algebraic semantics can be used for first-order fuzzy logics in a
rather straightforward way: just change the definition of a
\(\ast\)-structure of the previous section to a structure that,
instead of the interval [0,1] and \(\ast\) and its residuum, uses an
arbitary MTL-chain and its operation to compute truth-values of
formulas. Given a propositional fuzzy logic, one can use this
generalized semantics to obtain a recursively enumerable set of
tautologies and a strong completeness theorem for a corresponding
first-order Hilbert-style proof system.


## 8. Proof theory



It has been a considerable challenge to come up with analytic proof
systems for fuzzy logics. These are systems that share important
features, like the eliminability of cuts and the subformula property,
with Gentzen's sequent calculi for classical and intuitionistic
logic (see entry on
 the development of proof theory).
 A major breakthrough has been achieved with the introduction of a
hypersequent calculus for Godel-Dummett logic by Arnon
Avron (1991). Hypersequent calculi arise from sequent calculi by
defining inference rules that refer to finite multisets of sequents,
rather than to single sequents. Hypersequents are interpreted, at the
meta-level, as disjunctions of sequents. This interpretation implies
that it is sound to add additional sequents (external weakening) or to
replace multiple copies of the same sequent within a given
hypersequent by a single copy of that sequent (external contraction).
In the case of Godel-Dummett logic, the rules for
introducing logical connectives arise from the rules of
Gentzen's intuitionistic sequent calculus by simply adding
side-hypersequents the upper and lower sequents of the original rule.
For example, the sequent rule for introducing conjunction on the
right-hand side 

\[\frac{\Gamma\_1 \Rightarrow \phi \hspace{3ex} \Gamma\_2 \Rightarrow
\psi}{\Gamma\_1,\Gamma\_2 \Rightarrow \phi \wedge \psi}\]

 where \(\Gamma\_1\) and \(\Gamma\_2\) are
finite sequences of formulas, is turned into the following
hypersequent rule: 

\[\frac{H \mid \Gamma\_1 \Rightarrow \phi \hspace{3ex} H' \mid \Gamma\_2
\Rightarrow \psi}{H \mid H' \mid \Gamma\_1,\Gamma\_2 \Rightarrow \phi
\wedge \psi }\]

 where \(H\) and \(H'\) denote the
side-hypersequents, i.e., finite multisets of sequents added
disjunctively to the exhibited sequents. This by itself does not
change the corresponding logic (intuitionistic logic, in this case).
The crucial additional structural rule is the so-called communication
rule: 

\[\frac{H \mid \Gamma\_1,\Pi\_1 \Rightarrow \Delta\_1 \hspace{3ex} H'
\mid \Gamma\_2,\Pi\_2 \Rightarrow \Delta\_2}{H \mid H' \mid
\Gamma\_1,\Gamma\_2 \Rightarrow \Delta\_1 \mid \Pi\_2,\Pi\_2 \Rightarrow
\Delta\_2}\]

 Here \(\Gamma\_1, \Gamma\_2,\Pi\_1, \Pi\_2\) are finite
sequences of formulas; \(\Delta\_1\) and \(\Delta\_2\) are either single
formulas or remain empty; \(H\) and \(H'\) denote the
side-hypersequents, like above. To understand the impact of the
communication rule, we present a derivation of the crucial axiom
\((\varphi \to \psi) \lor (\psi \to \varphi)\), which features an
application of this rule immediately below the intial (hyper)sequents:

\[
\frac{\displaystyle\frac{\displaystyle\frac{\displaystyle\frac{\displaystyle\frac{\displaystyle\frac{\varphi \Rightarrow \varphi \hspace{5ex} \psi \Rightarrow \psi}{\varphi \Rightarrow \psi \mid \psi \Rightarrow \varphi}}{\Rightarrow \varphi \to \psi \mid \psi \Rightarrow \varphi}}{\Rightarrow \varphi \to \psi \mid\, \Rightarrow \psi \to \varphi}}{\Rightarrow (\varphi \to \psi) \vee (\psi \to \varphi) \mid\, \Rightarrow \psi \to \varphi}}{\Rightarrow (\varphi \to \psi) \vee (\psi \to \varphi) \mid\, \Rightarrow (\varphi \to \psi) \vee (\psi \to \varphi)}}{\Rightarrow (\varphi \to \psi) \lor (\psi \to \varphi)}
\]

 



To obtain a hypersequent calculus for the fundamental fuzzy logic MTL
one has to add the communication rule to a sequent system for the
contraction-free version of intuitionistic logic (a well-studied
example of a
 substructural logic).
 Analytic proof systems for other fuzzy logics, in particular
Lukasiewicz logic, call for a more radical departure from
traditional calculi, where the sequent components of hypersequents are
interpreted differently than intuitionistic or classical sequents.
Also labeled proof systems and various tableau calculi have been
suggested. A detailed presentation of the corresponding state of the
art can be found in Metcalfe, Olivetti, & Gabbay 2008 and Metcalfe
2011.


## 9. Semantics justifying truth-functionality



It is desirable, not only from a philosophical point of view, but also
to a get a better grip on potential applications of fuzzy logics to
relate the meaning of intermediary truth-values and corresponding
logical connectives to basic models of reasoning with vague and
imprecise notions. A number of such semantics that seek to justify
particular choices of truth-functional connectives have been
introduced. Just two of them are briefly described here.



Voting semantics is based on the idea that different agents (voters)
may coherently judge the same proposition differently. The proportion
of agents that accept a proposition \(\varphi\) as true may be seen as
a truth-value. Without further restrictions this does not lead to a
truth-functional semantics, but rather to an assignment of
probabilities to propositions. But if one assigns a fixed *level of
skepticism* to each agent and imposes some natural conditions that
keep the judgments on logically complex statements consistent with
those levels, then one can recover \(\min\), \(\max\), and \(1-x\) as
truth functions for conjunction, disjunction and negation,
respectively. Details can be found in Lawry 1998.



Another intriguing model of reasoning that provides a justification
for all propositional connectives of standard Lukasiewicz logic
has been introduced by Robin Giles (1974). It consists in a game,
where two players, I and you, systematically reduce logically complex
assertions (formulas) to simpler ones according to rules like the
following:


* If I assert \(\varphi \lor \psi\), then I have to assert either
\(\varphi\) or \(\psi\).
* If I assert \(\varphi \land \psi\), then you choose one of the
conjuncts and I have to assert either \(\varphi\) or \(\psi\),
accordingly.
* If I assert \(\varphi \to \psi\), then I have to assert \(\psi\)
if you assert \(\varphi\).



The rules for quantified statements refer to a fixed domain, assuming
that there is a constant symbol for each domain element one
stipulates:


* If I assert \(\forall x\, \varphi(x)\), then I have to assert
\(\varphi(c)\), for a constant \(c\) chosen by you.
* If I assert \(\exists x\, \varphi(x)\), then I have to assert
\(\varphi(c)\), for a constant \(c\) chosen by myself.



The rules for your assertions are dual. At each state of the game an
occurrence of a non-atomic formula in either the multiset of current
assertions by me or by you is chosen and gets replaced by subformulas,
as indicated by these rules, until only atomic assertions remain. A
final game state is then evaluated according to the following betting
scheme.



For each atomic formula there is a corresponding experiment which may
either fail or succeed, but may show dispersion, i.e., it may yield
different results when repeated. A fixed failure probability, called
risk value, is assigned to each experiment and thus to each atomic
formula. The players have to pay \(\$\)1 to the other player for each
of their atomic assertion where the associated experiments fails. For
any game starting with my assertion of \(\varphi\) my expected overall
loss of money, if we both play rationally, can be shown to correspond
to the inverse of the truth-value of \(\varphi\) evaluated in an
interpretation of Lukasiewicz logic that assigns the inverse of
the risk values as truth-values to atomic formulas. In particular, a
formula is valid in Lukasiewicz logic if and only if, for every
risk value assignment, I have a strategy that guarantees that my
expected overall loss at the end of game is \(0\) or negative.



Christian Fermuller and George Metcalfe (2009) have pointed out a
correspondence between optimal strategies in Giles's game and
cut-free proofs in a hypersequent system for Lukasiewicz logic.
The game has also been extended in Fermuller & Roschger 2014
to characterize various types of (semi-)fuzzy quantifiers, intended to
model natural language expressions like "about half" or
"almost all".



Jeff Paris (2000) provided an overview over other semantics supporting
various choices of truth functions; in particular, re-randomizing
semantics (Hisdal 1988), similarity semantics (e.g., Ruspini 1991),
acceptability semantics (Paris 1997), and approximation semantics
(Paris 2000). Let us also mention the resource-based semantics
proposed in Behounek 2009. Moreover there are different forms of
evaluation games for various fuzzy logics besides the one of Giles for
Lukasiewicz logic, outlined above. An overview over those
semantic games was given in Fermuller 2015.


## 10. Fuzzy logic and vagueness



Modeling reasoning with vague predicates and propositions is often
cited as the main motivation for introducing fuzzy logics. There are
many alternative theories of
 vagueness,
 but there is a general agreement that the susceptibility to the
 sorites paradox
 is a main feature of vagueness. Consider the following version of the
paradox:


* (1)\(10^{100}\)
is a huge number.
* (2)If \(n\) is
a huge number, then \(n-1\) is also huge.



On the face of it, it seems not to be unreasonable to accept these two
assumptions. By instantiating \(n\) with \(10^{100}\) in
 (2)
 and applying *modus ponens* with
 (1)
 as the other premise we conclude that \(10^{100}-1\) is huge. By
simply repeating this type of inference we arrive at the unreasonable
statement


* (3)\(0\) is a
huge number.



Fuzzy logic suggests an analysis of the *sorites paradox* that
respects the intuition that statement
 (2),
 while arguably not totally true, is almost true.



There are various ways to model this form of reasoning in t-norm-based
fuzzy logics that dissolve the paradox. For example, one may declare
that any instance of *modus ponens* is sound if the degree of
truth of the conclusion is not lower than that of the strong
conjunction of its premises.[5]
 As indicated, one stipulates that every instance of
 (2)
 is true to degree \(1-\epsilon\), for some very small number
\(\epsilon\). Even if we declare
 (1)
 to be perfectly true, the statement that \(10^{100}-1\) is huge, too,
might then be less than perfectly true without sacrificing the
soundness of instantiation and *modus ponens*. If, moreover,
the degree of truth of the conjunction of two not perfectly true (and
not perfectly false) statements is less than that of each conjunct, we
may safely declare that statement
 (3)
 is perfectly false and nevertheless insist on the soundness of each
step in the indicated chain of inferences. Informally speaking, the
paradox disappears by assuming that repeatedly decreasing some
perfectly huge number by a small amount leads to numbers of which it
is less and less true that they are huge too.



An alternative truth-degree-based solution to the *sorites
paradox* has been proposed in Hajek & Novak
2003. They introduce a new truth-functional connective modeling the
expression "it is almost true that". In this manner they
formalize *sorites*-style reasoning within an axiomatic theory
of an appropriate t-norm-based fuzzy logic.



Nicholas J.J. Smith (2005 and 2008) has argued that the so-called
*closeness principle* captures the essence of vagueness. It
expresses that statements of the same form about indistinguishable
objects should remain close in respect of truth. It is a hallmark of
many approaches to the sorites paradox that employ fuzzy logic that
they are compatible with this
 principle.[6]