ABSTRACT

This paper makes an original rhetorical contribution about the nature of logical negation. Following Peirce (1933), who reduced classical logic to its negation operation — distinction — this paper offers that negation should be presented as the central concept of logic for its pedagogical value. This is somewhat different from the prevailing view that logical consequence is the central topic of logic. Vagueness of logical consequence is the “common” reason (Russell 2019) to assert logical pluralism. Logical negation can have other meanings (Wittgenstein 1976), and here it is offered that logical negation is vague, which leads to logical pluralism as well. Logical pluralism gained salience in recent years in academic circles, beginning with Beall & Restall’s work culminating in a book (2006). There is now a large variety of logics cropping up. As a secondary part of the paper, a web app program called “Logic Puzzle,” coded by the author, aimed to make the abstract conceptual work about logical negation and vagueness more specific and concrete, and allows visual comparison between different logical negation operations. This program could be used in late high school math classes, though perhaps it is more appropriate for undergraduate math education classes. Finally, a preliminary empirical study combining qualitative and quantitative methods was conducted with the triangulable result that the technology increased awareness of choice in mathematics. Any curriculum modifications are left for the teachers to develop.

Keywords: Logical Pluralism, Education, Philosophy of Mathematics, Educational Technology, Vagueness

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TABLE OF CONTENTS

The Pedagogy of Logical Pluralism
Many Roads from the Axiom of Completeness
Vagueness as the Embodiment of Inquiry
1 Introduction
2 Literature Review and Conceptual Work
3 Setting up Empirical Problem to be Solved
4 Methodology
5 Data and Results
6 Conclusion
Poetry

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BACKGROUND

Logical pluralism is the view that there is more than one correct deductive logic, or way to sharpen the concept of logical consequence (the “if, then”), and gained prominence in academic circles recently. Since a number of papers by Beall and Restall, culminating in a book (2006), many other kinds of logic cropped up (Russell 2019). Differing ones include Varzi (2002) who adds more logical constants than other logicians do, Russell (2008) offers another logical pluralism between truth-bearers (the entities that can bear a truth-like property) such as sentences and propositions. Then there are varieties of logical pluralism holding that logics are models of language (Shapiro 2008; Cook 2010; Shapiro 2014), with the correct logic changing according to your goal or reason for modeling. The Stoics began logical pluralism, in a sense, in the variety of views that arose in their debates on logical consequence (Mates, 1961). Carnap upheld a particular kind of pluralism (Carnap 1937) that could be practiced by changing the language, the “syntactical rules” and “methods” (Ibid. §17). Lakatos argued (1962) that logic is traditionally used to justify many branches of knowledge, starting with definitions and “proving” the rest, following the Euclidean form. Euclid’s axiomatic form is historically very important for teaching. For thousands of years, Euclid’s books have been important textbooks, and, after the Renaissance, they are the most widely read textbooks in the world. (Hartshorne 2000, p1) Euclid’s innovation was not geometry but the form of proving propositions using classical logic, based on simple axioms. When non-Euclidean geometry was discovered, doubt was cast, not just on Euclidean geometry as true, but Euclid’s axiomatic form based on classical logic.

The change from classical logic to logical pluralism is a fundamental one. Not only is it gaining salience in academia, the shift portends a reorganization of many branches of knowledge, adding other valid ways to justify knowledge (while not subtracting the already “Euclidean” forms of knowledge). The prominence of logical pluralism leaves an opening to discover another fundamental way of teaching and presenting logic, different from Euclid’s. Logical pluralism changes the landscape of scholarship, the teacher’s curriculum, and of course the student’s understanding of mathematics in a fundamental way.

A firm reason for logical pluralism involves the recognition of probability as a fuzzy logic, a logic that is not simply used, but asserted as objects that exist in quantum physics.

"...probability theory might provide a canon for evaluating degrees of belief, ... Nonetheless, probability theory cannot be a complete answer here, for... In particular, we hold that it is a mistake to assert the premises of a valid argument while denying the conclusion..." (Beall & Restall 2006: loc 292)

Statistical reasoning allows a null hypothesis to be rejected based on a probability. This shows how two logics — a two-valued accept/reject logic, and an uncountably-many-valued fuzzy logic — are joined.

The goal of this paper is to offer a pedagogy of logical pluralism that can be appropriate for high school teachers to understand and adapt for high school education. As such it is addressed to people who know classical logic well, and begins the long journey from classical logic to poetry. [Note: If you are more interested in vagueness or poetry, skip to the middle or end of the book.] When adapting a topic for pedagogy, the concepts have to change into something rhetorically easy to accept. The basic pedagogical idea is a didactic between vagueness and the logical negation operation — distinction. Russell (1923) and Shapiro (2008, Loc 988-995) argued that vagueness is the opposite of distinction, and thus what vagueness is, exactly, must be left indistinct. Against this, classical logic assumes what will here be called negation absolutism: that a single, stable negation operation underlies all reasoning. It is seldom argued for, because it is seldom noticed.

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THE CONTRIBUTION: DISTINCT KINDS OF DISTINCTION

Peirce defined vagueness in terms of the law of non-contradiction, which is an exercise in applying the classical negation operation: ~(P and ~P). Peirce says that we are in a vague situation when this law fails to be true. (Peirce 1960, 505). For example, most people with a drive to avoid vagueness can easily agree that a neutron star is different from a question in a different way from how a lemon is different from a lime. However, agreeing to distinct kinds of distinctions (call these distinctions “A”) makes the law of non-contradiction fail, and puts people trying to avoid vagueness into vague territory.

One may say that having different ways of negating P does not indicate an indistinctness, it indicates greater distinctness. This depends on how we use the word distinctness. First of all, with the idea of “greater distinctness” we have already shifted our use of mere distinctness, to an undefined idea of degrees of distinctness, which implies a vagueness of our use of distinctness. Further, suppose there were type A distinctions between the same thing — for example, a right isosceles triangle with two equal sides measuring 1cm (call this triangle x) and a square with sides of 1cm (called y). x and y are different in that one has 3 sides and the other has 4. They are also different in that x is one triangle, and y is two x triangles put together. To simply call the triangle and the square distinct is to be vague on how they are distinct. The claim here is that that is exactly what we do, and because of this, distinctness is indistinct/vague. We are not being precise about our kinds of distinctions.

There are many attempts to solve vagueness. One attempt is to argue that we could find all the precisifications of a given vague use of the word distinction, and it makes no difference if we use distinction for all these distinctions for the sake of economy. To counter, suppose for example we want to find exactly where the end of my nose is (this is a common example of the problem of vagueness). We want to be very precise about all the parts of the line that distinguish my nose from not-my-nose, so we magnify. Unfortunately, this magnification makes the line harder and harder to describe, and eventually we get into quantum physics and can’t find the line anymore. The vagueness of distinction is just like any other vagueness, it can’t be solved by being more precise. In looking for all the distinctions between distinctions we will become unsure of whether they are distinctions at all (see the section “On Vagueness” below).

However, we can be sure of some distinctions between distinctions (as in distinction “A” above).

There are different ways to make the negation operation precise. The negation or distinction is not universal, but classical logical negation attempts to be universal, and the defining characteristic of classical logic negation is that ~~P implies P. The only way this can be true is if there is only one kind of negation operation, and it is universal/absolute. If the first “” in “~~P” means one thing and the second “” means another, we don’t know if we have P, or another kind of “~P.” Distinguishing the two occurrences as ~₁ and ~₂, the classical principle is ₁₂P → P, which holds only if ~₁ and ~₂ are the same operation. Wittgenstein (1976, p 80) already argued that there are other ways logical negation can be used. In his example, in effect a second negation on top of the first doesn’t do anything except add emphasis to a single negation. Here it is offered that a double negation can have many logical significances beyond mere emphasis, and beyond the total reversal of classical logical negation. The actual differences between differences, or logical negation operations, observed here are being brought to their natural conclusion: that the concept of difference is vague.

Derrida recognized one kind of distinction, different from the mathematical understanding of difference as a kind of absolute, unqualified ideal. It may be that Derrida’s différance is a conflation of the absolute difference/not-equals of mathematics, with the idea of meaning being deferred. It may also be that the connotation of deferred in différance is a qualification that makes différance more precise than difference. Either way, the reason difference can be qualified is that the idea of difference as an unqualified absolute suffers from vagueness, that any observed difference is already conflated with other differences. And this vagueness is natural, in the same way that there are natural kinds, there is a natural failure to classify, to name. As in measuring the plank of wood, the length is never perfectly measured. The idea that there is an “actual” length “out there,” or that the ultimate meaning (in the correspondence sense) of a length of a wooden plank is deferred to the completion of a never-ending measurement task is contrasted with the idea that the “actual” length is naturally vague, and the task need not be completed. For finite beings, all measurements are vague (people tend to call this “error”, but error implies that the vagueness people encounter is essentially unreal, even though it is encountered constantly whether we use precise measuring tools or our human senses). Derrida’s différance is not the negation or difference operation in constructive logic, or the paraconsistent definition of negation in Costa’s C1 (1977), or the many other formulations of the logical negation axiom, and these are all different from the absolute mathematical not-equals, and classical logical negation.

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SIGNIFICANCE FOR PHILOSOPHY OF MATH

There are two ways to handle the recognition that distinctness in its present form is indistinct. One way is to introduce new words for different kinds of distinctness, or make standard the use of qualifying words for this difference or that difference. This path would change the mathematical use of the not-equals symbol and the negation operation, and make logical pluralism standard. Another way to handle it is to accept that vagueness is not something we can avoid by pursuing a never-ending task of refining language. I am in favor of the second path — one of embracing vagueness. Ullman (1970) has a section called “Words with blurred edges” (1970, p 116) where he notices that vagueness is seen as a strength by poets, but seen as a bad thing by natural philosophers (the old name for scientists). Speaking as a published poet, my experience of education in the USA did not favor the poet’s temperament to embrace vagueness. To trouble the first path of handling the indistinctness of distinctness, it could be pointed out that there are higher-order varieties of the same problem of distinctions between distinctions — that there can be distinctions between the distinctions between distinctions, etc. and these would lead us back into a realm where vagueness must be accepted as, if not real and good, as constantly present in everyday life and troubling attempts to make distinctions. Distinction is problematic; higher-order distinction is not any less problematic than higher-order vagueness — the problem that solutions to vagueness are themselves vague.

There is no reason to privilege distinctness over vagueness, as Feyerabend (1975/2010) argued that there is no reason to privilege the telescope over the naked eye. In this, the ideas here place themselves under ancient skepticism, not nihilism nor relativism. Ancient skepticism guards a person against turning their sense impression into dogma that is not immediately apparent. (Sextus Empiricus 1996) The impressions the senses make to you are not open to question. This is not to say that knowledge is impossible, but simply that knowledge found in poetry “weighs in” just as much as knowledge found in the sciences.

In Verlaine (1884, p23-25):
Rien de plus cher que la chanson grise
Où l’Indécis au Précis se joint.

l’Indécis can be translated as vagueness and Précis as precision, “Nothing is more valuable than the grey song where vagueness and precision join.”

In Nightingale (2018) a study of teaching vagueness found that students who were perceived as “bad” or uncooperative were the most engaged and pro-active when the topic of empirical investigation was vagueness. This seems to indicate that education in what Nightingale coined as “precision knowledge” is hostile to certain temperaments. There are further social effects to the pursuit of “precision knowledge” in mathematics, and sciences who follow mathematical reasoning. Precision knowledge has a divisive effect on people and discourse, as is illustrated in the story of the Tower of Babel.

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ON VAGUENESS

There is too much to say about the many attempts to define vagueness. For this paper, we have adopted Peirce’s definition of vagueness, as the failure of the law of noncontradiction. However, Peirce made this definition when only one logic was recognized. For vagueness to take on its role in logical pluralism, the concept of vagueness was repurposed in Nightingale (2018). In general, the description of vagueness is better left to poetry. Rowland (2000) offered that the “question” itself is an expression of vagueness. If that is the case, describing vagueness would be a description of everything we can inquire about. The most “complete” attempt at doing this, outside the field of poetry, would be in real number analysis, although it is argued in Nightingale (2013) that this attempt, while beautiful to some, still fails.

Ullmann points out that the term “vague” is vague, or ambiguous (p117-118) (although Russell (1923) gave one definition of vagueness as ambiguity: that there is more than one possible referent for a given word-use). Ullmann offers a disambiguation of the term vague, which includes the general. On the other hand, Peirce is adamant that there be a clear distinction between general and vague (reprinted in Peirce 1934). The indeterminacy of generality (general knowledge is a useful kind of knowledge) lies in variation within a class, while indeterminacy of vagueness lies at the boundaries of classes. (Rowland 2000, Kindle Locations 1697-1700) The distinction, it must be noticed, suffers from vagueness, because there are individuals that lie near the boundary of a class that may or may not also be one of the variations allowed by a generality. Not only is it easy to miss the difference between vagueness and generality, the difference may be vague.

Wandering among different definitions (vague comes from the Latin vagus which means to wander), the roads seem to lead to Russell’s definition of “having more than one meaning… language is a ‘garden of forking paths'” to quote Borges’ story about a book that actually succeeds in chronicling the passage of time, forking as all possible choices are made. Russell noticed that these paths can be seen in experience — as things come closer, a multitude of details become available, all calling us to follow them. Sometimes I refer to this “general law of physics” (Russell 1923) as vagueness:

"...in the case when you are looking at a star in the sky through a telescope, and you can almost make out that it looks like two stars close together, we can say that 'that light is one star' or 'that light is two stars'. Holding both these statements is classically illogical by noncontradiction, and both can be held because the situation is vague — both in Peirce's sense and Russell's physical law." (Nightingale 2018, p 21-22)

Whatever vagueness is, the borderline case is where vagueness has made the most trouble for logicians. For example, Shapiro (2008) describes the borderline case with imagining an array of 2000 men from entirely bald on one side and gradually (imperceptibly, even) changing to a very hairy Jerry Garcia on the other end of the array. He attacks this ancient problem with what he calls the “forced march,” done by a group of “competent” speakers. They start with Jerry Garcia and investigate the next man, reaching a consensus each time, and the inference “if n is not bald, then n+1 is not bald” appears valid. Shapiro is concerned with the eventuality that the inference will break down and the competent speakers will be filled with conflict until they decide that one of the men is bald. This is because the difference between the words “bald” and “not bald” are vague, or there are cases where the word bald can be applied to a man that is arguably bald or not bald, if inquiry becomes more sensitive. This is how the borderline case relates to our definitions of vagueness in the previous paragraph.

The “competent speakers” or “masters” of meaning decide man n is bald when they had just agreed man n-1 was not bald. Shapiro seems to say that the speakers can stop there, even though, as Shapiro mentions, the competent speakers will be aware that the same change or jump at n from not bald to bald spreads to some indeterminate n-k other people who were previously validly inferred to be not bald and are now bald, because the change from n-1 to n is imperceptible to the naked eye (a similar example of vagueness can be produced regardless of the power of the microscope or telescope; regardless of the sensitivity of any scientific instrument). If the purpose of the conversation were to figure out who was bald and who wasn’t, wouldn’t the speakers turn back and start inferring that n-1 is bald, etc. ad infinitum? Since Shapiro chooses classical logic “without argument” (Shapiro 2008, Kindle Location 1023), and classical logic requires the Law of Excluded Middle, the target of conversation is necessarily settled “out there.” To know where (and if) this is settled, the speakers would have to enter into a never-ending conversation. Facing vagueness between bald and not bald, the committee has a choice to deliberate on a vague situation forever, or allow classical logic to come into question. Here Shapiro upholds a latent ideal of work as a saving power against contradiction, but that power only saves as long as work continues, and returns to contradiction if the “masters” give up.

Nightingale (2018) put forth the idea that vagueness is a material part of inquiry. When inquiring into the length of a wooden plank, one takes centimeters, then millimeters, etc, but ultimately the measuring tool fails to take the “actual” length of the plank — the body of the plank is vague (Nightingale 2013). Dewey defended the thesis that logic arises from inquiry:

"the view here expressed, they (logical laws) represent conditions which have been ascertained during the conduct of continued inquiry to be involved in its own successful pursuit." (Dewey 1938 Loc 284-286)

Inquiry with the senses, as has long been recognized, is fraught with vagueness. Vagueness and logic are complementary; they are part of the same activity: inquiry. Vagueness is when reality spills over out of our word-containers in the chase and capture of reality with words, or it is what escapes us when involved in any inquiry.

Probability is often misunderstood as a solution to vagueness, and without vagueness, there is no reason to assert logical pluralism. The way probability deals with vagueness is by giving vague situations real-valued degrees between opposites.

"One serious objection ... is that it really replaces vagueness with the most refined and incredible precision. Set membership, as viewed by the degrees of truth theorist, comes in precise degrees, ... The result is a commitment to precise dividing lines that is not only unbelievable but also thoroughly contrary to what I [call] 'robust' or 'resilient' vagueness. For ... it seems an essential part of the resilient vagueness of ordinary terms such as 'bald', 'tall', and 'overweight' that in Sorites sequences ... there is indeterminacy with respect to the division between the conditionals that have the value 1, and those that have the next highest value, whatever it might be. It is this central feature of vagueness which the degrees of truth approach, in its standard form, fails to accommodate, regardless of how many truth-values it introduces." (Tye 1994: p 14)

The measurement of the wooden plank is a good example of why vagueness persists in the face of the theory of real numbers, because the progression of measurement can be seen as an infinite and bounded sequence, which allows it to fall under the definition of the axiom of completeness: the property that distinguishes the real numbers from other numbers (Abbott 2001). It has already been argued elsewhere that this property fails in answering the problem of vagueness (Nightingale 2018, Nightingale 2013).

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INTRODUCING “LOGIC PUZZLE”

The idea that there is more than one way to make the negation operation precise, and thus that the negation operation is vague, was coded into a computer program by the author to allow a more specific, concrete presentation of the abstract conceptual work done above. The web app program “Logic Puzzle” could be used to instruct undergraduate math, math education, or late high school students. It involves fitting puzzle pieces together to create logical statements. There are three settings: classical, paraconsistent, and constructive logic. The program automatically generates the logical statement in normal notation, along with graphs of classical logic, paraconsistent truth tables, and separate graphs for the constructive logic setting.

A user can mouse over puzzle pieces, graphs and truth tables to get a description of each object. To build a logical statement, one merely drags-and-drops puzzle pieces in ways they fit together. If you would like to play with Logic Puzzle, go to http://159.89.203.232/LogicPuzzle-Dev/.

When a green piece is dragged to the right place, it modifies the logical statement by adding a proposition labeled “A,” “B,” or “C.” When a blue puzzle piece is dragged to the right place, it acts as both the parenthesis and the negation operation, so a left and a right blue piece is needed to close the parenthesis and indicate what is being negated. There are two possible instances of “A,” “B,” or “C,” one on the left and one on the right; this creates the possibility of building two statements with puzzle pieces that are in contradiction. When a user builds a contradiction, the program asks about changing to paraconsistent logic, and draws a paraconsistent truth table under the Peircian graphs.

CLASSICAL LOGIC SETTING

The Peircian graphs (Peirce 1933) reduce logic to a single symbol: the circle. The circle notation distinguishes between its inside and outside; it is effectively the negation operation. The Peircian graphs suggest a way of using space, or a lack of notation, instead of the AND operation.

This symbol is equivalent to “~(A AND ~B)”, since the outside circle suggests an A on the left, and a ~B on the right of its inside. This symbol is equivalent to “If A, then B.” It may be noticed that there is a “latent” symbol in use here because the ellipse also suggests the use of the operation AND. One may say that this notation is only possible with a loose definition of the circle that includes ellipses. Nightingale introduces the other notation in the Peircian graphs as the same effect, but there is no variation in the basic symbol.

The mark required for the spirit of the Peircian graph is so elemental it is almost simply the requirement that any mark be written.

PARACONSISTENT LOGIC SETTING

The paraconsistent logic chosen supports the idea that negation is the fundamental idea in logic. Costa’s C1 (1977) is defined by allowing a new line in the truth table anytime something true (represented with a “1” in the truth table above) is negated in another column in the truth table; the new line allows the negation to be true (“1”), while the old line will be the expected “false” (“0”) value. So we have a situation where if A is true, ~A may be true or false, and the law of non-contradiction fails. Moving from the Peircian graphs that reduce classical logic to negation, this particular paraconsistent logic is entirely defined by a difference in the definition of negation, emphasizing the importance of negation (or distinction) as the fundamental operation of logic.

CONSTRUCTIVE LOGIC SETTING

The constructive logic setting is similar in that you build statements the same way, by dragging and dropping puzzle pieces. There are the same green pieces that are the “Propositions” and the same blue piece for both negation and parenthesis. The meaning of “A,” “B,” or “C” is explained, so “A” is a connected ring, “B” is a star-shape with rays coming from a central point, and “C” is a “mesh.” This is because the “truth-value” of a constructive logic statement is not mere truth, but “constructed” and this makes negation calculate differently.

The possibilities are: 1) A or a ring is constructed, if that is all the user has built by dragging and dropping a green puzzle piece, then the graph will be a picture of a connected ring of nodes. 2) If ~A is constructed with puzzle pieces, the graph will show a configuration of nodes that makes it impossible to connect them into a ring. 3) If ~~A is constructed with puzzle pieces, the graph will show a configuration of unconnected nodes where it is possible to connect them into a ring, e.g. “A is constructable but not constructed.” If “A and B” is constructed, there will be nodes in a ring, and the nodes will also be connected in a way that forms a star shape. Negation has yet another fundamental difference in constructive logic. Starting with “constructed” (A), “not constructed (and constructable)” is (~~A) and “not constructable” is (~A).

The pedagogy presented by the “Logic Puzzle” program should be understandable to undergraduate students in math or math education. While all these concepts may not be explicitly learned by high-school students using the logic puzzle, the intent is that students will be exposed, or immersed, in a teaching tool with an important message: there are different logics, these logics are defined by their differing negation operations, and these differences can be compared, side by side, within a technological setting. The pedagogy of logical pluralism can be felt by students, even if they do not master the teachers’ understanding presented here, compressed as it is from the wider context available to a scholar. It is often the case that mathematical arguments are not made to young students, such as the argument for real numbers, as numbers, or as real.

The argument for real numbers isn’t presented fully unless a student majors in mathematics in college, where a student can find at the very end (covered?) of an undergraduate text on real numbers: “We all grow up believing in the existence of real numbers, but it is only through study of classical analysis that we become aware of their elusive and enigmatic nature.” (Abbott 2001, p 244) Unfortunately for many students, such arguments are often not of much help. This is one purpose of making “Logic Puzzle;” it presents an important mathematical idea with automatic graph generation, help messages as needed, and the mathematical rigor calculated “under the hood.” With the pedagogy of logical pluralism and its conception of the program “Logic Puzzle” explained, we will turn to how the logic puzzle application is to be studied, then to exactly what the students are able to grasp using this tool.

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METHODOLOGY

"...the field indicated, that of inquiries, is already pre-empted. There is, it will be said, a recognized subject which deals with it. That subject is methodology; and there is a well recognized distinction between methodology and logic, the former being an application of the latter." (Dewey 1938 Loc 166-168)

Dewey writes that the reason methodology and logic are seen as separate is logic is needed as an external and certain standard to judge methods of scientific inquiry.

"How can inquiry originate logical forms (as it has been stated that it does) and yet be subject to the requirements of these forms?" (Dewey 1938 Loc 179-180)
"The problem reduced to its lowest terms is whether logical standards and forms are intrinsic to inquiry or whether they are imported from outside... One might reply by saying that it can because it has. One might even challenge the objector to produce a single instance of improvement in scientific methods not produced in and by the self-corrective process of inquiry." (Dewey 1938 Loc 182-185)

How does one make assumptions about a methodology when the standard of judging methodologies is under scrutiny? This is why a qualitative approach is preferred. Induction is a very different logic from the deductive logics being examined. The study inquires using Grounded theory (Glaser & Strauss) continuing to explore multiple deductive logics as a topic for empirical study in education, except Nightingale (2018).

There were three instruments to the Logic Puzzle: a pre-post attitudinal survey (quantitative), a worksheet that guides the students to try certain constructions in classical, paraconsistent, and constructive logic, asking for their feedback (qualitative), and a post test asking them to select a logic given a situation. The test was partially quantitative and partially qualitative, with a correct answer of which logic, and any information the student felt they wanted to add.

The worksheet instructed the students to construct the statement “A and ~A,” a contradiction, so that Logic Puzzle drew Peircian graphs and a paraconsistent truth table. The students were then asked questions like what the graphs mean, how classical Peircian and constructive graphs differ, about how classical logic truth tables and paraconsistent truth tables are different. At the end there were open ended questions about what they thought about the program.

Logical pluralism was introduced to students with the Web App called “Logic Puzzle” in one language: Thai. In Thailand, Western (the usual truth-table classical logic) logic has colonized the national curriculum, and teachers begin instruction of logic to students around age 16. However, a requirement throughout the mathematics curriculum is “suitable reasoning.” With this requirement, education authorities in Thailand present mathematics as the model for how students should reason.

The students were generally college and late high school students in front of a computer doing an investigation into logic, not for fun. The worksheet-style direction and feedback part were coded and then the codes were investigated using basic qualitative questioning. The codes were compared to further define them and their relationships to each other to better understand the students’ experience of Logic Puzzle. Certain words were selected for meaning exploration. The quantitative results are the most tentative form of research applied in this study, because probability is under scrutiny.

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DATA AND RESULTS

QUALITATIVE FINDINGS

Because there were only 7 subjects in this study, the presentation of data can be more fine-grained; 10 worksheet questions makes 70 answers in the worksheet. There were a small number of coded “insightful answers” (8/70) that go beyond these types of merely “correct” (28/70) answers. Outside the two questions that almost nobody understood (what is a contradiction and how are the constructive and classical graphs alike/different) there were 7 misunderstand/blank. Having a graph gave students a visual to connect to the definition of negation. Maybe this contributed to the idea that the tool (or logical pluralism?) “made it easier” (5/70).

“Choice” (6/70) was another code. Students are generally not given a choice about mathematical facts. Giving students choices helps them to see that whatever math fact they are looking at was a choice made by mathematicians, not an absolute truth. They do not have to accept them or “hang on a word” or symbol. Students can think more freely because mathematics is seen as more hypothetical. There was indication that giving students choices “made it easier”; there was no indication that giving students choices in math made things harder. Another student sees that a different logic has the potential to be more concrete (constructive), instead of more general (paraconsistent).

“Relates to everyday life” (2/70) was another code. It was strange to see this code at all. I did not expect that students would be able to relate abstract A’s and B’s, logical operations, and geometrical graphs to everyday life. “Paraconsistent logic is possible with 2 answers. Can be used in everyday life more than Classical logic.” Here we see a connection made by a student between choice (having possible answers) and everyday life. Is math seen as difficult simply because it is asserted as without choice, unlike life?

Insightful students noticed that the paraconsistent truth table is “Different with many more meanings than traditional logic.” or “Rejection [negation] in constructive logic can be done in two ways, but for classical logic there is only one way” in response to how classical logic is different from constructive logic. This is the main pedagogical message of Logic Puzzle: that there are choices of logics because of distinctions between kinds of negation.

QUANTITATIVE RESULTS

For the attitudinal questionnaire, because of the small sample size, only large effect sizes were tested using a matched pair, one-tailed Student’s t-test. There were three questions (out of 12 in the survey) with a large Cohen’s D effect size (approx .89); two of these questions had interesting statistical significance. The first was “If I have a thought that is not mathematical, I know that it is not worthwhile.” This changed from being neutral and agreeing somewhat, to leaning towards disagreeing after the activity. The statistical significance of this large and important effect size was p = .051. While this is technically not a significant finding, in a strict orthodox view, it is still a tantalizing finding that indicates a fruitful direction for further study, especially given the small sample size of 7.

The next question was “Mathematics is not important in everyday life.” This also changed from neutral and somewhat agreeing, to somewhat disagreeing. The Cohen’s D effect size was large (.89) so a t-test was conducted and a p value of p = .052 was found. Another interesting, if unorthodox finding. It is the author’s view that if schools had allowed logical pluralism into their classrooms so that a large sample size would have been available, there would be a significant finding that the logical pluralism activity helped students to see a relationship between logic and everyday life.

SUMMARY OF FINDINGS

The strongest result — that of having a choice — is found in the test, the survey (more thoughts are worthwhile), and the worksheet, so it is triangulated. Emphatically, the choices given to students in Logic Puzzle are epistemologically fundamental. They are not trivial choices about how to express the same answer, or different ways of getting essentially the same answer. These are real choices. There was no indication that having possibilities made things harder, but there was indication that having possibilities made things easier.

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CONCLUSION

Students found more options with logical pluralism, and that comes as no surprise. They felt this freedom as a connection to life, and a sense of ease. Math could have been done in a different way, and it’s not as if traditional math does not have contradictions (Weber N.D.). It is supposed that students can feel this, even if they don’t know it — in the same way that real numbers are commonly presented intuitively without precise, mathematically rigorous arguments.

The message of Logic Puzzle as a technology “endowed or entrusted with meaning” (Latour 2002, p 29) is that logical deduction is not final. The fact that there is a variety of logics bends the blade of distinction against itself, rendering traditional taxonomies and bodies of knowledge dependent on an absolute idea of distinction open to question. And with honest inquiry there is an opening for constructing new meanings, finding new patterns and for new readings of ancient texts.

It has been observed that “the question” is how to express vagueness (Rowland 2000), and logical pluralism, as a response to the vague concept of logical negation, evokes questions more easily than classical logic alone, which presents itself as final.

The logic (with probability presented as a “theory” not a logic) in Thai schools today is consistent with the Law of Excluded Middle: that there is no real question whether it is A or ~A, “out there”; this question is always settled. With Excluded Middle, vagueness is unreal or error, and the reality is a world of external answers, already settled “out there.” (Brouwer 1981) Classical logic puts learners in situations where they have to deal with contradictions, such as the indistinctness of distinctness, and does not teach them a freedom of thought that allows them to deal with contradictions. One example is instantaneous velocity, which is a contradiction in terms. It is better to not assert that such a concept is precise and in contradiction; rather, instantaneous velocity is vague and that is why the law of non-contradiction fails. “How can an object have velocity if it doesn’t have time to move?” This is how to think about instantaneous velocity. Firstly, asking a question about a contradiction just feels better to students than compelling them to accept something conflicting. Secondly, a question has the property of being neither true nor false, which is the kind of proposition you need for a contradiction. A real question is one that is particularly stubborn and keeps coming up after generations of trying to answer it. Motion, consciousness and vagueness are examples of real questions.

With logical pluralism, teachers can show students a rejection of the logical terms used to create the contradiction, not in a negative way, but by introducing new logical terms in other logics. So instantaneous velocity uses the mathematical construct of points to describe movement. Rejection of the logical objects used to describe movement does not reject the question about movement, but leads to looking for other logics to ask about it.

Even if logical pluralism is not adopted in a curriculum, teachers ought to give their students permission to reject contradictions, and how would they do that safely without giving them alternative logical negation operations, allowing the failure of the law of noncontradiction by giving alternatives? Compelling assent to a contradiction without alternatives is the opposite of education. On the other hand, appealing to unending inquiry can be paralyzing and equally unhelpful. Introducing a variety of logics is a middle-ground solution, allowing students to articulate their questions with respect to a particular logic, and adapt to contradictions by recognizing that any logic has vague aspects.

There is no intent here to endorse the idea that anything can be true. Maybe a limited relativism is supported because absolutism is rejected as a result of the honest recognition that the logical negation operation is vague. People still need (vague) interaction, and have (vague) standards of reasoning to judge and believe ideas and writing. The perspective of logical pluralism is that there are many deductive standards to use, and perhaps some yet uninvented. Relativism isn’t really a serious point of view because logical standards, conversations, ideologies, paradigms, cultures, etc. are vague. The idea that something can be relatively true from the point of view of a culture or something else requires that this culture be fully defined the way classical logic is believed to be. Unfortunately, classical logic is not fully defined, nor is any culture or anything else (this is a direct result of the indistinctness of distinctness). Neither does this mean nothing is true. Ancient skepticism asserts that sense impressions are true and not open to question, even though they are vague, so nihilism is rejected.

Science education focuses on teaching a precise concept, even in alternative approaches like Philosophy for Children (P4C) (Ferriera 2012). However, vagueness is more interesting to students (Nightingale 2018), and pluralism helps understanding, against the idea that there is a scientific “true/real” precise concept, without contending concepts. The latent ideal of work that Shapiro upholds as a saving power against contradiction is contrasted with an ideal of rest, offered in Pyrrhonism as Ataraxia, because vagueness is found both after tremendous amounts of work and investigation using big science apparatus, as well as without work, immediately to our human senses.

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REFERENCES

Abbott, S. (2001). Understanding analysis. New York: Springer.

Beall, J., & Restall, G. (2006). Logical pluralism. Oxford: Clarendon Press.

Borges, J. L. (2002). Other inquisitions: 1937-1952. Austin: University of Texas Press.

Brouwer, L. E., & Dalen, D. V. (1981). Brouwer’s Cambridge lectures on intuitionism. Cambridge: Cambridge University Press.

Brouwer, L. E. J. (1907). On the Foundations of Mathematics. Thesis, Amsterdam; English translation in Heyting (ed.) 1975: 11-101.

Brouwer, L. E. J. (1908). The Unreliability of the Logical Principles. English translation in Heyting (ed.) 1975: 107-111.

Carnap, R. (1937). The Logical Syntax of Language. London: Kegan Paul.

Cook, R., 2010, “Let a thousand flowers bloom: a tour of logical pluralism,” Philosophy Compass, 5(6): 492-504.

Corbin, J. M., & Strauss, A. L. (2015). Basics of qualitative research: Techniques and procedures for developing grounded theory. Los Angeles: SAGE.

Costa, N. C., & Alves, E. H. (1977). A semantical analysis of the calculi C_n. Notre Dame Journal of Formal Logic, 18(4), 621-630.

Dewey, J. (1938). Logic, the theory of inquiry. New York: H. Holt and Company.

Drago, A. (2018). Suggestion for Teaching Science as a Pluralist Enterprise. Transversal: International Journal for the Historiography of Science, (5), 66.

Sextus Empiricus. (1996). The skeptic way: Sextus Empiricus’ “Outlines of Pyrrhonism.” Translated by B. Mates. New York: Oxford University Press.

Ferreira, L. B. (2012). Philosophy for Children in Science Class. Thinking: The Journal of Philosophy for Children, 20(1), 73-81.

Feyerabend, P. (2010). Against method. London: Verso.

Foucault, M. (1973). The order of things: An archaeology of the human sciences. New York: Vintage Books.

Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research.

Hartshorne, R. (2000). Geometry: Euclid and beyond. New York: Springer.

Lakatos, I. (1962). “Infinite Regress and Foundations of Mathematics,” Aristotelian Society Supplementary Volume, 36: 155-94.

Latour, B. (2002). We have never been modern. Cambridge, MA: Harvard University Press.

Mates, B. (1961). Stoic logic. Berkeley: University of California Press.

Nightingale, A. (2013). Many roads from the axiom of completeness. Questions Are Power. https://questionsarepower.files.wordpress.com/2016/03/many_roads_from_the_axiom_of_completeness-2.pdf

Nightingale, A. (2018). Vagueness as the Embodiment of Inquiry: An account of vagueness as it pertains to logic, and a study of teaching vagueness to young Thai (P4) students. https://questionsarepower.files.wordpress.com/2018/09/nightingale-dissertation-vagueness-as-the-embodiment-of-inquiry.pdf

Peirce, C. S. (1933). Collected Papers — IV Chapter 3 — Existential Graphs., pp. 4.397-4.417, edited by Charles Hartshorne and Paul Weiss, Harvard University Press, Cambridge.

Peirce, C. S., Hartshorne, C., & Weiss, P. (1934). Collected papers. Cambridge: The Belknap Pr. of Harvard University Press.

Peirce, C. S., Hartshorne, C., Weiss, P., & Burks, A. W. (1960). Collected papers of Charles Sanders Peirce. (Vol 5). Cambridge: Belknap Press of Harvard University Press.

Rowland, Tim. (2000). The Pragmatics of Mathematics Education: Vagueness and Mathematical Discourse. Taylor and Francis.

Russell, B.A.W. (1923). ‘Vagueness’, Australasian Journal of Psychology and Philosophy, 1, pp. 84-92.

Russell, G., (2008), “One true logic?” Journal of Philosophical Logic, 37(6): 593-611.

Russell, G. (2019). “Logical pluralism.” In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Spring 2019 Edition). https://plato.stanford.edu/archives/spr2019/entries/logical-pluralism/

Shapiro, S. (2008). Vagueness in context. Oxford University Press.

Shapiro, S. (2014). Varieties of Logic. Oxford University Press.

Tarski, A. “On the Concept of Logical Consequence.” In Logic, Semantics, Metamathematics: papers from 1923 to 1938, chapter 16, pages 409-420. Clarendon Press, Oxford, 1956. Translated by J. H. Woodger.

Tye, M. (1994). Sorites Paradoxes and the Semantics of Vagueness. Philosophical Perspectives, 8, 189.

Ullmann, S. (1972). Semantics: An introduction to the science of meaning. Oxford: Blackwell.

Verlaine, P. (1884). Jadis et naguère. Paris: Léon Vanier.

Varzi, A. C., 2002, “On logical relativity,” Philosophical Issues, 12: 197-219.

Weber, Z. (n.d.). Inconsistent Mathematics. Retrieved from https://www.iep.utm.edu/math-inc/#H6

Wittgenstein, L., Bosanquet, R., & Diamond, C. (1976). Lectures on the foundations of mathematics: Cambridge, 1939. Hassocks: Harvester.

Wyatt, N., & Payette, G. (2019). Against logical generalism. Synthese.