Issues of Vagueness 1

Some people, like 6 7 Gina Biggerly, are just plain tall Other people, like 4 7 Tina

Littleton, are just as plainly not tall But now consider Mary Middleford, who is

5 7 Is she tall? Well, kind of, but not really—certainly not as clearly as Gina is tall.

The statement "Mary Middleford is tall" is neither true nor false, presenting a counterexample to the Principle of Bivalence, which asserts that every declarative sentence must be either true or false This ambiguity arises from the vagueness of the predicate "tall," as it applies clearly to some individuals, fails to apply to others, and neither clearly applies nor fails to apply to borderline cases like Mary Middleford Consequently, the concept of tallness includes individuals whose height does not fit neatly into the categories of true or false.

Vague predicates differ from precise ones, which have clear boundaries in their application Mathematicians often use precise predicates to classify numbers, such as "even," which applies strictly to positive integers that are multiples of 2 This means that for any positive integer \( n \), the statement "n is even" is definitively true or false—1 is false, 2 is true, 3 is false, 4 is true, and so forth Therefore, "even" is classified as a precise predicate However, it's important to note that there are also vague predicates that can apply to positive integers, such as "large."

Classical logic, commonly taught in philosophy and mathematics, operates on the Principle of Bivalence, which states that every statement must be either true or false However, vagueness poses a significant challenge to this framework, as sentences with vague predicates may not fit neatly into these binary categories, leading to situations where they cannot be definitively classified as true or false.

In this article, we will use italics to denote sentences and terms when discussing them, following standard logical vocabulary conventions Unlike some alternatives, we do not employ quotation marks for this purpose Additionally, italics will be used for emphasis, with the context providing clarity on the distinction.

Classical logic falls short in representing sentences with vague predicates, as noted by philosopher Bertrand Russell, who argued that traditional logic relies on precise symbols, making it unsuitable for real-life situations Fuzzy logic was created to address this limitation, allowing for the inclusion of vague language and admitting an infinite range of truth-values beyond just true and false Unlike classical logic, fuzzy logic does not adhere to the Principle of Bivalence, making it a more flexible approach to reasoning in complex, uncertain scenarios.

Some will say,Why bother? Logic is the study of reasoning, and good reasoning— whether it be in the sciences or in the humanities—exclusively involves precise terms.

So we are justified in pursuing classical logic alone, tossing aside as don’t-cares any sentences that contain vague expressions But as Bertrand Russell pointed out in

In 1923, vagueness became a prevalent aspect of our communication, leading Max Black to assert in 1937 that an effective logic must be developed to tackle this vagueness in natural language discourse This necessity applies to both scientific discussions and everyday conversations, highlighting the importance of clarity in our interactions.

Deviations from logical or mathematical precision are common in symbolism, and labeling them as subjective creates a divide between formal laws and real-world experience, rendering the practical utility of formal sciences a mystery An adequate formal system eliminates the perception of vagueness as a language flaw Those who criticize vagueness often reference an ideal of scientific precision; however, indeterminacy, a hallmark of vagueness, is also inherent in scientific measurement Thus, vagueness is a characteristic present in both scientific and other forms of discourse.

Vague predicates are prevalent in both academic and everyday language, with examples including terms like hot, round, red, audible, and rich For instance, coffee can transition from hot to borderline hot over time, and the moon's shape can appear borderline round near the full moon Colors like red can also be subjective, as they blend into orange or purple at the edges of the spectrum Similarly, sound can shift from loud to borderline audible, and the concept of wealth can place individuals at the borderline of being rich Even numerical descriptions, often viewed as precise, can include vague terms like large and small.

Even in the absence of widespread vagueness, there are compelling reasons to create logics capable of addressing vague statements Exploring the necessary modifications to classical logic in this context proves to be both an intriguing and enlightening endeavor.

3 Black (1937), p 429 Black’s article is a gem, with its appreciation of the pervasiveness and use- fulness of vague terms and its attempt to formalize foundations for a logic that includes vague terms.

The Principle of Bivalence is challenged by issues of vagueness, leading to a reevaluation of the classical Law of Excluded Middle, which asserts that every statement of the form "A or not A" is true In classical logic, this law is a foundational concept, emphasizing precision However, the vagueness in statements, such as "Either Gina Biggerly is tall or she isn’t," raises questions about the applicability of this principle, highlighting the need for alternative approaches to address the logical complexities introduced by vagueness.

Tina Littleton's height can be classified as either tall or not tall, which aligns with logical reasoning However, the situation with Mary Middleford is more complex; it cannot be definitively stated that she is either tall or not tall, as neither assertion holds true.

The Principle of Bivalence and the Law of Excluded Middle are closely related concepts in logic The Principle of Bivalence asserts that every statement is either true or false Negation, indicated by "not," transforms a true statement into a false one and vice versa Consequently, for any statement A, either A is true or not A is true, which leads to the conclusion that the statement "either A or not A" must also be true, illustrating the Law of Excluded Middle.

Vagueness challenges classical logic and gives rise to the Sorites paradoxes For example, consider the predicate "tall." We start with the premise that Gina Biggerly is tall The second premise states that a height difference of 1/8 does not affect tallness; thus, someone who is 1/8 shorter than a tall person is still considered tall Consequently, this reasoning leads to the conclusion that Tina Littleton, who is 4/7 shorter than Gina, must also be tall.

Biggerly's height is categorized as tall, indicating that individuals measuring 6 feet 6 inches or taller are considered tall By applying this reasoning, we can deduce that those who are 6 feet 6 inches, 6 feet 5 inches, and progressively shorter heights are also classified as tall Ultimately, this leads us to the conclusion that Tina Litteleton, along with others who are 4 feet 7 inches tall, falls into the category of being tall.

The Sorites paradox, derived from the Greek word for "heap," explores the concept of vagueness in terms It begins with the premise that a large pile of sand, such as one that is four feet deep, qualifies as a heap The paradox suggests that if you remove a single grain of sand from this heap, what remains should still be considered a heap Through this iterative reasoning, one could absurdly conclude that even a solitary grain of sand is a heap, leading to the implication that even zero grains could also be classified as a heap This illustrates the challenges of defining vague terms like "heap."

Premise 1 x is T (where x is something of which T is clearly true).

Premise 2 Some type of small change to a thing that is T results in something that is also T.

4 It is possible to retain the Law of Excluded Middle while rejecting bivalence; this is the case for supervaluational logics For references see footnote 1 to Chapter 5.

Vagueness Defined 5

As the population grows, it can initially appear that a slight increase, such as 0.01 percent, will not significantly impact the current living standards However, this assumption is misleading, as continuous population growth, even at minimal rates, can ultimately lead to unsustainable conditions It is crucial to recognize that even small increases can accumulate over time, jeopardizing the quality of life and resources available to future generations.

Max Black describes the vagueness of a term as the presence of objects within its application field that make it inherently impossible to determine whether the term applies or not.

The field of application of a term defines the specific entities it relates to; for example, the term "tall" applies to people and buildings but not to integers or colors, as these cannot be described as tall Conversely, the term "even" encompasses integers while excluding people, colors, and buildings Understanding the field of application is crucial for accurately using terms in various contexts.

It is intrinsically impossible to definitively categorize Mary Middleford as tall or not tall, as this situation transcends mere ignorance about her height Even with precise knowledge of her height, we cannot apply the label of "tall" or "not tall" to her This contrasts with situations where uncertainty arises from a lack of information, such as when assessing the height of the author's brother, Barrie Bergmann In this case, the inability to determine if Barrie is tall stems from ignorance, rather than an intrinsic limitation of the term itself.

The concept of borderline cases refers to objects that fall within a term's field of application but cannot definitively be classified under that term For example, Mary Middleford is considered a borderline case for the term "tall," as her height does not clearly fit the definition In contrast, individuals like Gina, Tina, and Barrie do not fall into this fringe category and can be distinctly categorized as tall or not tall.

The term "precise" stands in stark contrast to "vague." For instance, the description "exactly 6'2" is precise, as it clearly defines a specific height In any given context, this term can definitively apply or not apply to an object based on whether it meets the exact measurement Similarly, the term "even," when referring to positive integers, is also precise; it applies if the integer is a multiple of 2 and does not apply otherwise This clarity in definition underscores the importance of precision in language.

8 Black (1937, p 430) For the most part we will restrict our attention to terms that are adjectives

In addition to adjectives (such as "tall"), common nouns (like "chair"), and verbs (for example, "to smile"), there are various parts of speech that can be ambiguous in their usage This topic will be explored further in Chapter 16.

English speakers often use the term "vague" to describe terms that lack specificity, but it's more accurate to label such terms as "general." For instance, the word "interesting" is general because it can refer to various attributes, such as unique facts, compelling arguments, or unusual writing styles Unlike the specific term "tall," which clearly indicates height, "interesting" is not precise While general terms can also be vague, it's crucial to differentiate between generality and vagueness Furthermore, vagueness should not be confused with ambiguity; a term is ambiguous if it has multiple distinct meanings For example, "light" can refer to color or weight, allowing for interpretations that can overlap or be mutually exclusive Philosopher W V O Quine even suggested that the existence of an object to which a term both applies and does not apply serves as a test for ambiguity.

Ambiguity does not create borderline cases; however, an ambiguous term can possess vagueness in various meanings Certain objects can be considered borderline cases for being light in color, while others may be borderline cases for being light in weight.

Vagueness differs from relativity, as a term is considered relative when its applicability changes based on specific subclasses within its field Many vague terms are also relative; for instance, when describing a woman as tall, we often mean tall relative to women in general, and even more specifically, tall for a particular race or ethnicity Therefore, the term "tall" varies in its applicability depending on the class of individuals being referenced.

The Problem of the Fringe 6

A term is considered vague when there is a fringe area within its applicability Max Black highlighted a logical issue that emerges from borderline cases where a term neither applies nor fails to apply This raises the question of whether objects exist within the fringe of a term's definition.

There are objects that are neither tall nor not tall, or equivalently

There are objects that are both not tall and not not tall.

Preview of the Rest of the Book 7

The Principle of Double Negation asserts that a doubly negated expression is equivalent to its affirmative form, meaning that two negations cancel each other out For example, the phrase "not not tall" is equivalent to simply stating "tall." Consequently, we can also claim that there are objects that exist within the fringe of the term "tall."

There are objects that are not tall and also tall.

The assertion that a term meets the criteria for vagueness, particularly the existence of borderline cases, leads to a contradiction, violating the Law of Noncontradiction This law states that no proposition can be both true and false simultaneously, and in this context, it indicates that an object cannot simultaneously possess and lack a property This dilemma, which we refer to as the Problem of the Fringe, highlights a significant challenge that must be resolved within a comprehensive logic framework for vagueness.

1.4 Preview of the Rest of the Book

This article explores the intersection of logic and philosophy, focusing on various logical systems, including fuzzy logic It emphasizes the importance of critically analyzing these systems, highlighting that students may be surprised to discover that logic can be questioned The discussion aims to clarify that a critical examination of logical systems is essential for developing a logic capable of addressing vague statements Additionally, the article notes that choosing not to pursue such a logic reflects a philosophical stance on the nature and purpose of logic, potentially limiting it to precise reasoning only.

In Chapters 2 and 3, we will examine classical propositional logic and first-order logic, establishing a foundational framework and introducing essential notation and terminology for the upcoming chapters Chapter 4 will focus on Boolean algebras, which are systems designed to encapsulate these logical principles.

Classical logic imposes an "algebraic" structure on truth-values, yet Boolean algebras are often overlooked in introductory symbolic logic courses This chapter introduces the topic, as algebraic analyses play a significant role in understanding formal fuzzy logic systems.

The earlier assertion that there are objects that are neither tall nor not tall contradicts the Law of Noncontradiction However, by following Black's approach of eliminating the double negation, we can clarify the argument.

In Chapters 5 and 6, we will discuss several prominent systems of three-valued propositional logic that abandon the Principle of Bivalence Chapters 7 and 8 will focus on first-order versions of these three-valued systems Chapter 9 will delve into the algebraic structures associated with three-valued logic We will consider these logical systems as potential frameworks for addressing vagueness, with some readers finding them sufficient while others may not Regardless of individual perspectives, the exploration of three-valued systems will reveal many principles that seamlessly transition into the realm of fuzzy logic.

Chapter 10 presents two significant issues related to vagueness within three-valued logical systems These challenges serve as a catalyst for transitioning from three-valued logic to fuzzy logic, where formulas can possess an infinite range of truth values.

Chapters 11 and 12 introduce fuzzy propositional logic, covering both semantics and derivation systems, while Chapter 13 focuses on algebras for fuzzy logics In Chapters 14 and 15, the discussion shifts to fuzzy first-order logic Chapter 16 explores the enhancement of fuzzy logic by incorporating fuzzy qualifiers, such as determining the meaning of "very tall," and fuzzy linguistic truth values that assess the degree of truth in statements Finally, Chapter 17 delves into the challenges of defining membership functions for vague concepts within fuzzy logic.

History and Scope of Fuzzy Logic 8

In the 1920s, Polish logician Jan Lukasiewicz pioneered formal infinite-valued logics, laying the groundwork for formal fuzzy logic He created a range of many-valued logical systems, progressing from three-valued to infinite-valued systems, each expanding on the previous to accommodate more truth-values While many prominent fuzzy logics are derived from Lukasiewicz's infinite-valued system, his philosophical focus was on indeterminism rather than vagueness, a topic that will be explored in Chapter 5.

In 1965, Lotfi Zadeh introduced the concept of fuzzy sets, which allow for varying degrees of membership, contrasting with classical sets where membership is binary An example provided by Zadeh is the set of tall men, highlighting the connection between vague terms and fuzzy sets Two years later, Joseph Goguen expanded on Zadeh's work by linking fuzzy sets to broader algebraic structures and relating them to infinite-valued logic, laying the groundwork for formal fuzzy logic Goguen's contributions marked the emergence of fuzzy logic in a more formal context, particularly through his analysis of Sorites arguments.

In 1979, Jan Pavelka published a three-part article that established a comprehensive framework for fuzzy logic, building on the work of Goguen He introduced a complete and consistent axiomatic system for fuzzy logic, contributing significantly to the field.

Fuzzy logic has evolved significantly, beginning with propositional fuzzy logic that utilizes "graded" rules of inference These two-part rules establish how one formula can be derived from others while defining the minimal degree of truth for the derived formula based on the original formulas' truth degrees Notably, Pavelka's work introduced several crucial metatheoretic results In 1990, Vilém Novák expanded this foundation to first-order fuzzy logic, and between 1995 and 1997, Petr Hájek made substantial simplifications to these systems, enhancing their applicability and understanding.

In 1998, H´ajek introduced an axiomatic system known as BL (basic logic), which effectively encapsulates the shared features of prominent formal fuzzy logics This system is accompanied by a corresponding algebraic structure, referred to as BL-algebra.

Nov´ak and H´ajek have dominated the field of fuzzy logic (in the narrow sense) with several texts and numerous articles, more of which will be cited later.

This article focuses specifically on fuzzy logic in its narrow sense, distinguishing it from fuzzy set theory and the broader interpretation of fuzzy logic While fuzzy set theory is integral to fuzzy logic, it remains a separate field The concept of fuzzy logic in its broader context was introduced by Zadeh in a 1975 article, where he described it as a logic characterized by fuzzy truth-values expressed in linguistic terms, imprecise truth tables, and rules of inference that are approximate rather than exact However, the term "logic" as applied here may not align with the traditional understanding of the term among logicians.

In this article, we will briefly explore Zadeh's linguistic truth-values, which can address certain philosophical objections to fuzzy logic Zadeh suggests that approximate reasoning leads to conclusions like: if object 'a' is small and object 'b' is approximately equal to 'a', then it follows that 'b' is also considered more or less small.

As is evident, the logic behind these rules allows us to conclude that if two objects are “approximately” equal and one has a certain property, then the other object

“more or less” has that property The rules used in computational systems based on

Zadeh's fuzzy logic, often described as rules of thumb, is expressed in natural language and proves to be highly effective in applications like expert systems A common example of a rule in a fuzzy expert system illustrates its practical use.

In conditions where temperature is high and humidity is low, the garden tends to become dry This relationship is analyzed using fuzzy sets, where temperature and humidity are treated as data points Zadeh refers to this approach as linguistic logic, highlighting its relevance in the broader context of fuzzy logic.

10 This and further work of Nov´ak’s appears in Nov´ak, Perfilieva, and Moˇckoˇr (1999).

10 Introduction research 11 Ruspini, Bonissone, and Predrycz (1998) is a good introduction to fuzzy logicin the broadsense.

Fuzzy logic technology has been utilized in various appliances for over a decade, including rice cookers that mimic a real cook's intuition by adjusting heat based on rice type, volume, and cooking time This technology is also found in washing machines, blood pressure monitors, and automatic transmission systems in cars, where microchip circuit logic manages imprecise measurements For further insights into fuzzy technologies, refer to Hirota (1993).

Tall People 10

Visit the Web site http://members.shaw.ca/harbord/heights.html This is fun and will get you thinking about whattallmeans.

Exercises 10

In his article "Vagueness," Max Black argues that sensory terms, such as color and shape descriptors, are inherently vague For instance, the term "green" can apply to the sea, which may appear greenish in certain lighting, creating a borderline case that is neither fully green nor entirely not green Similarly, the moon is considered round when full, but during its quarter phases, it presents borderline characteristics, making it difficult to classify as definitively round or not.

Vague terms often challenge our understanding of sensory experiences For hearing, the term "loud" can be subjective; for instance, a sound that is loud to one person may be merely audible to another, especially in different environments When it comes to smell, the word "fragrant" can be ambiguous, as some may find a particular scent delightful while others perceive it as overpowering In taste, the term "sweet" can vary significantly; a dish may be considered sweet by one individual yet bland or even savory by another, depending on personal preferences and cultural influences Lastly, the term "soft" in relation to touch can lead to confusion, as a fabric may feel soft to the touch but rough against the skin of someone with heightened sensitivity These examples illustrate how vague terms can create uncertainty in sensory descriptions.

2 Show that each of the following terms is vague by giving an example of a bor- derline case:young, fun, husband, sport, stale, chair, many, flat, book, sleepy.

3 Are any of the terms in question 2 also ambiguous? General? Relative? Give examples to support your claims.

Introducing a specific term to differentiate formal fuzzy logic, based on Goguen's work, from Zadeh's interpretation would clarify discussions surrounding fuzzy logic This distinction would help identify when critiques from logicians, like Susan Haack in 1979, are aimed at the broader notion of fuzzy logic as a form of logic, rather than the formalized work within the field.

4 Produce a version of the Sorites paradox using the termrich.

Sorites arguments can be constructed for terms exhibiting multidimensional vagueness, although they are most commonly associated with unidimensional vagueness The nature of vagueness allows for the formulation of Sorites paradoxes in both contexts, as the gradual shifts in meaning can lead to similar logical challenges However, the complexity of multidimensional vagueness may introduce additional nuances that complicate the construction of these arguments Ultimately, while unidimensional vagueness provides a clearer framework for Sorites arguments, multidimensional vagueness also presents opportunities for their development.

2 Review of Classical Propositional Logic

The Language of Classical Propositional Logic 12

In propositional logic, the fundamental linguistic units are simple sentences and the logical connectives that link them We represent atomic formulas, which denote these simple sentences, using uppercase Roman letters, and we employ integer subscripts when necessary Additionally, we symbolize English connectives to facilitate understanding and application in logical expressions.

English Connective Logical Operation Symbol not negation ơ and conjunction ∧ or disjunction ∨ if then conditional → if and only if biconditional ↔

The negation connective is classified as a unary connective because it operates on a single formula, while the other connectives are considered binary as they combine pairs of formulas For instance, in symbolized English sentences, "J" represents "John is a mathematician," "C" stands for "Christy is a mathematician," and "P" indicates "Christy is a philosopher."

John is not a mathematician ơJ

John is a mathematician and so is Christy J∧C Christy is either a mathematician or a philosopher C∨P

If John is a mathematician, then Christy’s a philosopher J→P John is a mathematician if and only if Christy is as well J↔C

The negation connective has the highest binding priority among logical connectives, meaning that in the absence of parentheses, it applies solely to the formula it directly precedes For instance, in the expression J∨C, the negation applies to J, resulting in the interpretation that either John isn't a mathematician or Christy is To change this default priority, parentheses are necessary.

1 Some common alternative symbols are ∼, −, ! for negation; &, ã for conjunction; |, + for disjunc- tion; ⊃, ⇒ for the conditional operation; and = , ≡, ⇔ for the biconditional operation.

Semantics of Classical Propositional Logic 13

and to indicate grouping among connectives of the same priority So, for example, ơ(J∨C)symbolizesIt’s not true that John or Christy is a mathematician(i.e.,Neither

In the logical expression (J∧C)∨P, the parentheses clarify that either both John and Christy are mathematicians, or Christy is a philosopher This distinction is crucial, as it prevents misinterpretation of the statement, which could otherwise suggest that John is a mathematician while Christy could be either a mathematician or a philosopher.

In logical expressions, the use of parentheses is essential to clarify the order of evaluation for binary connectives For instance, while the expression J∨C∨P can be interpreted in multiple ways, it is crucial to specify the order by using parentheses, such as in (J∨C)∨P or J∨(C∨P), to ensure accurate interpretation.

The rules for formingformulasin the language of classical propositional logic are as follows:

1 Every uppercase roman letter, with or without an integer subscript, is a formula.

3 IfPandQare formulas, so are (P∧Q), (P∨Q), (P→Q), and (P↔Q) 2

In logic, single roman letters such as A and B are referred to as atomic formulas, while formulas that utilize one or more connectives are known as compound formulas The primary connectives used in these formulas are defined in earlier sections For instance, in the formula \((A∨B)∨C\), which is derived from the atomic formulas \((A∨B)\) and \(C\) through the application of connectives, the main connective is identified as the second disjunction Additionally, it is standard practice to omit outermost parentheses in compound formulas, allowing \((A∨B)∨C\) to be expressed simply as \(A∨B∨C\).

A compound formula is defined by the operation represented by its main connective, such as a negation when the main connective is "not." The context will clarify whether we refer to the operation or the formula itself In a conjunction, the immediate subformulas are represented by P and Q.

P∧Qare called its conjuncts , and the immediate subformulasPandQof a disjunc- tionP∨Qare called its disjuncts Pis theantecedentof the conditionalP→Qand

Qis itsconsequent (At times we will also refer to connectives by the name of the operation they symbolize; e.g., we will callơanegation.)

2.2 Semantics of Classical Propositional Logic

The five logical connectives discussed are truth-functional, meaning the truth-values of the formulas they create depend on the truth-values of the individual constituent formulas.

In our discussion, we utilize boldface letters such as P, Q, and others to represent arbitrary formulas within the language, distinguishing them from specific formulas, which we denote with nonboldface letters This convention clarifies when we refer to a general formula, represented by boldface letters, known as metavariables.

14 Review of Classical Propositional Logic

The truth-functional operations in classical propositional logic are captured by the followingtruth-tables:

In classical logic, truth-values are assigned to atomic formulas, with "true" (T) and "false" (F) being the only options By using truth-value assignments, we can evaluate the truth-values of various logical formulas For instance, if a truth-value assignment indicates that J is false and C is true, then the truth-values of related expressions can be determined: the negation of J (¬J) is true, the negation of C (¬C) is false, the conjunction (J ∧ C) is false, the disjunction (J ∨ C) is true, the implication (J → C) is true, the reverse implication (C → J) is false, and the biconditional (C ↔ J) is also false.

More generally, a truth-table can be used to display the values that a formula will have on all truth-value assignments Here’s a truth-table for the formula ơJ∧(C∨R):

The article outlines the truth-value combinations for variables C, J, and R, displaying the resulting values of the expression ơJ∧(C∨R) alongside its subformulas Each atomic subformula's truth-value is indicated directly beneath it, while the truth-values for compound subformulas and the overall formula are shown under their main connectives Specifically, the first row indicates that the subformula ơJ is false when C, J, and R are assigned true values, leading to the conclusion that the entire formula is also false under this assignment, although the disjunction yields a true value.

2.2 Semantics of Classical Propositional Logic 15

The semantics of a language pertains to its meaning or interpretation In classical propositional logic, this semantics involves bivalent truth-value assignments and the definitions of truth-functional operations, which are essential for constructing a truth table for any given formula.

Logicians often identify certain connectives as primitive while defining others in relation to them A typical approach involves treating the logical connectives "or" (ơ) and "and" (∧) as primitive, subsequently defining the remaining connectives based on these fundamental ones.

The symbol "= def" signifies "is defined as." This definition is supported by logical reasoning, as demonstrated by the formula ơ(ơ P ∧ ơ Q), which holds true when ơ P ∧ ơ Q is false This occurs when either or both of ơ P and ơ Q are false, meaning that at least one of P or Q is true, which aligns with the definition of disjunction P ∨ Q being true Additionally, the accuracy of this definition can be confirmed through a truth table.

The truth-values for the disjunction match those of the first negation in the second formula, demonstrating that the two formulas are equivalent Additionally, the accuracy of the definitions for both the conditional and biconditional can be confirmed through a similar verification process.

Another common (and similar) way of dividing the connectives into primitive and defined ones is to takeơand∨as primitive and then to introduce the others as:

The two connectivesơand→can also be taken as primitive, as will be confirmed in an exercise.

In classical propositional logic, formulas that are universally true across all truth-value assignments are known as tautologies, while those that are universally false are referred to as contradictions The truth-values listed in the truth-table beneath a formula's main connective reveal whether the formula qualifies as a tautology, a contradiction, or neither.

16 Review of Classical Propositional Logic of the Law of Excluded Middle is A ∨ ơA, and this is a tautology in classical logic:

The column of truth-values under the logical operator ∨ contains only Ts, demonstrating that the formula A ∨ ¬A is a tautology It's important to note that a truth-value assignment must assign truth-values to all atomic formulas within the language Consequently, the truth-table we analyzed does not represent complete truth-value assignments, but every truth-value assignment must ultimately assign one of the two values, T or F.

FtoA, the truth-table shows us that the formulaA∨ ơAis true onalltruth-value assignments.

The formulaA∧ ơAis a contradiction, while its negation, which is often called the Law of Noncontradiction , is a tautology:

The second formula's primary connective is the initial negation connective, which is accompanied solely by Ts Importantly, none of the previous formulas representing assertions about John and Christy qualify as tautologies or contradictions For instance, we can observe this through specific examples.

F F F F F F F F T F and in each case the column under the main connective contains both Ts andFs 3

Two formulas of classical propositional logic are equivalent if they have the same truth-value on each truth-value assignment The formulasC→ Jand ơJ→ ơCare equivalent:

3 For uniformity we always list the atomic constituents of formulas in alphabetical order, even when that is not the order in which they appear in compound formulas.

2.2 Semantics of Classical Propositional Logic 17

The columns under the main connectives of the two formulas, the conditional connective in each case, are identical On the other hand, the formulasC→Jand

Normal Forms 18

This section introduces disjunctive and conjunctive normal forms for propositional logic formulas Every propositional formula can be represented in both disjunctive normal form (DNF) and conjunctive normal form (CNF), which standardizes logical expressions for various applications, including the computational proof technique known as resolution (Robinson 1965) We will utilize these normal forms to demonstrate functional completeness in Section 2.5 Additionally, these forms facilitate semantic connections between classical logic, three-valued logics, and fuzzy logics.

We’ll begin with disjunctive normal form First we define literals to include all atomic formulas and their negations:A,ơA,B,ơB, Next we define what a phrase is:

A phrase is either a single literal, or a conjunction of literals:A,ơA,B,ơB,A∧A,

A∧ ơB,(D∧ ơE)∧F, and so forth Finally, we define disjunctive normal form :

1 Every phrase is in disjunctive normal form

2 IfPandQare in disjunctive normal form, so is(P∨Q)

A formula is considered to be in disjunctive normal form if it utilizes only the three logical connectives: negation (¬), conjunction (∧), and disjunction (∨) In this form, negations are restricted to appearing solely in front of atomic formulas, and no conjunctive subformulas may include disjunctions Examples of such formulas include A and A ∨ ¬B.

A∧ ơB,(A∧ ơB)∨B,(A∧ ơB)∧B, and(C∧ ơE)∨(D∨(E∧F)).

We assert that every formula in classical propositional logic can be transformed into at least one equivalent formula in disjunctive normal form (DNF) To demonstrate this, we outline a method for converting any classical propositional logic formula into DNF, ensuring that each transformation step maintains equivalence This process relies on established logical equivalences.

P→Q is equivalent to ơP∨Q (Implication)

P↔Q is equivalent to (ơP∨Q)∧(ơQ∨P) (Implication) ơ(P∧Q) is equivalent to ơP∨ ơQ (DeMorgan’s Law) ơ(P∨Q) is equivalent to ơP∧ ơQ (DeMorgan’s Law) 4 ơơP is equivalent to P (Double Negation)

(Proof that these forms are equivalent is left as an exercise.) We’ll explain the trans- formation process using the formula ơ(ơ(P→Q)∨(ơR→(S↔T)))

First we use the Implication equivalences to eliminate all conditionals and bicon- ditionals, producing the formula ơ(ơ(ơP∨Q)∨(ơơR∨((ơS∨T)∧(ơT∨S))))

Next we use DeMorgan’s Laws to move negations deeper into the formula until all negations appear in front of atomic formulas In our example we use the second

DeMorgan Law first to obtain ơơ(ơP∨Q)∧ ơ(ơơR∨((ơS∨T)∧(ơT∨S))) and then ơ(ơơP∧ ơQ)∧(ơơơR∧ ơ((ơS∨T)∧(ơT∨S))

Next we can use the first DeMorgan Law twice to obtain

(ơơơP∨ ơơQ)∧(ơơơR∧(ơ(ơS∨T)∨ ơ(ơT∨S))) and then the second Law twice more to obtain

(ơơơP∨ ơơQ)∧(ơơơR∧((ơơS∧ ơT)∨(ơơT∧ ơS)))

Now Double Negation eliminates all double negations:

We are nearing completion, but it's important to recognize that this is a conjunction, and some of the components are disjunctions Therefore, the formula has not yet achieved disjunctive normal form, which would require all components to be conjunctions.

We can use the first Distribution equivalence to convert the overall formula into a disjunction, withơPin place ofP,Qin place ofQ, and(ơR∧((S∧ ơT)∨(T∧ ơS))) in place ofR:

(ơP∧(ơR∧((S∧ ơT)∨(T∧ ơS))))∨(Q∧(ơR∧((S∧ ơT)∨(T∧ ơS)))).

The two laws, named after the 19th-century mathematician Augustus DeMorgan, are foundational principles in set theory Although commonly attributed to DeMorgan, historical accounts suggest that these laws were recognized as far back as scholastic times and possibly even in ancient history.

20 Review of Classical Propositional Logic

Next we can apply the second Distribution equivalence to both occurrences of the subformula(ơR∧((S∧ ơT)∨(T∧ ơS)), withơRin place ofP,(S∧ ơT)in place of

Q, and(T∧ ơS)in place ofR, to obtain

(ơP∧((ơR∧(S∧ ơT))∨(ơR∧(T∧ ơS))))∨

This formula is in disjunctive normal form, and it is equivalent to the original formula ơ(ơ(P→Q)∨(ơR→(S↔T))) because each step of the transformation produces an equivalent formula.

To convert any formula to its equivalent in disjunctive normal form, we follow a systematic process: first, we eliminate conditional and biconditional connectives Next, we apply DeMorgan's Laws to push negations as far as possible and use Double Negation to remove all instances of double negation At this stage, all negations will be positioned before atomic formulas, with binary connectives reduced to conjunctions or disjunctions Finally, we utilize Distribution laws to transform the expression into disjunctive normal form by substituting each conjunction with a disjunctive conjunct, resulting in a disjunction of conjunctive disjuncts Additionally, Section 2.5 will introduce an alternative method for achieving disjunctive normal form.

In contrast, formulas in conjunctive normal form are conjunctions of disjunc- tions, rather than disjunctions of conjunctions We define a clause as:

2 IfPandQare clauses, so is(P∨Q) and conjunctive normal form as:

1 Every clause is in conjunctive normal form

2 IfPandQare in conjunctive normal form, so is(P∧Q)

Any formula in classical propositional logic can be transformed into an equivalent formula in conjunctive normal form This process mirrors the steps taken for disjunctive normal form, but it concludes with the application of specific distribution equivalences.

A formula(P∧Q)∨Ris equivalent to(P∨R)∧(Q∨R) (Distribution)

The formula P∨(Q∧R) is equivalent to (P∨Q)∧(P∨R), demonstrating the principle of distribution in logic Complementary literals are defined as a pair where one literal negates the other, such as S and ¬S Two significant general results arise from this understanding of complementary pairs in logical expressions.

Result 2.1: A clauseCof classical propositional logic is a tautology if and only ifCcontains a complementary pair of literals.

A tautology, represented by clause C, must be a disjunction because no single literal can be a tautology For every truth-value assignment, at least one disjunct is true, which implies that clause C must include a complementary pair of literals.

In an axiomatic system for classical propositional logic, the truth-values of literals can be independent, leading to scenarios where all literals are false For instance, the expression P ∨ (((Q ∨ Q) ∨ (¬R ∨ ¬S)) ∨ T) evaluates to false when P, Q, and T are false while R and S are true Conversely, when a disjunctive formula C includes a complementary pair of literals, at least one of these literals must be true in every truth-value assignment, ensuring that the formula C itself is true.

Result 2.2: A phrasePof classical propositional logic is contradictory if and only ifPcontains a complementary pair of literals.

Proof: Left as an exercise.

As a consequence of 2.1 and 2.2 we have

Result 2.3: A formulaPof classical logic that is in conjunctive normal form is a tautology if and only if each clause inPcontains a complementary pair of literals.

A formula in conjunctive normal form (CNF) can be classified as either a clause or a conjunction A clause is considered a tautology if it contains a complementary pair of literals, as established in Result 2.1 Conversely, a conjunction is a tautology only if all its individual conjuncts are tautologies, which occurs when the clauses forming the conjunctions are also tautologies According to Result 2.1, each of these clauses qualifies as a tautology if they each contain a complementary pair of literals.

Result 2.4: A formulaPof classical logic that is in disjunctive normal form is a contradiction if and only if each phrase inPcontains a complementary pair of literals.

Proof: Left as an exercise.

An Axiomatic Derivation System for Classical

Derivation systems, such as axiomatic and natural deduction systems, play a crucial role in assessing formulas and evaluating arguments Axiomatic systems, commonly used in fuzzy logic, form the foundation of this discussion In an axiomatic derivation system, a collection of formulas is designated as axioms, accompanied by rules for deriving new formulas from existing ones Numerous axiomatic systems have been explored within classical propositional logic, and the one presented here is a notable example.

Polish logician JanLukasiewicz (1930,1934) to simplify a system proposed by the

5 For examples of natural deduction derivation systems see Bergmann, Moor, and Nelson (2004).

22 Review of Classical Propositional Logic

The Frege-Lukasiewicz system, referred to as CLA (classical propositional logic axiomatic system), is based on the work of German logician Gottlob Frege (Frege 1879) This system is particularly noteworthy as it allows for simple modifications that can axiomatize both three-valued and fuzzy logics, which were also pioneered by Lukasiewicz CLA is comprised of three axiom schemata.

CL1.P→(Q→P) CL2 (P→(Q→R))→((P→Q)→(P→R)) CL3 (ơP→ ơQ)→(Q→P) and the single inference rule MP, which is short for the rule’s traditional name, Modus Ponens:

MP(Modus Ponens) FromPandP→Q, inferQ.

Anaxiom schema represents an infinite set of axioms, encompassing all formulas that fit a specific structural form These formulas are referred to as instances of the axiom schema An instance is defined as any formula derived from uniformly substituting formulas from the language for each of the letters P, Q, and R, where the same formula is used consistently for every occurrence of each letter For instance, various formulas can be considered instances of the axiom schema CL1.

Note that each of the axiom schemata has the form of a tautology For example, here is the truth-table for CL2:

It can be verified that the remaining two axiom schemata also possess tautological forms Furthermore, the rule of Modus Ponens is a truth-preserving rule, meaning that when applied to true formulas, it yields a true formula; specifically, if both P and P→Q are true, then Q must also be true.

2.4 An Axiomatic System for Classical Propositional Logic 23

A derivation consists of a series of formulas, each identified as an assumption, an instance of an axiom schema, or derived from previous formulas using the Modus Ponens (MP) rule Notably, all assumptions must initiate the derivation, a requirement that, while not theoretically essential, simplifies the formulation of rules in subsequent chapters without compromising derivational strength.

We have derived the conclusion M from the assumptions A and (B → A) →

In the derivation from A to M, each formula is sequentially numbered and accompanied by annotations that clarify the reasoning behind its inclusion The first two lines are based on assumptions, while line three is derived from an axiom schema, specifically CL1, with substitutions made for P and Q Lines four to six demonstrate the application of the Modus Ponens (MP) rule, referencing earlier formulas A formula is considered derived from a set of assumptions when it appears within that context, illustrating the logical progression of the derivation.

Based on the assumptions presented in lines 1 and 2, we can derive important conclusions If a formula can be established from a specific set of assumptions, it is also considered derivable from any larger set that includes those assumptions Therefore, the previous derivation confirms that the formula is valid within this broader context.

Mis derivable fromanyset that contains bothAand(B→A)→(A→M) As another example, the following derivation shows thatA→Bis derivable fromơA:

2 ơA→(ơB→ ơA) CL1, with A /P, B /Q

Some formulas, namely, the tautologies of classical logic, can be derived without using any assumptions For example,A→Ais a tautology and we can derive it without any assumptions as follows:

24 Review of Classical Propositional Logic

A proof is a derivation that lacks assumptions, while a formula is classified as a theorem if it has a proof concluding with that formula The recent proof demonstrates that A → A is a theorem, and we can refer to this proof as a proof of that theorem Another significant theorem is ơA → (A → B) Before we present its proof, it's important to note the relationship between this theorem and the derivation of A → B from ơA, where the antecedent and consequent of the theorem correspond to these two formulas.

Qis derivable from a formulaPin CLA there is a corresponding theoremP→Qin CLA, and vice versa This general fact is known as the Deduction Theorem: 6

Result 2.5 (The Deduction Theorem for Classical Propositional Logic):Qis deriv- able fromPin CLA if and only ifP→Qis a theorem in CLA.

The theorem states that if P → Q is established, then Q can be derived from P This is straightforward to prove: when P → Q is a theorem, a proof exists within the context of Classical Logic (CLA) By adding P as an assumption to this proof, we can apply Modus Ponens to conclude Q from both P and P → Q.

In this article, we will demonstrate how to transform any derivation of Q based on the assumption of P into a proof of the implication P → Q, thereby confirming the theoremhood of the latter We will also provide an example by converting our derivation of Q to illustrate this method effectively.

The goal is to transform a derivation of Q from the assumption P into a sequence of theorems, specifically P → P, P → R1, P → R2, , P → Rn, culminating in the theorem P → Q This approach demonstrates that each formula in the derivation, including the final theorem, stands as a theorem in its own right, as the new derivation does not rely on the assumption P or any other assumptions.

We begin the derivation by deriving the first new formulaP→Pexactly as we derivedA→Aon the previous page, usingPin place ofA Now, each ofR 1,

In a logical derivation, each formula R2 through Rn can either represent an instance of an axiom schema or be derived from preceding formulas using Modus Ponens (MP) For any formula Ri that is an instance of an axiom schema, we can extend the derivation by adding specific lines to support the conclusion.

P→R i in the new derivation: m R i {by relevant axiom schema} m+1 R i →(P→R i ) CL1, withR i /P,P/Q m+2 P→R i m+1,m+2 MP

IfR i followed by MP from earlier formulasR k andR k →R i in the sequenceR 1,

R 2, ,R i–1,R n , then we already haveP→R k andP→(R k →R i ) in the new derivation, say, on linesmandn, and three additional lines will deriveP→R i :

6 There is also a semantic version of the Deduction Theorem: the set { P } entails Q if and only if

The statement "P implies Q" is a tautology, which can be easily demonstrated through direct proof Additionally, this conclusion aligns with Result 2.5, highlighting the soundness and completeness of classical propositional logic (refer to page 26).

2.4 An Axiomatic System for Classical Propositional Logic 25 m P→R k

Using the method described in the proof of the Deduction Theorem and the earlier derivation

2 ơA→(ơB→ ơA) CL1, with A /P, B /Q

5 A→B 3,4 MP we construct the following derivation establishing the theoremhood ofơA→(A→B)

(to the left of the lines containing the conditionals whose consequents are formulas from the earlier derivation we write the line numbers from that derivation):

2 ơA → ((ơA → ơA) → ơA)) → ((ơA → (ơA → ơA)) → (ơA → ơA))

CL2, with ơA / P, ơA → ơA / Q, ơA / R

6 ơA → (ơB → ơA) CL1, with ơA / P, ơB /Q

7 (ơA → (ơB → ơA)) → (ơA → (ơA → (ơB → ơA))) CL1, with ơA → (ơB → ơA) / P, ơA / Q

9 ơ A → ( ơ A → ( ơ B → ơ A)) → ((ơA → ơA) → (ơA → (ơB → ơA)))

10 (ơA → ơA) → (ơA → (ơB → ơA)) 8,9 MP

16 (ơA → (ơB → ơA)) → (ơA → (A → B)) 14,15 MP

26 Review of Classical Propositional Logic

Shorter derivations are achievable, as demonstrated by the fact that the formula on line 11 was unnecessary since it is already presented on line 6 The main focus is to illustrate the mechanical conversion procedure introduced in the proof of the Deduction Theorem, which demonstrates that any derivation of Q from P can be transformed into a proof of P → Q.

A derivation system is considered sound for classical propositional logic if all theorems are tautologies and if any formula P that is derivable from a set of formulas is also entailed by that set The soundness of the system CLA is exemplified by the tautologies A→A and ơA→(A→B), which demonstrate the validity of the derivations within this logical framework.