Editing Three-valued logic (section)

== Motivation ==

Broadly speaking, the primary motivation for research of three valued logic is to represent the truth value of a statement that cannot be represented as true or false.<ref name="Cobreros">{{cite journal |last1=Cobreros |first1=Pablo |last2=Égré |first2=Paul |last3=Ripley |first3=David |last4=Rooij |first4=Robert van |title=Foreword: Three-valued logics and their applications |journal=Journal of Applied Non-Classical Logics |date=2 January 2014 |volume=24 |issue=1-2 |pages=1–11 |doi=10.1080/11663081.2014.909631}}</ref> Łukasiewicz initially developed three-valued logic for the [[problem of future contingents]] to represent the truth value of statements about the undetermined future.<ref name="Prior">{{cite journal |last1=Prior |first1=A. N. |title=Three-Valued Logic and Future Contingents |journal=The Philosophical Quarterly |date=1953 |volume=3 |issue=13 |pages=317–326 |doi=10.2307/2217099 |url=https://www.jstor.org/stable/2217099 |issn=0031-8094|url-access=subscription }}</ref><ref name="Taylor">{{cite journal |last1=Taylor |first1=Richard |title=The Problem of Future Contingencies |journal=The Philosophical Review |date=1957 |volume=66 |issue=1 |pages=1–28 |doi=10.2307/2182851 |url=https://www.jstor.org/stable/2182851 |issn=0031-8108|url-access=subscription }}</ref><ref name="Rybaříková">{{cite journal |last1=Rybaříková |first1=Zuzana |title=Łukasiewicz, determinism, and the four-valued system of logic |journal=Semiotica |date=1 May 2021 |volume=2021 |issue=240 |pages=129–143 |doi=10.1515/sem-2019-0115}}</ref> [[Bruno de Finetti]] used a third value to represent when "a given individual does not know the [correct] response, at least at a given moment."<ref name="de Finetti">{{cite journal |last1=de Finetti |first1=Bruno |title=The logic of probability (translated) |journal=Philosophical Studies |date=1 January 1995 |volume=77 |issue=1 |pages=181–190 |doi=10.1007/BF00996317 |quote=But there is a second possible way to conceive of many-valued logics: that while a proposition, in itself, can have only two values, true or false, that is to say two responses, yes or no, it may happen that a given individual does not know the [correct] response, at least at a given moment; therefore, for the individual there is a third attitude possible toward a proposition. This third attitude does not correspond to a distinct third value of yes or of no, but simply to a doubt between yes or no}}</ref><ref name="Cobreros" /> [[Hilary Putnam]] used it to represent values that cannot physically be decided:<ref name="Putnam">{{cite journal |last1=Putnam |first1=Hilary |title=Three-valued logic |journal=Philosophical Studies |date=1 October 1957 |volume=8 |issue=5 |pages=73–80 |doi=10.1007/BF02304905 |quote=However, it is not the case that 'middle' means "neither verified nor falsified at the present time". As we have seen, 'verified' and 'falsified' are epistemic predicates--that is to say, they are relative to the evidence at a particular time--whereas 'middle,' like 'true' and 'false' is not relative to the evidence.}}</ref>

{{Blockquote
|text=For example, if we have verified (by using a speedometer) that the velocity of a motor car is such and such, it might be impossible in such a world to verify or falsify certain statements concerning its position at that moment. If we know by reference to a physical law together with certain observational data that a statement as to the position of a motor car can never be falsified or verified, then there may be some point to not regarding the statement as true or false, but regarding it as "middle". It is only because, in macrocosmic experience, everything that we regard as an empirically meaningful statement seems to be at least potentially verifiable or falsifiable that we prefer the convention according to which we say that every such statement is either true or false, but in many cases we don't know which.
}}

Similarly, [[Stephen Cole Kleene]] used a third value to represent [[Predicate (mathematical logic)|predicates]] that are "undecidable by [any] algorithms whether true or false"<ref name="Kleene">{{cite book |last1=Kleene |first1=Stephen Cole |title=Introduction to metamathematics |date=1952 |publisher=North-Holland Publishing Co., Amsterdam, and P. Noordhoff, Groningen |page=336 |quote=The strong 3-valued logic can be applied to completely defined predicates Q(x) and R(x), from which composite predicates are formed using ̅, V, &, ->, ≡ in the usual 2-valued meanings, thus, (iii) Suppose that there are fixed algorithms which decide the truth or falsity of Q(x) and of R(x), each on a subset of the natural numbers (as occurs e.g. after completing the definitions of any two partial recursive predicates classically). Let t, f, u mean 'decidable by the algorithms (i.e. by use of only such information about Q(x) and R(x) as can be obtained by the  algorithms) to be true', 'decidable by the algorithms to be false', 'undecidable by the algorithms whether true or false'. (iv) Assume a fixed state of knowledge about Q(x) and R(x) (as occurs e.g. after pursuing algorithms for each of them up to a given stage). Let t, f, u mean 'known to be true', 'known to be false', 'unknown whether true or false'.}}</ref><ref name="Cobreros" />