Fuzzy Logic

First published Tue Nov 15, 2016; substantive revision Thu Nov 11, 2021

Fuzzy logic is intended to model logical reasoning with vague orimprecise statements like “Petr is young (rich, tall, hungry,etc.)”. It refers to a family ofmany-valued logics, where the truth-values are interpreted as degrees of truth. Thetruth-value of a logically compound proposition, like “Carles istall and Chris is rich”, is determined by the truth-value of itscomponents. In other words, like in classical logic, one imposestruth-functionality.

Fuzzy logic emerged in the context of the theory of fuzzy sets,introduced by Lotfi Zadeh (1965). A fuzzy set assigns a degree ofmembership, typically a real number from the interval $[0,1]$, toelements of a universe. Fuzzy logic arises by assigning degrees oftruth to propositions. The standard set of truth-values (degrees) isthe real unit interval $[0,1]$, where $0$ represents“totally false”, $1$ represents “totallytrue”, and the other values refer to partial truth, i.e.,intermediate degrees of truth.^[1]

“Fuzzy logic” is often understood in a very wide sensewhich includes all kinds of formalisms and techniques referring to thesystematic handling ofdegrees of some kind (see, e.g.,Nguyen & Walker 2000). In particular in engineering contexts(fuzzy control, fuzzy classification, soft computing) it is aimed atefficient computational methods tolerant to suboptimality andimprecision (see, e.g., Ross 2010). This entry focuses on fuzzy logicin a restricted sense, established as a discipline of mathematicallogic following the seminal monograph by Petr Hájek (1998) andnowadays usually referred to as “mathematical fuzzylogic”. For details about the history of different variants offuzzy logic we refer to Bělohlávek, Dauben, & Klir2017.

Mathematical fuzzy logic focuses on logics based on a truth-functionalaccount of partial truth and studies them in the spirit of classicalmathematical logic, investigating syntax, model-theoretic semantics,proof systems, completeness, etc., both, at the propositional and thepredicate level (see Cintula, Fermüller, Hájek, &Noguera 2011 and 2015).

1. Fuzzy connectives based on t-norms

The standard set of truth-values for fuzzy logics is the real unitinterval $[0,1]$ with its natural ordering $\leq$, ranging fromtotal falsity (represented by $0$) to total truth (represented by$1$) through a continuum of intermediate degrees of truth (seeSections5 and7 for alternative interpretations of truth constants and alternativesets of truth-values). A fundamental assumption of (mainstream)mathematical fuzzy logic is that connectives are to be interpretedtruth-functionally over the set of degrees of truth. Suchtruth functions are assumed, in the standard setting, to behaveclassically on the extremal values $0$ and $1$. A very naturalbehavior of conjunction, disjunction, and negation is achieved byimposing $x \land y = \min\{x,y\}$, $x \lor y = \max\{x,y\}$, and$\neg x = 1 - x$ for each $x,y \in [0,1]$.

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

Another, non-idempotent, conjunction $\&$ is typically added toaccount for the intuition that applying a partially true hypothesistwice might lead to a different degree of truth than using it onlyonce. Such a conjunction is usually interpreted by a binary operationon $[0,1]$, which is not necessarily idempotent, but stillassociative, commutative, non-decreasing in both arguments and has$1$ as neutral element. These operations are calledt-norms(triangular norms) and their mathematical properties havebeen 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 theŁukasiewicz t-norm: $x *_{Ł} y=\max\{x+y-1,0\}$. Thesethree t-norms are actually continuous functions and any othercontinuous t-norm can be described as anordinal sum of thesethree basic ones (Ling 1965; Mostert & Shields 1957).

Interestingly, each left-continuous t-norm determines a suitablechoice for implication. Indeed, it is known that a t-norm $\ast$ isleft-continuous if, and only if, there is a unique binary operation$\Rightarrow_\ast$ on $[0,1]$ satisfying the so-calledresiduation condition: \[x \ast y \leq z\ \text{ if and only if }\ x \leq y \Rightarrow_\ast z.\] The function$\Rightarrow_\ast$ is known asresiduum of $\ast$, and itcan be shown that $x \Rightarrow_\ast y = \max\{z \mid x \ast z \leqy\}$. 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 $x \Rightarrow_{Ł} y = \min\{1, 1 - x+y\}$.

In the general t-norm setting, the negation is interpreted using theresiduum. Namely, for a given left-continuous t-norm $\ast$ we set$\neg_\ast x = x\Rightarrow_\ast 0$. For the Łukasiewicz t-normwe obtain the negation $\neg_{Ł} x = 1 - x$ (known asŁukasiewicz negation or the standard involutive negation) whereas for theminimum and the product t-norms we obtain the same negation (known asGödel negation): $\neg_\mathrm{G} 0 = 1$ and $0$ otherwise.

2. MTL: A fundamental fuzzy logic

The weakest logic with connectives interpreted by truth functions ofthe type described above is MTL (Monoidal T-norm based Logic,Esteva & Godo 2001). It is a logic with the primitive binaryconnectives $\mathbin{\&}, \to, \wedge$ 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 semanticsgiven by all left-continuous t-norms. Namely, given a particularleft-continuous t-norm $\ast$ and a mapping from propositionalvariables to $[0,1]$, we define the $\ast$-evaluation of allformulas 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 eachleft-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 isthe conclusion. A formula $\varphi$ that always evaluates to $1$(i.e. $\emptyset\models_\mathrm{MTL}\varphi$) is called valid or atautology of MTL. Note that the formula $\varphi \mathbin{\&}\psi \to \varphi \land \psi$ is a tautology in MTL but the converseimplication $\varphi \land \psi \to \varphi \mathbin{\&} \psi$is not, i.e., the conjunction $\&$ is stronger than$\land$.

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

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

andmodus 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 offormulas which ends with $\varphi$ and its members are instances ofthe axioms, elements of $\Gamma$, or follow from previous ones bythe inference rule (modus ponens). This Hilbert-style systemis 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 bedecidable, however its computational complexity has not yet beendetermined.

3. Łukasiewicz logic

Łukasiewicz 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 toevaluations based on the Łukasiewicz t-norm (in symbols: for eachfinite set of formulas $\Gamma$ and each formula$\varphi$, we have $\Gamma \models_{Ł} \varphi$ iff $\Gamma\vdash_{Ł} \varphi$).^[2]

This logic was an early example of a many-valued logic, introduced byJan Łukasiewicz and Alfred Tarski (1930), well before theinception of the theory of fuzzy sets, by means of an equivalentaxiomatic system (withmodus ponens as the only inferencerule):

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

Łukasiewicz logic is the only t-norm based fuzzy logic where allconnectives are interpreted by continuous functions, including theimplication given by the function $x \Rightarrow_{Ł}y=\min\{1,1-x+y\}$. McNaughton’s theorem (1951) states thatreal-valued functions over [0,1] that interpret formulas ofŁukasiewicz logic are exactly the continuous piecewise linearfunctions with integer coefficients. In terms of computationalcomplexity, the validity problem for this logic is asymptotically notworse than in classical logic: it remains coNP-complete.

4. Gödel–Dummett logic

Gödel–Dummett logic, also known as Dummett’s LC orsimply as Gödel logic, is another early example of a many-valuedlogic with truth-values in $[0,1]$. It was introduced by MichaelDummett (1959) as the extension ofintuitionistic logic by the axiom \[(\varphi \to \psi) \lor (\psi \to \varphi).\] This formula enforces a linear order in theunderlying (Kripke-style as well as algebraic) semantics. It alsoappears in the context of Kurt Gödel’s observation that itis impossible to characterize intuitionistic logic by finite truthtables (Gödel 1932). In the fuzzy logic setting,Gödel–Dummett logic can alternatively be obtained as anaxiomatic extension of MTL by adding the axiom $\varphi \to \varphi\mathbin{\&} \varphi$, which amounts to requiring the idempotenceof $\&$, and hence the interpretation of both conjunctionscoincides. Semantically, Gödel–Dummett logic can also bedefined as the consequence relation given by the minimum t-norm. It isdistinguished as the only t-norm based logic where the validity of aformula in a given evaluation does not depend on the specific valuesassigned to the propositional variables, but only on the relativeorder of these values. In this sense, Gödel–Dummett logiccan be seen as a logic of comparative truth. Like for Łukasiewiczlogic, the computational complexity of testing validity remainscoNP-complete.

5. Other notable fuzzy logics

Besides MTL (the logic of all left-continuous t-norms) andŁukasiewicz and Gödel–Dummett logics (each induced bya particular t-norm), one can consider logicsinduced by setsof t-norms or, in general, arbitrary axiomatic extensions of MTL. Inparticular, the logic of allcontinuous t-norms(Hájek’s basic fuzzy logic BL) is obtained by adding theaxiom \[\varphi\mathbin{\&}(\varphi\to{{\psi}}) \to\psi\mathbin{\&}(\psi\to\varphi)\] to those of MTL. Actually, for any set of continuoust-norms there is a finite axiomatization of the corresponding logic(Esteva, Godo, & Montagna 2003; Haniková 2014); in mostcases the axiomatization captures semantic consequence fromfinite sets of premises. In particular, the logic of thethird prominent continuous t-norm (algebraic product), known asProduct logic, is the extension of BL by the axiom \[\neg\varphi \vee ((\varphi\to\varphi\mathbin{\&}{{\psi}})\to{{\psi}}).\] On theother hand, not all axiomatic extensions of MTL can be given a t-normbased semantics. For example, classical logic can be axiomatized asMTL $+$ $\varphi\vee\neg \varphi$, but the axiom of excludedmiddle 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 ofthe conjunction & to be a t-norm are too strong. In particular,the assumption that $1$ is its neutral element enforces a definitionof tautology as a formula always evaluated to $1$ and theconsequence relation as preservation of the value $1$ – thatis, $1$ is the onlydesignated value in the semantics.^[3] A natural way to introduce logics with more than one designatedtruth-degree is to assume that the neutral element for theinterpretation of & is a number $t <1$. It can be shown thatin this situation the designated truth-degrees are exactly thosegreater than or equal to $t$. Such interpretations of conjunctionsare calleduninorms. The logic of left-continuous uninormswas axiomatized in Metcalfe & Montagna (2007).

Analogously, one may argue against commutativity or even againstassociativity of &. Axiomatizations of resulting logics aredescribed in the literature (see Cintula, Horčík, &Noguera 2013; Jenei & Montagna 2003); an exception is the logic ofnon-commutative uninorms for which no natural axiomatic system isknown.

Finally, taking into account that fuzzy logics, unlike classicallogic, are typically not functionally complete, one can increase theirexpressive power by adding new connectives. The most commonlyconsidered connectives are: truth-constants $\bar r$ for eachrational 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, Hájek, & Navara 2000).

A thorough overview of all the kinds of propositional fuzzy logicsmentioned in this section (and a general theory thereof) can be foundin theHandbook 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 counterpartMTL$\forall$ in a predicate language $\mathcal{P\!L}$ (defined asin the classical case) in the following way. The semantics is given bystructures in which predicate symbols are interpreted as functionsmapping 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 $M$, afunction $f_{\mathbf M}\colon M^n\to M$ for each $n$-ary functionsymbol $f\in \mathcal{P\!L}$, and a function $P_{\mathbf M}\colonM^n\to [0,1]$ for each $n$-ary predicate symbol $P\in\mathcal{P\!L}$. Fixing a valuation ${\mathrm v}$ of objectvariables in $M$, one defines values of terms($\|f(t_1,\dots,t_n)\|^{\mathbf M}_{\mathrm v} = f_{\mathbfM}(\|t_1\|^{\mathbf M}_{\mathrm v},\dots,\|t_n\|^{\mathbf M}_{\mathrmv})$) and truth-values of atomic formulas($\|P(t_1,\dots,t_n)\|^{\mathbf M}_{\mathrm v} = P_{\mathbfM}(\|t_1\|^{\mathbf M}_{\mathrm v},\dots,\|t_n\|^{\mathbf M}_{\mathrmv})$). Truth-values of a universally/existentially quantified formulaare computed as infimum/supremum of truth-values of instances of theformula where the quantified variable runs over all elements of thedomain $M$. 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 ${\mathrm v}[x{:}a]$ isthe valuation sending $x$ to $a$ and keeping values of othervariables unchanged. The values of other formulas are computed byinterpreting, 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 theconsequence relation given by the preservation of total truth (value$1$) in $\ast$-structures (allowing $\ast$ to run over allleft-continuous t-norms). More precisely, we say that a first-orderformula $\varphi$ is a consequence of a set of formulas $\Gamma$(in symbols: $\Gamma \models_{\mathrm{MTL}\forall} \varphi$) if, foreach 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 proofsystem with the following axioms:

(P)The first-orderinstances of the axioms of MTL
$(\forall1 )$$\forall x\,\varphi(x)\to\varphi(t)$, where the term $t$ issubstitutable for $x$ in $\varphi$
$(\exists1)$$\varphi(t)\to \exists x\, \varphi(x)$, where the term $t$ issubstitutable 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$\forall$ aremodus ponens plusthe rule ofgeneralization: from $\varphi$ infer $\forallx\,\varphi$.

There are two ways of introducing first-order counterparts for otherpropositional t-norm based fuzzy logics. On the one hand, theaxiomatization of MTL$\forall$ can be extended by adding thefirst-order instances of propositional axioms like those seen in theprevious sections. In this manner one obtains syntactic presentationsof first-order variants of, e.g., Łukasiewicz logic,Gödel–Dummett logic, Product logic, or Hájek’sbasic fuzzy logic BL. On the other hand, the semantic definition ofMTL$\forall$ can be easily modified by defining the consequencerelations given by the corresponding (sets of) t-norms. Then, thenatural question is whether, in each case, these two routes lead tothe same logic (as it happened for MTL). For soundness there is noproblem, as the axiomatic systems are easily checked to be sound withrespect to their corresponding classes of structures. As forcompleteness, there is no general answer and the problem has to beconsidered case by case.

For Gödel–Dummett logic the axiomatic system is stronglycomplete with respect to its semantics. However, the set oftautologies of the semantics for Łukasiewicz logic is notrecursively axiomatizable as shown by Bruno Scarpellini (1962). EmilRagaz (1981) proved that it is actually $\Sigma_2$-complete in thesense of arithmetical hierarchy. The situation is even worse in thecase of Product logic and Hájek’s basic fuzzy logic BL,where the sets of first-order tautologies of all structures given bycontinuous t-norms are as complex as true arithmetics (Montagna 2001).Completeness can be achieved either by adding a suitable infinitaryinference rule to the Hilbert-style proof system, as done by LouiseHay (1963) for Łukasiewicz logic, or by generalizing the set oftruth-values (see next section).

First-order counterparts of weaker fuzzy logics can be studied inanalogous, syntactic and semantic, ways; see the survey presentationin Cintula, Horčík, & Noguera 2014.

7. Algebraic semantics

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

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

$\langle A, \wedge, \vee, \overline{0}, \overline{1} \rangle$ isa bounded lattice
$\langle A, \&, \overline{1} \rangle$ is a commutativemonoid
$(x\to y)\vee(y\to x) = \overline{1}$
$x\mathbin{\&} y \leq z$ 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 anMTL-chain. It is worth noting that MTL-algebras are a subvariety ofresiduated lattices which provide the algebraic semantics forsubstructural logics; the first pointer to the tight connection between these two familiesof logics.

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

MTL-chains are the basic building blocks of the whole class ofalgebras, in the sense that each MTL-algebra can be decomposed as asubdirect product of chains. This implies that the logic isalso complete with respect to the semantics of MTL-chains, which isthen used as the first step in the proof of its completeness withrespect to the t-norm based semantics (Jenei & Montagna 2002).

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

The algebraic semantics can be used for first-order fuzzy logics in arather 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 anarbitary MTL-chain and its operation to compute truth-values offormulas. Given a propositional fuzzy logic, one can use thisgeneralized semantics to obtain a recursively enumerable set oftautologies and a strong completeness theorem for a correspondingfirst-order Hilbert-style proof system.

8. Proof theory

It has been a considerable challenge to come up with analytic proofsystems for fuzzy logics. These are systems that share importantfeatures, like the eliminability of cuts and the subformula property,with Gentzen’s sequent calculi for classical and intuitionisticlogic (see entry onthe development of proof theory). A major breakthrough has been achieved with the introduction of ahypersequent calculus for Gödel–Dummett logic by ArnonAvron (1991). Hypersequent calculi arise from sequent calculi bydefining inference rules that refer to finite multisets of sequents,rather than to single sequents. Hypersequents are interpreted, at themeta-level, as disjunctions of sequents. This interpretation impliesthat it is sound to add additional sequents (external weakening) or toreplace multiple copies of the same sequent within a givenhypersequent by a single copy of that sequent (external contraction).In the case of Gödel–Dummett logic, the rules forintroducing logical connectives arise from the rules ofGentzen’s intuitionistic sequent calculus by simply addingside-hypersequents the upper and lower sequents of the original rule.For example, the sequent rule for introducing conjunction on theright-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 $\Gamma_2$ arefinite sequences of formulas, is turned into the followinghypersequent 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 theside-hypersequents, i.e., finite multisets of sequents addeddisjunctively to the exhibited sequents. This by itself does notchange the corresponding logic (intuitionistic logic, in this case).The crucial additional structural rule is the so-called communicationrule: \[\frac{H \mid \Gamma_1,\Pi_1 \Rightarrow \Delta_1 \hspace{3ex} H'\mid \Gamma_2,\Pi_2 \Rightarrow \Delta_2}{H \mid H' \mid\Gamma_1,\Gamma_2 \Rightarrow \Delta_1 \mid \Pi_2,\Pi_2 \Rightarrow\Delta_2}\] Here $\Gamma_1, \Gamma_2,\Pi_1, \Pi_2$ are finitesequences of formulas; $\Delta_1$ and $\Delta_2$ are either singleformulas or remain empty; $H$ and $H'$ denote theside-hypersequents, like above. To understand the impact of thecommunication rule, we present a derivation of the crucial axiom$(\varphi \to \psi) \lor (\psi \to \varphi)$, which features anapplication 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 MTLone has to add the communication rule to a sequent system for thecontraction-free version of intuitionistic logic (a well-studiedexample of asubstructural logic). Analytic proof systems for other fuzzy logics, in particularŁukasiewicz logic, call for a more radical departure fromtraditional calculi, where the sequent components of hypersequents areinterpreted differently than intuitionistic or classical sequents.Also labeled proof systems and various tableau calculi have beensuggested. A detailed presentation of the corresponding state of theart can be found in Metcalfe, Olivetti, & Gabbay 2008 and Metcalfe2011.

9. Semantics justifying truth-functionality

It is desirable, not only from a philosophical point of view, but alsoto a get a better grip on potential applications of fuzzy logics torelate the meaning of intermediary truth-values and correspondinglogical connectives to basic models of reasoning with vague andimprecise notions. A number of such semantics that seek to justifyparticular choices of truth-functional connectives have beenintroduced. 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 proportionof agents that accept a proposition $\varphi$ as true may be seen asa truth-value. Without further restrictions this does not lead to atruth-functional semantics, but rather to an assignment ofprobabilities to propositions. But if one assigns a fixedlevel ofskepticism to each agent and imposes some natural conditions thatkeep the judgments on logically complex statements consistent withthose levels, then one can recover $\min$, $\max$, and $1-x$ astruth functions for conjunction, disjunction and negation,respectively. Details can be found in Lawry 1998.

Another intriguing model of reasoning that provides a justificationfor all propositional connectives of standard Łukasiewicz logichas been introduced by Robin Giles (1974). It consists in a game,where two players, I and you, systematically reduce logically complexassertions (formulas) to simpler ones according to rules like thefollowing:

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 theconjuncts 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, assumingthat there is a constant symbol for each domain element onestipulates:

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 anoccurrence of a non-atomic formula in either the multiset of currentassertions by me or by you is chosen and gets replaced by subformulas,as indicated by these rules, until only atomic assertions remain. Afinal game state is then evaluated according to the following bettingscheme.

For each atomic formula there is a corresponding experiment which mayeither fail or succeed, but may show dispersion, i.e., it may yielddifferent results when repeated. A fixed failure probability, calledrisk value, is assigned to each experiment and thus to each atomicformula. The players have to pay $\$$1 to the other player for eachof their atomic assertion where the associated experiments fails. Forany game starting with my assertion of $\varphi$ my expected overallloss of money, if we both play rationally, can be shown to correspondto the inverse of the truth-value of $\varphi$ evaluated in aninterpretation of Łukasiewicz logic that assigns the inverse ofthe risk values as truth-values to atomic formulas. In particular, aformula is valid in Łukasiewicz logic if and only if, for everyrisk value assignment, I have a strategy that guarantees that myexpected overall loss at the end of game is $0$ or negative.

Christian Fermüller and George Metcalfe (2009) have pointed out acorrespondence between optimal strategies in Giles’s game andcut-free proofs in a hypersequent system for Łukasiewicz logic.The game has also been extended in Fermüller & Roschger 2014to characterize various types of (semi-)fuzzy quantifiers, intended tomodel natural language expressions like “about half” or“almost all”.

Jeff Paris (2000) provided an overview over other semantics supportingvarious choices of truth functions; in particular, re-randomizingsemantics (Hisdal 1988), similarity semantics (e.g., Ruspini 1991),acceptability semantics (Paris 1997), and approximation semantics(Paris 2000). Let us also mention the resource-based semanticsproposed in Běhounek 2009. Moreover there are different forms ofevaluation games for various fuzzy logics besides the one of Giles forŁukasiewicz logic, outlined above. An overview over thosesemantic games was given in Fermüller 2015.

10. Fuzzy logic and vagueness

Modeling reasoning with vague predicates and propositions is oftencited as the main motivation for introducing fuzzy logics. There aremany alternative theories ofvagueness, but there is a general agreement that the susceptibility to thesorites paradox is a main feature of vagueness. Consider the following version of theparadox:

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

On the face of it, it seems not to be unreasonable to accept these twoassumptions. By instantiating $n$ with $10^{100}$ in(2) and applyingmodus ponens with(1) as the other premise we conclude that $10^{100}-1$ is huge. Bysimply repeating this type of inference we arrive at the unreasonablestatement

(3)$0$ is ahuge number.

Fuzzy logic suggests an analysis of thesorites paradox thatrespects 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-basedfuzzy logics that dissolve the paradox. For example, one may declarethat any instance ofmodus ponens is sound if the degree oftruth of the conclusion is not lower than that of the strongconjunction of itspremises.^[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 thesoundness of instantiation andmodus ponens. If, moreover,the degree of truth of the conjunction of two not perfectly true (andnot perfectly false) statements is less than that of each conjunct, wemay safely declare that statement(3) is perfectly false and nevertheless insist on the soundness of eachstep in the indicated chain of inferences. Informally speaking, theparadox disappears by assuming that repeatedly decreasing someperfectly huge number by a small amount leads to numbers of which itis less and less true that they are huge too.

An alternative truth-degree-based solution to thesoritesparadox has been proposed in Hájek & Novák2003. They introduce a new truth-functional connective modeling theexpression “it is almost true that”. In this manner theyformalizesorites-style reasoning within an axiomatic theoryof an appropriate t-norm-based fuzzy logic.

Nicholas J.J. Smith (2005 and 2008) has argued that the so-calledcloseness principle captures the essence of vagueness. Itexpresses that statements of the same form about indistinguishableobjects should remain close in respect of truth. It is a hallmark ofmany approaches to the sorites paradox that employ fuzzy logic thatthey are compatible with this principle.^[6]

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Supplementary document:

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Hajek, Petr, “Fuzzy Logic”,The StanfordEncyclopedia of Philosophy (Fall 2016 Edition), Edward N. Zalta(ed.), URL = <https://plato.stanford.edu/archives/fall2016/entries/logic-fuzzy/>. [This was the previous entry on fuzzy logic in theStanfordEncyclopedia of Philosophy — see theversion history.]

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