Many-valued logic 1From Wikipedia, the free encyclopedia

Contents

1 Four-valued logic 11.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Many-valued logic 42.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Kleene (strong) K3 and Priest logic P3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.2 Bochvars internal three-valued logic (also known as Kleenes weak three-valued logic) . . . 52.2.3 Belnap logic (B4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.4 Gdel logics Gk and G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.5 ukasiewicz logics Lv and L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.6 Product logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.7 Post logics Pm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.1 Matrix semantics (logical matrices) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Proof theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Relation to classical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.5.1 Suszkos thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.7 Research venues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.11 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Problem of future contingents 103.1 Aristotles solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Leibniz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 20th century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

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3.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Three-valued logic 144.1 Representation of values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2.1 Kleene and Priest logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2.2 ukasiewicz logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2.3 Bochvar logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2.4 ternary Post logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2.5 Modular algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.3 Application in SQL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Chapter 1

Four-valued logic

In logic, a four-valued logic is used to model signal values in digital circuits: the four values are Z, X and the booleanvalues 1 and 0. Z stands for high impedance or open circuit, while X stands for unknown. There is also a 9-valuedlogic standard by the IEEE called IEEE 1164.There are other types of four value logic, such as Belnaps four-valued relevance logic: the possible values are 1)true, 2) false, 3) both true and false, and 4) neither true nor false. Belnaps logic is designed to cope with multipleinformation sources such that if only true is found then true is assigned, if only false is found then false is assigned,if some sources say true and others say false then both is assigned, and if no information is given by any informationsource then neither is assigned.

1.1 Applications

1.1.1 ElectronicsAmong the distinct logic values supported by digital electronics theory (as dened in VHDL's std_logic) are suchdiverse elements as:

1 or High, usually representing TRUE. 0 or Low, usually representing FALSE. X representing a Conict. U representing Unassigned or Unknown. - representing "Don't Care". Z representing "high impedance", undriven line. H, L andW are other high-impedance values, the weak pull to High, Low and Don't Know correspond-ingly.

The U value does not exist in real-world circuits, it is merely a placeholder used in simulators and for designpurposes. Some simulators support representation of the Z value, others do not. The Z value does exist inreal-world circuits but only as an output state.

Use of U value in simulation

Many hardware description language (HDL) simulation tools, such as Verilog and VHDL, support an unknown valuelike that shown above during simulation of digital electronics. The unknown value may be the result of a design error,which the designer can correct before synthesis into an actual circuit. The unknown also represents uninitialisedmemory values and circuit inputs before the simulation has asserted what the real input value should be.HDL synthesis tools usually produce circuits that operate only on binary logic.

1

2 CHAPTER 1. FOUR-VALUED LOGIC

Use of X value in digital design

When designing a digital circuit, some conditions may be outside the scope of the purpose that the circuit will perform.Thus, the designer does not care what happens under those conditions. In addition, the situation occurs that inputs toa circuit are masked by other signals so the value of that input has no eect on circuit behaviour.In these situations, it is traditional to use X as a placeholder to indicate "Don't Care" when building truth tables. Thisis especially common in state machine design and Karnaugh map simplication. The X values provide additionaldegrees of freedom to the nal circuit design, generally resulting in a simplied and smaller circuit.[1]

Once the circuit design is complete and a real circuit is constructed, the X values will no longer exist. They willbecome some tangible 0 or 1 value but could be either depending on the nal design optimisation.

Use of Z value for high impedance

Main article: three-state

Some digital devices support a form of three-state logic on their outputs only. The three states are 0, 1, and Z.Commonly referred to as tristate [2] logic (a trademark of National Semiconductor), it comprises the usual true andfalse states, with a third transparent high impedance state (or 'o-state') which eectively disconnects the logic output.This provides an eective way to connect several logic outputs to a single input, where all but one are put into thehigh impedance state, allowing the remaining output to operate in the normal binary sense. This is commonly usedto connect banks of computer memory and other similar devices to a common data bus; a large number of devicescan communicate over the same channel simply by ensuring only one is enabled at a time.It is important to note that while outputs can have one of three states, inputs can only recognise two. Hence thekind of relations shown in the table above do not occur. Although it could be argued that the high-impedance stateis eectively an unknown, there is absolutely no provision in the vast majority of normal electronics to interpret ahigh-impedance state as a state in itself. Inputs can only detect 0 and 1.When a digital input is left disconnected (i.e., when it is given a high impedance signal), the digital value interpretedby the input depends on the type of technology used. TTL technology will reliably default to a 1 state. On the otherhand CMOS technology will temporarily hold the previous state seen on that input (due to the capacitance of the gateinput). Over time, leakage current causes the CMOS input to drift in a random direction, possibly causing the inputstate to ip. Disconnected inputs on CMOS devices can pick up noise, they can cause oscillation, the supply currentmay dramatically increase (crowbar power) or the device may completely destroy itself.

Exotic ternary-logic devices

True three-valued logic can be implemented in electronics, although the complexity of design has thus far made ituneconomical to pursue commercially and interest has been primarily conned to research (see Setun); 'Normal'binary logic is simply cheaper to implement and in most cases can easily be congured to emulate ternary systems.There are, however, useful applications in fuzzy logic and error correction, and several true ternary logic devices havebeen manufactured.

1.1.2 Software

Vehicle technology

In the SAE J1939 standard, used for CAN data transmission in heavy road vehicles, there are four logical (boolean)values, False, True, Error Condition, and Not installed (represented by values 0-3). Error Condition means there isa technical problem obstacling data acquisition. The logics for that is for example True and Error Condition=ErrorCondition. Not installed is used for a feature which does not exist in this vehicle, and should be disregarded for logicalcalculation. On CAN, usually xed data messages are sent containing many signal values each, so a signal representinga not-installed feature will be sent anyway.

1.2. NOTES 3

1.2 Notes[1] Wakerly, John F (2001). Digital Design Principles & Practices. Prentice Hall. ISBN 0-13-090772-3.

[2] National Semiconductor (1993), LS TTL Data Book, National Semiconductor Corporation

Chapter 2

Many-valued logic

In logic, a many-valued logic (also multi- or multiple-valued logic) is a propositional calculus in which there aremore than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e.,true and false) for any proposition. Classical two-valued logic may be extended to n-valued logic for n greaterthan 2. Those most popular in the literature are three-valued (e.g., ukasiewiczs and Kleenes, which accept thevalues true, false, and unknown), the nite-valued (nitely-many valued) with more than three values, and theinnite-valued (innitely-many valued), such as fuzzy logic and probability logic.

2.1 HistoryThe rst known classical logician who didn't fully accept the law of excluded middle was Aristotle (who, ironically, isalso generally considered to be the rst classical logician and the father of logic[1]). Aristotle admitted that his lawsdid not all apply to future events (De Interpretatione, ch. IX), but he didn't create a system of multi-valued logic toexplain this isolated remark. Until the coming of the 20th century, later logicians followed Aristotelian logic, whichincludes or assumes the law of the excluded middle.The 20th century brought back the idea of multi-valued logic. The Polish logician and philosopher Jan ukasiewiczbegan to create systems of many-valued logic in 1920, using a third value, possible, to deal with Aristotles paradoxof the sea battle. Meanwhile, the American mathematician, Emil L. Post (1921), also introduced the formulation ofadditional truth degrees with n 2, where n are the truth values. Later, Jan ukasiewicz and Alfred Tarski togetherformulated a logic on n truth values where n 2. In 1932 Hans Reichenbach formulated a logic of many truth valueswhere ninnity. Kurt Gdel in 1932 showed that intuitionistic logic is not a nitely-many valued logic, and deneda system of Gdel logics intermediate between classical and intuitionistic logic; such logics are known as intermediatelogics.

2.2 ExamplesMain articles: Three-valued logic and Four-valued logic

2.2.1 Kleene (strong) K3 and Priest logic P3Kleene's "(strong) logic of indeterminacyK3 (sometimesKS3 ) and Priests logic of paradox add a third undenedor indeterminate truth value I. The truth functions for negation (), conjunction (), disjunction (), implication(K), and biconditional (K) are given by:[2]

The dierence between the two logics lies in how tautologies are dened. In K3 only T is a designated truth value,while in P3 both T and I are (a logical formula is considered a tautology if it evaluates to a designated truth value). InKleenes logic I can be interpreted as being underdetermined, being neither true nor false, while in Priests logic Ican be interpreted as being overdetermined, being both true and false. K3 does not have any tautologies, while P3has the same tautologies as classical two-valued logic.

4

2.2. EXAMPLES 5

2.2.2 Bochvars internal three-valued logic (also known as Kleenes weak three-valuedlogic)

Another logic is Bochvars internal three-valued logic ( BI3 ) also called Kleenes weak three-valued logic. Exceptfor negation and biconditional, its truth tables are all dierent from the above.[3]

The intermediate truth value in Bochvars internal logic can be described as contagious because it propagates ina formula regardless of the value of any other variable.[4]

2.2.3 Belnap logic (B4)

Belnaps logic B4 combines K3 and P3. The overdetermined truth value is here denoted as B and the underdeterminedtruth value as N.

2.2.4 Gdel logics Gk and G

In 1932 Gdel dened[5] a familyGk of many-valued logics, with nitely many truth values 0; 1k1 ; 2k1 ; : : : k2k1 ; 1 ,for exampleG3 has the truth values 0; 12 ; 1 andG4 has 0; 13 ; 23 ; 1 . In a similar manner he dened a logic with innitelymany truth values,G1 , in which the truth values are all the real numbers in the interval [0; 1] . The designated truthvalue in these logics is 1.The conjunction ^ and the disjunction _ are dened respectively as the maximum and minimum of the operands:

u ^ v := minfu; vg

u _ v := maxfu; vg

Negation :G and implication!G are dened as follows:

:Gu =(1; ifu = 00; ifu > 0

u!G v =(1; ifu v0; ifu > v

Gdel logics are completely axiomatisable, that is to say it is possible to dene a logical calculus in which all tautologiesare provable.

2.2.5 ukasiewicz logics Lv and L

Implication!L and negation :L were dened by Jan ukasiewicz through the following functions:

:Lu := 1 u

u!L v := minf1; 1 u+ vg

At rst ukasiewicz used these denition in 1920 for his three-valued logic L3 , with truth values 0; 12 ; 1 . In 1922 hedeveloped a logic with innitely many values L1 , in which the truth values spanned the real numbers in the interval[0; 1] . In both cases the designated truth walue was 1.[6]

By adopting truth values dened in the same way as for Gdel logics 0; 1v1 ; 2v1 ; : : : ; v2v1 ; 1 , it is possible to createa nitely-valued family of logics Lv , the abovementioned L1 and the logic L@0 , in which the truth values are givenby the rational numbers in the interval [0; 1] . The set of tautologies in L1 and L@0 is identical.

6 CHAPTER 2. MANY-VALUED LOGIC

2.2.6 Product logic In product logic we have truth values in the interval [0; 1] , a conjunction and an implication ! , dened asfollows[7]

u v := uv

u! v :=(1; ifu vvu ; ifu > v

Additionally there is a negative designated value 0 that denotes the concept of false. Through this value it is possibleto dene a negation : and an additional conjunction ^ as follows:

:u := u! 0 u ^ v := u (u! v)

2.2.7 Post logics PmIn 1921 Post dened a family of logics Pm with (as in Lv and Gk ) the truth values 0; 1m1 ; 2m1 ; : : : ; m2m1 ; 1 .Negation :P and disjunction _P are dened as follows:

:Pu :=(1; ifu = 0u 1m1 ; ifu 6= 0

u _P v := maxfu; vg

2.3 Semantics

2.3.1 Matrix semantics (logical matrices)

2.4 Proof theory

2.5 Relation to classical logicLogics are usually systems intended to codify rules for preserving some semantic property of propositions acrosstransformations. In classical logic, this property is truth. In a valid argument, the truth of the derived proposition isguaranteed if the premises are jointly true, because the application of valid steps preserves the property. However,that property doesn't have to be that of truth"; instead, it can be some other concept.Multi-valued logics are intended to preserve the property of designationhood (or being designated). Since there aremore than two truth values, rules of inference may be intended to preserve more than just whichever corresponds (inthe relevant sense) to truth. For example, in a three-valued logic, sometimes the two greatest truth-values (when theyare represented as e.g. positive integers) are designated and the rules of inference preserve these values. Precisely,a valid argument will be such that the value of the premises taken jointly will always be less than or equal to theconclusion.For example, the preserved property could be justication, the foundational concept of intuitionistic logic. Thus, aproposition is not true or false; instead, it is justied or awed. A key dierence between justication and truth, inthis case, is that the law of excluded middle doesn't hold: a proposition that is not awed is not necessarily justied;instead, its only not proven that its awed. The key dierence is the determinacy of the preserved property: Onemay prove that P is justied, that P is awed, or be unable to prove either. A valid argument preserves justicationacross transformations, so a proposition derived from justied propositions is still justied. However, there are proofsin classical logic that depend upon the law of excluded middle; since that law is not usable under this scheme, thereare propositions that cannot be proven that way.

2.6. APPLICATIONS 7

2.5.1 Suszkos thesisSee also: Principle of bivalence Suszkos thesis

2.6 ApplicationsKnown applications of many-valued logic can be roughly classied into two groups.[8] The rst group uses many-valued logic domain to solve binary problems more eciently. For example, a well-known approach to represent amultiple-output Boolean function is to treat its output part as a single many-valued variable and convert it to a single-output characteristic function. Other applications of many-valued logic include design of Programmable Logic Arrays(PLAs) with input decoders, optimization of nite state machines, testing, and verication.The second group targets the design of electronic circuits which employ more than two discrete levels of signals,such as many-valued memories, arithmetic circuits, Field Programmable Gate Arrays (FPGA) etc. Many-valuedcircuits have a number of theoretical advantages over standard binary circuits. For example, the interconnect onand o chip can be reduced if signals in the circuit assume four or more levels rather than only two. In memorydesign, storing two instead of one bit of information per memory cell doubles the density of the memory in the samedie size. Applications using arithmetic circuits often benet from using alternatives to binary number systems. Forexample, residue and redundant number systems can reduce or eliminate the ripple-through carries which are involvedin normal binary addition or subtraction, resulting in high-speed arithmetic operations. These number systems havea natural implementation using many-valued circuits. However, the practicality of these potential advantages heavilydepends on the availability of circuit realizations, which must be compatible or competitive with present-day standardtechnologies.

2.7 Research venuesAn IEEE International Symposium on Multiple-Valued Logic (ISMVL) has been held annually since 1970. It mostlycaters to applications in digital design and verication.[9] There is also a Journal of Multiple-Valued Logic and SoftComputing.[10]

2.8 See alsoMathematical logic

Degrees of truth Fuzzy logic Gdel logic Kleene logic Kleene algebra (with involution) ukasiewicz logic MV-algebra Post logic Principle of bivalence A. N. Prior Relevance logic

Philosophical logic

8 CHAPTER 2. MANY-VALUED LOGIC

False dilemma Mu

Digital logic

MVCML, multiple-valued current-mode logic IEEE 1164 a nine-valued standard for VHDL IEEE 1364 a four-valued standard for Verilog Noise-based logic

2.9 Notes

2.10 References[1] Hurley, Patrick. A Concise Introduction to Logic, 9th edition. (2006).

[2] (Gottwald 2005, p. 19)

[3] (Bergmann 2008, p. 80)

[4] (Bergmann 2008, p. 80)

[5] Gdel, Kurt (1932). Zum intuitionistischen Aussagenkalkl. Anzeiger Akademie der Wissenschaften Wien (69): 65f.

[6] Kreiser, Lothar; Gottwald, Siegfried; Stelzner, Werner (1990). Nichtklassische Logik. Eine Einfhrung. Berlin: Akademie-Verlag. pp. 41 45. ISBN 978-3-05-000274-3.

[7] Hajek, Petr: Fuzzy Logic. In: Edward N. Zalta: The Stanford Encyclopedia of Philosophy, Spring 2009. ()

[8] Dubrova, Elena (2002). Multiple-Valued Logic Synthesis and Optimization, in Hassoun S. and Sasao T., editors, LogicSynthesis and Verication, Kluwer Academic Publishers, pp. 89-114

[9] http://www.informatik.uni-trier.de/~{}ley/db/conf/ismvl/index.html

[10] http://www.oldcitypublishing.com/MVLSC/MVLSC.html

2.11 Further readingGeneral

Bziau J.-Y. (1997), What is many-valued logic ? Proceedings of the 27th International Symposium onMultiple-Valued Logic, IEEE Computer Society, Los Alamitos, pp. 117121.

Malinowski, Gregorz, (2001), Many-Valued Logics, in Goble, Lou, ed., The Blackwell Guide to PhilosophicalLogic. Blackwell.

Bergmann, Merrie (2008), An introduction to many-valued and fuzzy logic: semantics, algebras, and derivationsystems, Cambridge University Press, ISBN 978-0-521-88128-9

Cignoli, R. L. O., D'Ottaviano, I, M. L., Mundici, D., (2000). Algebraic Foundations of Many-valued Reason-ing. Kluwer.

Malinowski, Grzegorz (1993). Many-valued logics. Clarendon Press. ISBN 978-0-19-853787-8. S. Gottwald, A Treatise on Many-Valued Logics. Studies in Logic and Computation, vol. 9, Research StudiesPress: Baldock, Hertfordshire, England, 2001.

Gottwald, Siegfried (2005). Many-Valued Logics (PDF).

2.12. EXTERNAL LINKS 9

Miller, D. Michael; Thornton, Mitchell A. (2008). Multiple valued logic: concepts and representations. Syn-thesis lectures on digital circuits and systems 12. Morgan & Claypool Publishers. ISBN 978-1-59829-190-2.

Hjek P., (1998), Metamathematics of fuzzy logic. Kluwer. (Fuzzy logic understood as many-valued logic suigeneris.)

Specic

Alexandre Zinoviev, Philosophical Problems of Many-Valued Logic, D. Reidel Publishing Company, 169p.,1963.

Prior A. 1957, Time and Modality. Oxford University Press, based on his 1956 John Locke lectures Goguen J.A. 1968/69, The logic of inexact concepts, Synthese, 19, 325373. Chang C.C. and Keisler H. J. 1966. Continuous Model Theory, Princeton, Princeton University Press. Gerla G. 2001, Fuzzy logic: Mathematical Tools for Approximate Reasoning, Kluwer Academic Publishers,Dordrecht.

Pavelka J. 1979, On fuzzy logic I: Many-valued rules of inference, Zeitschr. f. math. Logik und Grundlagen d.Math., 25, 4552.

Metcalfe, George; Olivetti, Nicola; Dov M. Gabbay (2008). Proof Theory for Fuzzy Logics. Springer. ISBN978-1-4020-9408-8. Covers proof theory of many-valued logics as well, in the tradition of Hjek.

Hhnle, Reiner (1993). Automated deduction in multiple-valued logics. Clarendon Press. ISBN 978-0-19-853989-6.

Azevedo, Francisco (2003). Constraint solving over multi-valued logics: application to digital circuits. IOSPress. ISBN 978-1-58603-304-0.

Bolc, Leonard; Borowik, Piotr (2003). Many-valued Logics 2: Automated reasoning and practical applications.Springer. ISBN 978-3-540-64507-8.

Stankovi, Radomir S.; Astola, Jaakko T.; Moraga, Claudio (2012). Representation of Multiple-Valued LogicFunctions. Morgan & Claypool Publishers. doi:10.2200/S00420ED1V01Y201205DCS037. ISBN 978-1-60845-942-1.

2.12 External links Gottwald, Siegfried (2009). Many-Valued Logic. Stanford Encyclopedia of Philosophy. Stanford Encyclopedia of Philosophy: "Truth Values"by Yaroslav Shramko and Heinrich Wansing. IEEE Computer Society's Technical Committee on Multiple-Valued Logic Resources for Many-Valued Logic by Reiner Hhnle, Chalmers University Many-valued Logics W3 Server (archived) Yaroslav Shramko and Heinrich Wansing (2014). Suszkos Thesis. Stanford Encyclopedia of Philosophy. Carlos Caleiro, Walter Carnielli, Marcelo E. Coniglio and Joo Marcos, Twos company: The humbug ofmany logical values in Jean-Yves Beziau, ed. (2007). Logica Universalis: Towards a General Theory of Logic(2nd ed.). Springer Science & Business Media. pp. 174194. ISBN 978-3-7643-8354-1.

Chapter 3

Problem of future contingents

Aristotle: if a sea-battle will not be fought tomorrow, then it was also true yesterday that it will not be fought. But all past truths arenecessary truths. Therefore it is not possible that the battle will be fought

Future contingent propositions (or simply, future contingents) are statements about states of aairs in the futurethat are neither necessarily true nor necessarily false.The problem of future contingents seems to have been rst discussed by Aristotle in chapter 9 of his On Inter-pretation (De Interpretatione), using the famous sea-battle example.[1] Roughly a generation later, Diodorus Cronusfrom the Megarian school of philosophy stated a version of the problem in his notorious Master Argument.[2] Theproblem was later discussed by Leibniz.The problem can be expressed as follows. Suppose that a sea-battle will not be fought tomorrow. Then it was alsotrue yesterday (and the week before, and last year) that it will not be fought, since any true statement about whatwill be the case was also true in the past. But all past truths are now necessary truths; therefore it is now necessarilytrue that the battle will not be fought, and thus the statement that it will be fought is necessarily false. Therefore it isnot possible that the battle will be fought. In general, if something will not be the case, it is not possible for it to bethe case. For a man may predict an event ten thousand years beforehand, and another may predict the reverse; thatwhich was truly predicted at the moment in the past will of necessity take place in the fullness of time (18 b35).This conicts with the idea that of our own free choice: that we have the power to determine or control the course of

10

3.1. ARISTOTLES SOLUTION 11

events in the future, which seems impossible if what happens, or does not happen, is necessarily going to happen, ornot happen. As Aristotle says, if so there would be no need to deliberate or to take trouble, on the supposition thatif we should adopt a certain course, a certain result would follow, while, if we did not, the result would not follow.

3.1 Aristotles solutionAristotle solved the problem by asserting that the principle of bivalence found its exception in this paradox of the seabattles: in this specic case, what is impossible is that both alternatives can be possible at the same time: either therewill be a battle, or there won't. Both options can't be simultaneously taken. Today, they are neither true nor false; butif one is true, then the other becomes false. According to Aristotle, it is impossible to say today if the proposition iscorrect: we must wait for the contingent realization (or not) of the battle, logic realizes itself afterwards:

One of the two propositions in such instances must be true and the other false, but we cannot say determi-nately that this or that is false, but must leave the alternative undecided. One may indeed be more likely tobe true than the other, but it cannot be either actually true or actually false. It is therefore plain that it isnot necessary that of an armation and a denial, one should be true and the other false. For in the caseof that which exists potentially, but not actually, the rule which applies to that which exists actually doesnot hold good. (9)

For Diodorus, the future battle was either impossible or necessary. Aristotle added a third term, contingency, whichsaves logic while in the same time leaving place for indetermination in reality. What is necessary is not that there willor that there won't be a battle tomorrow, but the dichotomy itself is necessary:

A sea-ght must either take place tomorrow or not, but it is not necessary that it should take place tomorrow,neither is it necessary that it should not take place, yet it is necessary that it either should or should not takeplace tomorrow. (De Interpretatione, 9, 19 a 30.)

Thus, the event always comes in the form of the future, undetermined event; logic always comes afterwards. Hegelwould say the same thing by claiming that wisdom came at dusk. For Aristotle, this is also a practical, ethical question:to pretend that the future is determined would have unacceptable consequences for man.

3.2 LeibnizLeibniz gave another response to the paradox in 6 ofDiscourse on Metaphysics: That God does nothing which is notorderly, and that it is not even possible to conceive of events which are not regular. Thus, even a miracle, the Eventby excellence, does not break the regular order of things. What is seen as irregular is only a default of perspective, butdoes not appear so in relation to universal order. Possibility exceeds human logics. Leibniz encounters this paradoxbecause according to him:

Thus the quality of king, which belonged to Alexander the Great, an abstraction from the subject, is notsuciently determined to constitute an individual, and does not contain the other qualities of the samesubject, nor everything which the idea of this prince includes. God, however, seeing the individual concept,or haecceity, of Alexander, sees there at the same time the basis and the reason of all the predicates whichcan be truly uttered regarding him; for instance that he will conquer Darius and Porus, even to the pointof knowing a priori (and not by experience) whether he died a natural death or by poison,- facts whichwe can learn only through history. When we carefully consider the connection of things we see also thepossibility of saying that there was always in the soul of Alexander marks of all that had happened to himand evidences of all that would happen to him and traces even of everything which occurs in the universe,although God alone could recognize them all. (8)

If everything which happens to Alexander derives from the haecceity of Alexander, then fatalism threatens Leibnizsconstruction:

We have said that the concept of an individual substance includes once for all everything which can everhappen to it and that in considering this concept one will be able to see everything which can truly be said

12 CHAPTER 3. PROBLEM OF FUTURE CONTINGENTS

concerning the individual, just as we are able to see in the nature of a circle all the properties which can bederived from it. But does it not seem that in this way the dierence between contingent and necessary truthswill be destroyed, that there will be no place for human liberty, and that an absolute fatality will rule aswell over all our actions as over all the rest of the events of the world? To this I reply that a distinctionmust be made between that which is certain and that which is necessary. (13)

Against Aristotles separation between the subject and the predicate, Leibniz states:

Thus the content of the subject must always include that of the predicate in such a way that if oneunderstands perfectly the concept of the subject, he will know that the predicate appertains to it also.(8)

The predicate (what happens to Alexander) must be completely included in the subject (Alexander) if one un-derstands perfectly the concept of the subject. Leibniz henceforth distinguishes two types of necessity: necessarynecessity and contingent necessity, or universal necessity vs singular necessity. Universal necessity concerns universaltruths, while singular necessity concerns something necessary which could not be (it is thus a contingent necessity).Leibniz hereby uses the concept of compossible worlds. According to Leibniz, contingent acts such as Caesar cross-ing the Rubicon or Adam eating the apple are necessary: that is, they are singular necessities, contingents andaccidentals, but which concerns the principle of sucient reason. Furthermore, this leads Leibniz to conceive of thesubject not as a universal, but as a singular: it is true that Caesar crosses the Rubicon, but it is true only of thisCaesar at this time, not of any dictator nor of Caesar at any time (8, 9, 13). Thus Leibniz conceives of substanceas plural: there is a plurality of singular substances, which he calls monads. Leibniz hence creates a concept of theindividual as such, and attributes to it events. There is a universal necessity, which is universally applicable, and asingular necessity, which applies to each singular substance, or event. There is one proper noun for each singularevent: Leibniz creates a logic of singularity, which Aristotle thought impossible (he considered that there could onlybe knowledge of generality).

3.3 20th centuryOne of the early motivations for the study of many-valued logics has been precisely this issue. In the early 20thcentury, the Polish formal logician Jan ukasiewicz proposed three truth-values: the true, the false and the as-yet-undetermined. This approach was later developed by Arend Heyting and L. E. J. Brouwer;[3] see ukasiewicz logic.Issues such as this have also been addressed in various temporal logics, where one can assert that "Eventually, eitherthere will be a sea battle tomorrow, or there won't be. (Which is true if tomorrow eventually occurs.)The Modal FallacyThe error in the argument underlying the alleged Problem of Future Contingents lies in the assumption that X isthe case implies that necessarily, X is the case. In logic, this is known as the Modal Fallacy.[4]

By asserting A sea-ght must either take place tomorrow or not, but it is not necessary that it should take placetomorrow, neither is it necessary that it should not take place, yet it is necessary that it either should or should not takeplace tomorrow. Aristotle is simply claiming necessarily (a or not-a), which is correct.However, the next step in Aristotles reasoning seems to be: If a is the case, then necessarily, a is the case, whichis a logical fallacy.Expressed in another way: (i) If a proposition is true, then it cannot be false. (ii) If a proposition cannot be false, thenit is necessarily true. (iii) Therefore if a proposition is true, it is necessarily true.That is, there are no contingent propositions. Every proposition is either necessarily true or necessarily false. Thefallacy arises in the ambiguity of the rst premise. If we interpret it close to the English, we get:(iv) P entails it is not possible that not-P (v) It is not possible that not-P entails it is necessary that P (vi) Therefore,P entails it is necessary that PHowever, if we recognize that the original English expression (i) is potentially misleading, that it assigns a necessityto what is simply nothing more than a necessary condition, then we get instead as our premises:(vii) It is not possible that (P and not P) (viii) (It is not possible that not P) entails (it is necessary that P)From these latter two premises, one cannot validly infer the conclusion:

3.4. SEE ALSO 13

(ix) P entails it is necessary that P

3.4 See also Logical determinism Free will Principle of distributivity Principle of plenitude Truth-value link In Borges' The Garden of Forking Paths, both alternatives happen, thus leading to what Deleuze calls incom-possible worlds

3.5 Notes[1] Dorothea Frede, The sea-battle reconsidered, Oxford Studies in Ancient Philosophy 1985, pp. 31-87.

[2] Dialectical School entry by Susanne Bobzien in the Stanford Encyclopedia of Philosophy

[3] Paul Tomassi (1999). Logic. Routledge. p. 124. ISBN 978-0-415-16696-6.

[4] Norman Swartz, The Modal Fallacy

3.6 Further reading Dorothea Frede (1985), The Sea-Battle Reconsidered, Oxford Studies in Ancient Philosophy 3, 31-87. Peter hrstrm; Per F. V. Hasle (1995). Temporal logic: from ancient ideas to articial intelligence. Springer.ISBN 978-0-7923-3586-3.

Richard Gaskin (1995). The sea battle and the master argument: Aristotle and Diodorus Cronus on the meta-physics of the future. Walter de Gruyter. ISBN 978-3-11-014430-7.

Melvin Fitting; Richard L. Mendelsohn (1998). First-order modal logic. Springer. pp. 3540. ISBN 978-0-7923-5335-5. attempts to reconstruct both Aristotles and Diodorus arguments in propositional modal logic

John MacFarlane (2003), Sea Battles, Futures Contingents, and Relative Truth and Future Contingent andRelative Truth, The Philosophical Quarterly 53, 321-36

Jules Vuillemin, Le chapitre IX du De Interpretatione d'Aristote - Vers une rhabilitation de l'opinion commeconnaissance probable des choses contingentes, in Philosophiques, vol. X, n1, April 1983 (French)

3.7 External links Future Contingents entry by Peter hrstrm and Per Hasle in the Stanford Encyclopedia of Philosophy Medieval Theories of Future Contingents entry by Simo Knuuttila in the Stanford Encyclopedia of Philosophy The Master Argument: The Sea Battle in De Intepretatione 9, Diodorus Cronus, Philo the Dialectician with abibliography on Diodorus and the problem of future contingents

Chapter 4

Three-valued logic

In logic, a three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is anyof several many-valued logic systems in which there are three truth values indicating true, false and some indeter-minate third value. This is contrasted with the more commonly known bivalent logics (such as classical sentential orBoolean logic) which provide only for true and false. Conceptual form and basic ideas were initially created by Janukasiewicz and C. I. Lewis. These were then re-formulated by Grigore Moisil in an axiomatic algebraic form, andalso extended to n-valued logics in 1945.

4.1 Representation of valuesAs with bivalent logic, truth values in ternary logic may be represented numerically using various representations ofthe ternary numeral system. A few of the more common examples are:

in balanced ternary, each digit has one of 3 values: 1, 0, or +1; these values may also be simplied to , 0, +,respectively.[1]

in the redundant binary representation, each digit can have a value of 1, 0, 0, or 1 (the value 0 has twodierent representations)

in the ternary numeral system, each digit is a trit (trinary digit) having a value of: 0, 1, or 2 in the skew binary number system, only most-signicant non-zero digit has a value 2, and the remaining digitshave a value of 0 or 1

1 for true, 2 for false, and 0 for unknown, unknowable/undecidable, irrelevant, or both.[2]

0 for false, 1 for true, and a third non-integer maybe symbol such as ?, #, ,[3] or xy.

Inside a ternary computer, ternary values are represented by ternary signals.This article mainly illustrates a system of ternary propositional logic using the truth values {false, unknown, and true},and extends conventional Boolean connectives to a trivalent context. Ternary predicate logics exist as well; these mayhave readings of the quantier dierent from classical (binary) predicate logic, and may include alternative quantiersas well.

4.2 LogicsWhere Boolean Logic has 4 monadic operators, the addition of a third value in ternary logic leads to a total of 27distinct operators on a single input value. Similarly, where Boolean logic has 16 distinct diadic operators (operatorswith 2 inputs), ternary logic has 19,683 such operators. Where we can easily name a signicant fraction of the Booleanoperators (not, and, or, nand, nor, exclusive or), it is unreasonable to attempt to name all but a small fraction of thepossible ternary operators.[4]

14

4.2. LOGICS 15

4.2.1 Kleene and Priest logicsSee also: Kleene algebra (with involution)

Below is a set of truth tables showing the logic operations for Kleene's strong logic of indeterminacy and Priestslogic of paradox.In these truth tables, the unknown state can be thought of as neither true nor false in Kleene logic, or thought of as bothtrue and false in Priest logic. The dierence lies in the denition of tautologies. Where Kleene logics only designatedtruth value is T, Priest logics designated truth values are both T and U. In Kleene logic, the knowledge of whetherany particular unknown state secretly represents true or false at any moment in time is not available. However, certainlogical operations can yield an unambiguous result, even if they involve at least one unknown operand. For example,since true OR true equals true, and true OR false also equals true, one can infer that true OR unknown equals true,as well. In this example, since either bivalent state could be underlying the unknown state, but either state also yieldsthe same result, a denitive true results in all three cases.If numeric values, e.g. balanced ternary values, are assigned to false, unknown and true such that false is less thanunknown and unknown is less than true, then A AND B AND C... = MIN(A, B, C ...) and A OR B OR C ... =MAX(A, B, C...).Material implication for Kleene logic can be dened as:A! B def= NOT(A) OR B , and its truth table iswhich diers from that for ukasiewicz logic (described below).Kleene logic has no tautologies (valid formulas) because whenever all of the atomic components of a well-formedformula are assigned the value Unknown, the formula itself must also have the value Unknown. (And the onlydesignated truth value for Kleene logic is True.) However, the lack of valid formulas does not mean that it lacksvalid arguments and/or inference rules. An argument is semantically valid in Kleene logic if, whenever (for anyinterpretation/model) all of its premises are True, the conclusion must also be True. (Note that the Logic of Paradox(LP) has the same truth tables as Kleene logic, but it has two designated truth values instead of one; these are: Trueand Both (the analogue of Unknown), so that LP does have tautologies but it has fewer valid inference rules.)[5]

4.2.2 ukasiewicz logicFurther information: ukasiewicz logic

The ukasiewicz 3 has the same tables for AND, OR, and NOT as the Kleene logic given above, but diers in itsdenition of implication. This section follows the presentation from Malinowskis chapter of the Handbook of theHistory of Logic, vol 8.[6]

In fact, using ukasiewiczs implication and negation, the other usual connectives may be derived as:

A B = (A B) B A B = (A B) A B = (A B) (B A)

Its also possible to derive a few other useful unary operators (rst derived by Tarski in 1921):

MA = A A LA = MA IA =MA LA

They have the following truth tables:M is read as it is not false that... or in the (unsuccessful) Tarskiukasiewicz attempt to axiomatize modal logicusing a three-valued logic, it is possible that... L is read it is true that... or it is necessary that... Finally I is readit is unknown that... or it is contingent that...

16 CHAPTER 4. THREE-VALUED LOGIC

In ukasiewiczs 3 the designated value is True, meaning that only a proposition having this value everywhere isconsidered a tautology. For example A A and A A are tautologies in 3 and also in classical logic. Not alltautologies of classical logic lift to 3 as is. For example, the law of excluded middle, A A, and the law ofnon-contradiction, (A A) are not tautologies in 3. However, using the operator I dened above, it is possible tostate tautologies that are their analogues:

A IA A [law of excluded fourth] (A IA A) [extended contradiction principle].

4.2.3 Bochvar logicMain article: Many-valued_logic Bochvar.27s_internal_three-valued_logic_.28also_known_as_Kleene.27s_weak_three-valued_logic.29

4.2.4 ternary Post logic

4.2.5 Modular algebrasSome 3VL modular algebras have been introduced more recently, motivated by circuit problems rather than philo-sophical issues:[7]

Cohn algebra Pradhan algebra Dubrova and Muzio algebra

4.3 Application in SQLMain article: Null (SQL)

The database structural query language SQL implements ternary logic as a means of handling comparisons withNULL eld content. The original intent of NULL in SQL was to represent missing data in a database, i.e. theassumption that an actual value exists, but that the value is not currently recorded in the database. SQL uses acommon fragment of the Kleene K3 logic, restricted to AND, OR, and NOT tables. In SQL, the intermediate value isintended to be interpreted as UNKNOWN. Explicit comparisons with NULL, including that of another NULL yieldsUNKNOWN. However this choice of semantics is abandoned for some set operations, e.g. UNION or INTERSECT,where NULLs are treated as equal with each other. Critics assert that this inconsistency deprives SQL of intuitivesemantics in its treatment of NULLs.[8] The SQL standard denes an optional feature called F571, which adds someunary operators, among which IS UNKNOWN corresponding to the ukasiewicz I in this article. The addition ofIS UNKNOWN to the other operators of SQLs three-valued logic makes the SQL three-valued logic functionallycomplete,[9] meaning its logical operators can express (in combination) any conceivable three-valued logical function.

4.4 See also Aymara language a Bolivian language famous for using ternary rather than binary logic[10]

Binary logic (disambiguation) Boolean algebra (structure) Boolean function

4.5. REFERENCES 17

Digital circuit Four-valued logic Setun - an experimental Russian computer which was based on ternary logic Ternary numeral system (and Balanced ternary) Three-state logic

4.5 References[1] Knuth, Donald E. (1981). The Art of Computer Programming Vol. 2. Reading, Mass.: Addison-Wesley Publishing Com-

pany. p. 190.

[2] Hayes, Brian (NovemberDecember 2001). Third Base. American Scientist (Sigma Xi, the Scientic Research Society)89 (6): 490494. doi:10.1511/2001.6.490.

[3] The Penguin Dictionary of Mathematics. 2nd Edition. London, England: Penguin Books. 1998. p. 417.

[4] Douglas W. Jones, Standard Ternary Logic, Feb. 11, 2013

[5] http://www.uky.edu/~{}look/Phi520-Lecture7.pdf

[6] Grzegorz Malinowski, Many-valued Logic and its Philosophy in Dov M. Gabbay, John Woods (eds.) Handbook of theHistory of Logic Volume 8. The Many Valued and Nonmonotonic Turn in Logic, Elsevier, 2009

[7] Miller, D. Michael; Thornton, Mitchell A. (2008). Multiple valued logic: concepts and representations. Synthesis lectureson digital circuits and systems 12. Morgan & Claypool Publishers. pp. 4142. ISBN 978-1-59829-190-2.

[8] Ron van der Meyden, "Logical approaches to incomplete information: a survey" in Chomicki, Jan; Saake, Gunter (Eds.)Logics for Databases and Information Systems, Kluwer Academic Publishers ISBN 978-0-7923-8129-7, p. 344; PS preprint(note: page numbering diers in preprint from the published version)

[9] C. J. Date, Relational database writings, 1991-1994, Addison-Wesley, 1995, p. 371

[10] El idioma de los aymaras (in Spanish). Aymara Uta. Retrieved 2013-08-20.

4.6 Further reading Bergmann, Merrie (2008). An Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras, and Deriva-tion Systems. Cambridge University Press. ISBN 978-0-521-88128-9. Retrieved 24 August 2013., chapters5-9

Mundici, D. The C*-Algebras of Three-Valued Logic. Logic Colloquium 88, Proceedings of the Colloquiumheld in Padova 6177 (1989). doi:10.1016/s0049-237x(08)70262-3

4.7 External links Introduction to Many-Valued Logics by Bertram Fronhfer. Handout from a Technische Universitt Dresden2011 summer class. (Despite the title, this is almost entirely about three-valued logics.)

18 CHAPTER 4. THREE-VALUED LOGIC

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Four-valued logicApplications Electronics Software

Notes

Many-valued logicHistoryExamples Kleene (strong) K3 and Priest logic P3 Bochvars internal three-valued logic (also known as Kleenes weak three-valued logic) Belnap logic (B4) Gdel logics Gk and G ukasiewicz logics Lv and LProduct logic Post logics Pm

Semantics Matrix semantics (logical matrices)

Proof theory Relation to classical logicSuszkos thesis

Applications Research venues See alsoNotesReferencesFurther reading External links

Problem of future contingentsAristotles solution Leibniz 20th century See also Notes Further reading External links

Three-valued logicRepresentation of valuesLogics Kleene and Priest logics ukasiewicz logic Bochvar logic ternary Post logic Modular algebras

Application in SQL See alsoReferencesFurther reading External links Text and image sources, contributors, and licensesTextImagesContent license