Iconic Logic: The Visual Art of Drawing the Right Conclusion

Abstract

Most people, evidence suggests, have a hard time thinking straight. Symbolic logic is a tool that can help remedy this problem. Unfortunately, it is highly abstract and uses symbols whose meanings depend on unintuitive arbitrary conventions. Without sacrificing rigor, iconic logic is more concrete and uses icons that resemble what they stand for and whose meanings are thus easier to picture, process, and remember. Here I review and critique iconic existential graphs and concept diagrams—the former link iconic logic to iconic mathematics; the latter expand popular Euler or Venn diagrams, and to some degree, their user-friendliness has been empirically investigated. I lay out how expertise in perception, cognition, and genetics can inform and improve this empirical research to help render iconic logic more ergonomic. After all, logic is a tool, and tools should not only suit their use but also their user.

Full text

Iconic Logic: The Visual Art of Drawing the Right Conclusion
Peter Kramer1
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1Department of General Psychology, University of Padua, Italy
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* Correspondence:
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orcid.org/0000-0003-4807-7077
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peter.kramer@unipd.it
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Keywords: existential graph, concept diagram, iconic logic, logical graph, iconic mathematics,
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diametric theory, social exchange, reasoning ability.
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Abstract
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Most people, evidence suggests, have a hard time thinking straight. Symbolic logic is a tool that can
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help remedy this problem. Unfortunately, it is highly abstract and uses symbols whose meanings rely
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on unintuitive arbitrary conventions. Without sacrificing rigor, iconic logic is more concrete and uses
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icons that resemble what they stand for and whose meanings are thus easier to picture, process, and
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remember. Here I review and critique iconic existential graphs and concept diagrams—the former
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link iconic logic to iconic mathematics; the latter expand popular Euler or Venn diagrams and have,
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to some degree, been empirically investigated for user-friendliness. I lay out how expertise in
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perception, cognition, and genetics can inform and improve such empirical research to help make
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iconic logic more ergonomic. After all, logic is a tool, and tools should not only suit their use but also
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their user.
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No, no, you are not thinking, you are just being logical
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—Niels Bohr (Frisch, 1979)
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1 Introduction
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1.1 Toward a rigorous but more user-friendly logic
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Formal logic offers rules for how to reason in a watertight, step-by-step manner to derive
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indisputable conclusions from a given set of starting assumptions. It is a great tool but typically not
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very user friendly. Here, I present a review of attempts to make it more ergonomic without
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sacrificing its rigor. I discuss two important solutions: existential graphs (particularly interesting
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because a precursor of them inspired a more general “iconic mathematics”) and concept diagrams
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(particularly interesting because they inspired behavioral experiments). To my knowledge, no
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behavioral scientists have been involved in designing these behavioral experiments, but I argue that it
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would have been good—and still is—to get experts on perception and cognition on board. By
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uncovering the logician’s typical psychological profile, I explain why most logicians seem to
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underappreciate the need for a more ergonomic kind of logic, just like most mathematicians seem to
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underappreciate the need for a more ergonomic mathematics (Kramer, 2022). I refer to gene
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expression to explain why many otherwise intelligent people are poor logicians and why, particularly
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for them, iconic logic is a better tool than is symbolic logic.
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1.2 Icons versus symbols
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Logic is expressed with the help of signs. A sign is something that stands for something else.
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According to Charles Sanders Peirce, one of the founders of semiotics (the study of signs) a sign is
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either a token or a type of tokens (Peirce, 1992; Nöth, 1990). A token stands for an instance (e.g., a
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particular woman) of a type of tokens (the set of women in general). According to Peirce, a sign is
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also either an index, icon, or symbol (Peirce, 1992; Nöth, 1990). An index directs attention to what it
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stands for, like smoke to fire or a finger to what it points at. An icon resembles what it stands for, like
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a picture of a train resembles a train, the word “splash” the sound of a splash, and six dots on a die a
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quantity of six. A symbol refers to what it stands for by an arbitrary convention or chance
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association, like for no apparent reason “+” means “plus”, “\” means “therefore”, and “6” and “six”
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a quantity of six again.
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The extent to which a sign is an index, icon, or symbol is a matter of degree. The more a sign is a
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symbol, the less intuitive, harder to remember, and less reader-friendly it is. Ironically, rather than on
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icons or indices, formal logic relies most heavily on precisely this kind of signs. Many symbols,
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moreover, are assigned only a temporary meaning that can change from context to context. The
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letters x and y, for example, may stand for “ewes” and “rams” in one context but for, say, “midgets”
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and “elves” in another. Especially when there are many of them, symbols burden readers’ memory
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more than do other signs. Symbolic logic is therefore typically only used by experts with, as I shall
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argue later, particularly strong memories of exactly the right type.
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Iconic logic offers an alternative to symbolic logic. Without sacrificing rigor, it relies less on symbols
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and more on icons. Iconic logic still is partly symbolic. Yet just like the hallmark of the chocolate bar
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is that it has chocolate, even though it also has other ingredients, the hallmark of iconic logic is that it
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is more iconic than symbolic logic is, even though it is also to some extent symbolic (Kralemann &
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Lattmann, 2013; Stjernfelt, 2007; Stjernfelt, 2022). Calling chocolate bars “candy bars” is accurate
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but misses what is important about them. Likewise, calling iconic logic “graphical logic” or
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“diagrammatic logic” also misses the point. It is its superior iconicity that makes iconic logic worth
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its name (Bricken, 2019a) and more intuitive than symbolic logic can ever be. The icons that are the
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easiest to process, picture in one’s mind, and remember, represent only the essential features or gist
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of what they stand for and leave out irrelevant details (Kramer, 2023). So, although a photo of a train
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resembles a train more than a highly simplified drawing of it does, iconic logic uses the latter,
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minimalist type of icons rather than the former, more complex ones.
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To some extent, symbolic logic is also iconic. For example, in P Ì Q, by convention, P and Q
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represent sets, but Ì is narrow on the extreme left and wide on the right, which vaguely suggests that
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P is part of something bigger, that P is a subset of Q. Symbolic logic is also somewhat iconic where it
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presents similar things in similar form, like when it presents variables in lowercase but predicates in
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uppercase (Kralemann & Lattmann, 2013; Stjernfelt, 2007; Stjernfelt, 2022). Euler diagrams (Euler
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& Brewster, 1833), however, represent sets and basic relationships between sets in a much more
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iconic and thus intuitive way. In such diagrams, by convention, delineated regions (encircled ones in
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Figures 1A-1C) represent sets or collections. (By definition, a set contains its members, like a field of
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sheep contains sheep, whereas a collection consists of its members, like a flock of sheep consists of
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sheep; Moktefi, 2015. Furthermore, any member can appear only once in a set but multiple times in a
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collection; Bricken, 2019a.) Although it is not self-evident that such a delineated region as a circle
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should represent a set or a collection, thinking of it as a highly simplified drawing of a field or a flock
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certainly renders it more iconic and intuitive than does a capital letter P or Q.
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Figure 1: Euler and Venn diagrams. The first three diagrams express: (A) “All ewes are sheep”. (B) “Some sea
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creatures are mammals”, (C) “Things can be sheep or goats but not both”. The last three diagrams represent
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three sets of numbers in symbolic notation (D) in an Euler diagram (E) and in a Venn diagram (F).
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In Euler diagrams, the positions of the regions relative to one another express the logical
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interrelationships between sets or collections, and in this way they are also iconic (Moktefi, 2015).
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For example, by representing the set of ewes inside the set of sheep (rams and ewes), it becomes
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immediately obvious that all ewes are sheep (Figure 1A); by letting the set of mammals overlap with
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the set of sea creatures, it becomes immediately obvious that some, but not all, mammals are sea
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creatures and vice versa (Figure 1B); and, by representing sheep separately from goats, it becomes
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immediately clear that no sheep is a goat and vice versa (Figure 1C). Venn diagrams (Venn, 1881)
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resemble Euler diagrams but, instead of manipulating the relative position of delineated regions to
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express set relationships, these regions are compartmentalized (compare Figures 1D–1F). Although
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Euler and Venn diagrams were developed in respectively the 18th and 19th century, they are still
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popular today. As they become more complex, however, they quickly become hard to read, and the
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range of logical problems they can handle is also rather limited.
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2 Existential graphs
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Around the beginning of the 20th century, Charles Sanders Peirce, who was not only a founder of the
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study of signs but also an important contributor to symbolic logic, proposed a more powerful iconic
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alternative to Euler and Venn diagrams: existential graphs. He thought that not his symbolic logic but
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this iconic alternative “ought to be the logic of the future” (Peirce, 2020, p. 12; see also Peirce,
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2021a; Peirce, 2021b). The iconic logic of existential graphs is particularly interesting because a very
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similar precursor of these graphs (entitative graphs) has been developed into a more general iconic
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mathematics (for a psychologist’s take of iconic mathematics, see Kramer, 2022; for mathematicians’
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and computer scientists’ expositions, see Bricken, 2019a; Bricken, 2019b; Bricken, 2021; Kauffman,
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1995; Kauffman et al., 2023; Spencer Brown, 1969).
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Peirce offered three kinds of existential graphs, which he rather unhelpfully—and, as we shall see
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later, rather tellingly—called “alpha”, “beta”, and “gamma”. I call them instead, in line with their
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symbolic counterparts, existential propositional, existential predicate, and existential higher-order
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and modal logic. Peirce did not complete the latter, and I will only discuss the former two (but see
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Peirce, 2021a and Dau, 2006; Roberts, 1973). Along the way, I will revive some of Peirce’s ideas
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that he himself deemed impractical but that, thanks to modern technology, no longer are and that—as
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a surprising byproduct—render his graphs quite decorative; suitable, quite frankly, to hang on a wall
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as art.
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2.1 Black-and-white thinking
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2.1.1 Sketching out premises
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To construct his existential graphs, Peirce drew on a sheet of paper (sheet of assertion) regions
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enclosed by circles, ovals, or other shapes. These regions could be nested within each other but, for
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the graphs to be considered “well-formed”, they were not allowed to overlap
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(Figure 2C). Within
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any region or subregion one can write propositions (statements that are either true or false) or capital
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letters like P, Q and R that stand for such propositions. Those written within the most encompassing
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region (the sheet of assertion itself) mean what they say (P is P); those within its subregions mean the
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negation of what they say (P becomes “not P”); those within its sub-subregions, the negation of the
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negation of what they say (P becomes “not not P”), and so on. Such drawings of regions with
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propositions written in them form the starting points (premises) for rigorous logical arguments that
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are valid if they strictly follow a bespoke set of transformation rules and sound if, in addition, the
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premises happen to be true. Sound arguments render conclusions indisputably true that otherwise
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may at best merely seem to be true.
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Peirce did allow the overlapping of the boundaries of regions, but this is an unnecessary complication (Dau, 2006).

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Figure 2: Existential graphs. Several logical relationships are expressed in English (A), symbols (B),
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traditional existential graphs (C), and black-and-white existential graphs (D). Note that some logical
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relationships can be expressed in two different ways in both English (A) and symbols (B) but in just one way
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in traditional (C) or black-and-white (D) existential graphs. The bottom half of the figure shows examples of
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what I call frames (which Peirce called “double cuts”, with each individual boundary called a “cut”). Here
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“or” means “and/or” (inclusive, rather than exclusive, or).
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In his graphs, Peirce almost always drew only the outlines of regions (Figure 2C); by and large,
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modern iconic logicians follow him in this practice. Occasionally, however, he shaded regions that
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negate something (Figure 3A; Peirce, 2020, p. 571). Perhaps because we are diurnal rather than
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nocturnal, across cultures we tend to associate positive things with the light color of white and
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negative things with the dark color of black (Jonauskaite et al., 2020). Peirce’s idea to use shading for
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negation is thus quite intuitive and ergonomic. Still, reminiscent of the fact that the negation of a
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negation amounts to a positive affirmation, a region’s subregions were left unshaded, sub-subregions
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were shaded again, sub-sub-subregions unshaded, and so forth.
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Unfortunately, Peirce deemed the shading practice “insufferably inconvenient” (Peirce, 2020, p. 571)
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for the scribe (the person who draws graphs and adds text to them). And although Peirce later did use
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shading again after all, his followers today still tend not to. With modern software, however, scribing
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convenience is no longer an issue. To better serve the reader, therefore, it would seem preferable to
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reinstate Peirce’s discarded idea (Figure 2D), which has the added benefit of rendering nested regions
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easier to distinguish and to refer to. When shading is not used, regions are now referred as being
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either oddly or evenly enclosed. Whether a region is surrounded by an odd or even number of
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boundaries, however, is rarely as obvious as whether this region is shaded or not. Using shading is
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thus more ergonomic.
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Figure 3: Rare graphs and pictures by Peirce’s own hand (Peirce, 2020, p. 483-484, 571). (A) A graph with
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“not A” expressed with the help of shading. The expression in its entirety reads: “not A and both not not B and
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not not C” or equivalently: “not (A and both not B and not C)” or equivalently: “A implies (B or C)”. (B) A
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picture of plant roots. (C) A picture of a house that has been encircled to express the negation of its meaning.
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(D) A logical argument that, instead of words, uses a picture of an open book—here intended to visualize
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“tautology”. For the logic of the argument, and an explanation of the mentioned rules, see the main text below.
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Some of Peirce’s work has been lost, but in what has survived he nearly always expresses
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propositions with the help of words or letters like P and Q (Figure 2C and Figure 3A). In exceedingly
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rare snippets of text, however, Peirce did not use such abstract tokens but instead tiny little pictures
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(Figures 3B-3D; Peirce, 2020, p. 483-484). Most likely he deemed drawing these pictures
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“insufferably inconvenient” for scribes too. Yet pictures are literally easier to picture, and also easier
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to remember, than are abstract words or letters (Kramer, 2023). Using pictures is thus more
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ergonomic. Modern technology can also make scribing pictures much less trouble than before. It
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would thus seem preferable to reinstate this idea as well (compare Figures 5 and 6). Wherever this
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seems helpful, I represent propositions and concepts not with abstract tokens but with black-on-white
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icons and their negation with white-on-black ones (see also Kramer, 2023).
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2.1.2 Drawing conclusions
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To draw valid conclusions from premises expressed with existential graphs, inference rules need to
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be followed. These rules allow one to transform a graph that expresses the premises of an argument
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into one that validly expresses its conclusion (Figures 4-7). To keep these rules simple, I define as a
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graph anything consisting of one or more regions and/or propositions, and as a frame any region
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without content that completely encloses another region that itself may, or may not, have any content
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(Figures 2C and 2D, bottom half). Originally, the rules refer to regions as either oddly or evenly
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enclosed; reviving Peirce’s use of shading, my reformulation of them refers to these same regions as,
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respectively, either black (shaded) or white (unshaded). Now the rules of existential propositional
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logic permit nothing except the following (reformulated from Dau, 2006; for proof of the soundness
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of these rules, see Dau, 2006; see also Peirce, 2020):
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(1) Insert/erase frame: Draw or erase a frame inside any region.
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(2) Insert/erase graph: Draw any graph inside a black region or erase any inside a white region.
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(3) Iterate/deiterate graph: Draw a copy of a subgraph inside an encompassing graph in either the
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same region or one further inward, or erase such a copy (for visibility, draw any copy in the color
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opposite to that of its local background). Do not copy any part of the subgraph onto this subgraph
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itself.
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Figure 4: Example applications of the inference rules of existential propositional logic (adapted, except for
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Figures 4A-4C and 4F, from Roberts, 1973, p. 42 and 43, in turn adapted from similar examples by Peirce,
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2020, #93981). The Insert/erase frame rule permits one to draw a frame on a blank sheet of assertion (A) or
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any other region (B) or (C) or to erase such a frame. The Insert graph rule permits one to transform (D) into
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(E) or (F), and the Erase graph rule to transform (G) into (H) or (I). The Iterate graph rule permits one to
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transform (J) into any of the graphs shown in (K) through (P), and the Deiterate graph rule allows one to do
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the converse. Not permitted is the transformation of (S) into (T). This transformation copies a subgraph of (S),
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consisting of the innermost region of (S) and its content (the graph P), onto the subgraph itself (in reversed
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colors and redimensioned). Such a transformation would lead to a contradiction, most clearly seen when the
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erase frame rule is used to transform (T) into (U).
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Figure 5: Basic existential propositional logic with letters. Modus Ponens argument: given, as premises, that P
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is true and that “P implies Q” is valid, we can infer that Q is true. Rephrased, the premise states that P is true
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and that it is not the case that P is false and Q is true. Given that P is true, the Deiterate rule eliminates from
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further consideration the idea that P could be false (i.e., the white-on-black P is erased). The Erase frame rule
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then rephrases “not not Q” as just “Q” (i.e., the black frame around the black-on-white Q is erased). And the
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Erase graph rule finally allows us to conclude that given that P and Q is true, Q on its own is true as well (i.e.,
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P can be erased). Modus Tollens argument: given, as premises, that “P implies Q” is valid, but that Q is false,
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we can infer that P is false. Given that Q is false, the Deiterate graph rule eliminates the idea that Q could be
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true from further consideration (i.e., the black-on-white Q is erased). The Erase graph rule now allows us to
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conclude that given that P is false and Q is false, P on its own is false too (the white-on-black Q can be
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erased). (Here, unlike the Modus Ponens argument, the Modus Tollens one does not need the Erase frame
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rule.)
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Figure 6: Basic existential propositional logic with icons instead of words or letters. (A) Illustration of a
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Wason selection task (Wason, 1968; Wason, 2013). Four cards are shown with either an adult or a baby on one
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side (on yellow) and either a cognac glass or baby formula on the other (on blue). The task is to indicate which
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of these four cards must be turned over to ascertain that a card with a baby on one side depicts baby formula
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on the other. The correct answer is that the card depicting the baby must be turned over, to check whether—
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per Modus Ponens—the other side correctly features baby formula, and the card depicting the cognac glass
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must be turned over, to check whether—per Modus Tollens—the other side correctly depicts an adult and not
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a baby. (B) Modus Ponens and Modus Tollens arguments with concrete icons instead of abstract words or
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letters. (Icons from thenounproject.com; for acknowledgements, see Section 6.)
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After impressionism, expressionism, abstract art, computer art, and numerous other artistic genres,
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some have argued that we have seen it all, and that art has nothing new to offer anymore. Yet it
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seems unprecedented that logical inferences can become as decorative as paintings. Some logical
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diagrams do contain illustrations alongside ordinary text (Berger, 2017; Even-Ezra, 2021) and many
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diagrams have some artistic aspects (Elkins, 1995; Stjernfelt, 2017). Yet, Figure 7B, say, could be
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mistaken for an example of lettrism, in which letters are used purely for esthetic reasons; if its letters
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were replaced by icons, the figure would be even less recognizable as an inference or practically
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useful diagram. Only half in jest, therefore, I am tempted to call such iconic proofs inferential art and
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to entitle Figure 7B “composition of dark versus light”—or more provocatively, but meaningfully—
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“black and white thinking”.
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Figure 7: Relatively complex existential propositional logic (adapted from Roberts, 1973, p. 46). (A) The
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iconic form of the Distributive Law of Implication: [P ® (Q ® R)] ® [(P ® Q) ® (P ® R)]. In symbolic
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logic, the distributive law has a hierarchical structure, typically elucidated with brackets within brackets. In
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iconic logic, the nested brackets are replaced with nested regions. To clarify the hierarchical structure of these
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nested regions, I first show those involved at the highest level in black and white (line 1), those at the
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intermediate level in dark and light gray (line 2), and those at the lowest level in blue and yellow (line 3).
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Subregions within light regions (the white, light gray, and yellow regions) are represented in the color opposite
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to those within dark regions (the black, dark gray, and blue regions). Likewise, propositions are also shown in
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the opposite color as the regions in which they are placed (line 4). The final representation of the distributive
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law in existential propositional logic is obtained by painting all dark regions black and all light regions white
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(line 5). (B) Proof of the distributive law in ten steps, from top to bottom and left to right, using: (1) the Insert
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frame rule, (2) the Insert graph rule, (3) the Iterate graph rule (and reversing fore- and background colors), (4)
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the Erase frame rule (and redimensioning a bit), (5) some reorganization, (6) some more reorganization, (7)
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the Insert frame rule, (8) the Iterate rule, (9) the Insert frame rule, and (10) some final adjustment.
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2.1.3 Drawing logic into algebra
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Besides uniting logic with art, one can also unite it with math. That is, one can reformulate logical
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derivations as equations and turn logic into algebra (Boole, 2009; Dunn & Hardegree, 2001). This
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Boolean algebra is popular in electrical engineering and its fuzzy logic extension in artificial
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intelligence. An iconic alternative to it has been developed that is based on entitative graphs (Figure
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8; Bricken, 2023; Spencer Brown, 1969). Entitative graphs can be converted into existential graphs
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and vice versa but in the former, unlike the latter, P Q does not express P and Q but P or Q (with
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“or” being inclusive rather than exclusive, meaning “and/or”). With a pair of brackets delineating a
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region, it follows that ((P) (Q)) expresses not (not P or not Q), which equals P and Q. P implies Q is
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expressed as (P) Q or, in other words, as not P or Q. The desired goal of a logical derivation is that it
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ends with a conclusion, the desired goal of its algebraic equivalent that it ends with a tautology—an
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equation that, under all circumstances, is trivially and undeniably correct.
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Figure 8: An iconic alternative to Boolean algebra. (A) Traditional, symbolic logical notation (left) and its
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entitative-graph equivalent (right). In the latter, truth is replaced by the confirmation of the existence of
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something, or more precisely, by the denial of the existence of nothing (i.e., by an empty region enclosed by a
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pair of brackets) and falsehood by literally nothing. A proposition is expressed with a capital letter (e.g., P)
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and its negation by enclosing this letter inside a region (inside a pair of brackets). (B) Basic equations (axioms)
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assumed to be valid. (Note how much more concise and intuitive they are, compared to Peirce’s inference
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rules.) (C) A Modus Ponent argument expressed in traditional, symbolic logic (the triangle of three dots means
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“therefore”). (D) The same argument expressed as an equation. (E) Proof of the Modus Ponent argument: First
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line: the argument reformulated as an entitative graph, using either brackets (middle column) or the black-and-
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white notation of previous figures (right column). Second line: using the axiom ((A)) = A to simplify the
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equation. Third line: rearranging terms. Fourth line: using the axiom A (A B) = A (B) to simplify the equation
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further. Fifth line: using the axiom A ( ) arrive at the obvious tautology that proves the argument.
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2.2 Adding color to our thoughts
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In order to reason about not assertions but individual persons, objects, or other items that are
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mentioned within such assertions, Peirce enriched his existential graphs with networks (ligatures)
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consisting of one or more line segments (lines of identity) that bind (ligare, in Latin) items to their
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properties or relationships with other items (Figure 9, middle column). To one’s liking, lines of
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identity can be drawn straight, curved, or even meandering. Properties and relationships are described
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in attached labels (predicates). The ligatures with their predicates effectively turn existential
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propositional logic into existential predicate logic. To help the reader recall what the predicate labels
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stand for, I once again use icons instead of words or letters (Figure 9, right column).
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Figure 9: Ligatures in existential graphs. Symbolic logical expressions (left column) and equivalent existential
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graphs (middle and right columns) that feature ligatures (continuous, dashed, or dotted lines or networks of
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lines) and predicates (capital letters in the middle column, icons in the right one). $x, "x, ¬, Ù, Ú, ®
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respectively mean “there exists at least one x”, “all or every x”, “not”, “and”, “or”, “implies”. P means “is a
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person”, H “has hands”, F “has feet”, C “has a car”, M “is a man”, L “loves”, W “is a woman”. Note, in line
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K, that some existential graphs can be translated into symbolic logic in two or more mutually equivalent ways.
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(Icons from thenounproject.com; for acknowledgements, see Section 6.)
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2.2.1 Making sense logically but not psychologically
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Although existential graphs are more iconic, and thus likely easier to picture and more intuitive than
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traditional symbolic predicate logic, they are not yet as intuitive as they could be. What is intuitive is
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that a single line of identity may represent a single item (Figure 9A, middle). Its equivalent in
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symbolic predicate logic uses, instead of this single line, a single variable like x or y (Figure 9A,
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left). Yet if a line of identity completely passes through a subregion (which expresses negation), then
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even though it is just one line, it is nonetheless supposed to represent two distinct items (Figure 9B,
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middle). This is quite counterintuitive, because as any introduction to perceptual organization attests,
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humans strongly tend to perceive a uniformly connected, uniformly colored, and smoothly continuing
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line as forming one single whole and not two different ones (Kimchi, Behrmann, & Olson, 2003;
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Wagemans, 2015). Indeed, in symbolic predicate logic (Figure 9B, left), instead of one single line,
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two distinct variables are used (say, x and y) and it is then stated that these two variables are not
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identical (x ≠ y).
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Conversely, intuitively enough, two unconnected lines of identity may represent two different items
333
(Figure 9C, middle). In symbolic predicate logic, instead of two unconnected lines, two different
334

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variables are used, like x and y (Figure 9C, left). Yet even though the two lines of identity are
335
unconnected, they may nonetheless potentially represent just one item. This can occur, for example,
336
when it is not known that the two items represented by these two lines are in fact one and the same
337
(compare Figures 9B and 9B, middle; only Figure 9B unambiguously represents two items and never
338
just one). Likewise, in symbolic logic, when two variables are used (say, x and y), then as long as it
339
is not known that these two variables are unequal (x ≠ y), they may nonetheless potentially represent
340
one and the same item as well—that is, it is possible that x = y (compare Figures 9B and 9C, left).
341
342
2.2.2 Making sense logically and psychologically
343
Although Peirce’s system works well from a logical point of view, as just laid out, it does not do so
344
quite as well from a psychological one. To remedy the problem, I therefore propose a superficial
345
change that makes the system look more intuitive but does not affect how it works.
346
347
Of a line of identity that completely passes through a subregion, one piece represents one item and
348
another piece another item. I propose that these two parts are given different appearances rather than
349
the same one (Figures 9B, 9J, 9L). The difference in appearance can be achieved by changing line
350
type (e.g., from continuous to dashed or dotted) and/or color (e.g., from brownish yellow to blue
2
). I
351
propose that, in all other cases, two interconnected lines of identity are given the same appearance.
352
353
Consider three particularly instructive examples. First, in Figure 9D, not all lines of identity represent
354
the same item, but because it is not clear which ones represent different items and which ones do not,
355
they are not given different appearances. Second, in Figure 9K, there are two lines of identity that
356
represent respectively a man and a woman. From a purely logical rather than a biological point of
357
view, the man and the woman could be one and the same person. Hence, they are not given different
358
appearances. Third, Figure 9L is similar to Figure 9K, but now the man and the woman are explicitly
359
stated to be different. In this case, therefore, the lines of identity associated with the man and the
360
woman do get distinct appearances.
361
362
The rules of inference (discussed a little later) consider the connections between lines of identity but
363
not their appearances. By keeping it that way, the system thus continuous to work as before, but
364
now—thanks to its new appearance—it is easier to read.
365
366
2.2.3 Linking individual existence to universal truth
367
Regardless of whether it is attached to a predicate, the outermost endpoint of a ligature indicates
368
whether the item represented by this ligature exists or not (Figure 10). If the outermost endpoint is
369
located on the white sheet of assertion, the item’s existence is asserted; if located in a black
370
subregion, it is denied; if located in a white sub-subregion, its denial is denied, and so on. A
371
combination of denial (negation) and existential quantification (assertion of an item’s existence) can
372
amount to a universal quantification (assertion of a universal truth). Unlike in ordinary symbolic
373
logic, in existential predicate logic there is no separate notation for universal quantification, and the
374
same existential graph can thus sometimes have more than one symbolic interpretation (Figure 10A).
375
In fact, keeping things as simple as possible, existential graphs often have the same representation for
376
various syntactically different, but logically equivalent, symbolic expressions. For example, in
377
2
To accommodate the colorblind, besides black, white, and gray, I use either at most one other color or blue and yellow
or brown but never both red and green.

ICONIC LOGIC
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existential graphs, the following three symbolic expressions are all represented in exactly the same
378
way: $x; $x$y(x=y); and $x$y$z(x=y Ù y=z Ù x=z) (Bellucci & Pietarinen, 2021).
379
380
381
382
Figure 10: Individual existence and universal truth in existential graphs with ligatures. $x means “there exists
383
at least one x”, "x “all or every x”, ¬ “not”, Ù “and”, ® “implies”, Mx “x is a man”, Wy “y is a woman”, Lx
384
“is in love”, Ly “is loved”. (A) “No man is not in love”, or equivalently, “all men are in love”. The two
385
equivalent assertions share the same notation in existential predicate logic but two different ones in their
386
symbolic counterpart. (B) “Some woman is loved”. (C) “There is some woman, and all men love her”. (D)
387
“All men love some woman” (Figure 10C asserts the existence of something; Figure 10D does not). (E)
388
“There is something that is not both a man and not in love” or equivalently: “there is a man who is in love”.
389
(F) “There is not something that is both a man and not in love” or equivalently: “there is no man who is not in
390
love” (Figure 10E asserts the existence of something; Figure 10F does not). (G) “There is some woman, and
391
all men love her” (alternative to Figure 10C, assuming the woman-icons are tokens, rather than types of
392
tokens, and refer to the same woman and not to potentially different ones). (H). “There is some woman, and all
393
men love her” (alternative to Figure 10C). (I) “There is some woman, and all men love her” (alternative to
394
Figure 10C). (J) “All men love some woman” (copy of Figure 10D). (Icons from thenounproject.com; for
395
acknowledgements, see Section 6.)
396
397
2.2.4 Drawing conclusions
398
With minor adjustments, the rules of inference for existential predicate logic are identical to those for
399
existential propositional logic but two rules are added that specifically regulate line-of-identity
400
transformations. For existential propositional logic, I defined as a graph anything consisting of one or
401
more regions and/or propositions. For existential predicate logic, I define it as anything consisting of
402
one or more regions and/or lines of identity and/or predicates. Moreover, whereas in existential
403

ICONIC LOGIC
14
propositional logic a frame may not have any content, in existential predicate logic an exception is
404
made for lines of identity that completely traverse the frame and do not terminate, or form junctions,
405
within its boundaries (Dau, 2006). In Figure 11B, for example, the graph that follows the label
406
“Deiterate graph rule” contains a line of identity that goes from a white region to another white
407
region and in the process it completely traverses a black region. This black region has no other
408
content and is thus considered a frame. With these minor issues settled, the rules of existential
409
predicate logic now permit nothing except the following (reformulated from Dau, 2006; for proof of
410
the soundness of the rules, see Dau, 2006; see also Peirce, 2020):
411
412
(1) Insert/erase frame: Draw or erase a frame inside any region.
413
(2) Insert/erase graph: Draw any graph inside a black region or erase any inside a white region.
414
(3) Iterate/deiterate graph: Draw a copy of a subgraph inside an encompassing graph in either the
415
same region or one further inward, or erase such a copy (for visibility, draw any copy in the color
416
opposite to that of its local background). Do not copy any part of the subgraph onto this subgraph
417
itself.
418
(4) Extend/retract line: Extend a line of identity (or branch of it) inward or retract it outward.
419
(5) Join/disjoin line: Join lines of identity in a black region or disjoin them in a white one. Join a
420
copy of a graph further inward that represents the same item with a ligature or disjoin the two.
421
422
The Extend/retract line rule permits replacing some assertions with similar, weaker ones but not the
423
converse. For example, the Retract line rule permits replacing the assertion “there is something that is
424
not both a man and not in love” (Figure 10E) with the weaker one “there is not something that is both
425
a man and not in love” (Figure 10F). In simpler words, the former (Figure 10E) says “there is a man
426
who is in love”, the latter (Figure 10F) “there is no man who is not in love”; the former asserts the
427
existence of a man, the latter does not.
428
429
The Join line rule permits replacing an assertion about things that are potentially distinct (represented
430
by lines of identity that are unconnected in white regions; Figure 9K) with a similar assertion in
431
which these things that are definitely distinct (represented by lines of identity that are connected in a
432
black region; Figure 9L). The converse is forbidden. For example, one may not join the lines of
433
identity in Figure 9K or disjoin them in Figure 9L. Additionally, the Join line rule permits replacing
434
Figure 10G with Figure 10H.
435
436
Both Peirce (2020) himself and Dau (2006) have offered various versions of these inference rules that
437
differ in their details. Roberts (1973) presents a version of Peirce’s rules with an extra constraint on
438
one of them —obtained from Peirce’s writing elsewhere—to stop it from being too permissive. Shin
439
(2002) presents a version that is interestingly permissive in some respects but too permissive Dau
440
(2006) in others. Using graph theory, Dau offers a mathematical reformulation of his rules and
441
proves that these rules are sound (cannot lead to contradictions; see also Gangle & Caterina, 2015).
442
Yet, they may be overly restrictive. For example, both Dau und Shin have a version of the Deiterate
443
graph rule that permits replacing Figure 10H with Figure 10I, but unlike Dau, Shin not only permits
444
retracting a loose end of a line of identity outward into any region (Retract line rule) but also inward
445
into a white one (as in the transformation of Figure 10I into Figure 10J). Indeed, the assertion that all
446
men love some woman (Figure 10J) is similar but weaker than the assertion that there is some
447
woman who is loved by all men (Figure 10I). One would think, therefore, that replacing the latter
448
(Figure 10I) by the former (Figure 10J) should be permitted. Indeed, Peirce (2019, p. 309) does
449
permit this, but it is not obvious how Dau’s rules would do so.
450
451
Besides the fact that the rules of existential predicate logic are not fully crystalized out yet, at least
452

ICONIC LOGIC
15
one minor extension of the notation is also needed to allow the system to express all of “first order”
453
predicate logic (Bellucci, Liu, & Pietarinen, 2020). By and large, however, existential graphs seem a
454
promising alternative to traditional propositional and predicate logic (for more elaborate examples
455
than those in Figure 10, see Figures 11 and 12; for the translation of existential logic to symbolic
456
logic, see (Dau, 2006)). An iconic alternative to Boolean algebra that can handle predicate logic may
457
be even more promising and is in preparation (W. Bricken, personal communication, 12 September
458
2023).
459
460
461
462
Figure 11: Basic existential predicate logic (adapted from Dau, 2006, pp. 13-14, in turn adapted from similar
463
examples by Peirce, 2020, #93981). (A) Icon meanings. (B) An iconic proof of a syllogism. Premise 1: A
464
Peterbilt is a trailer truck. Premise 2: Every trailer truck has eight wheels. Here, the Extend line rule expresses
465
the fact that the item that has the property of being both a Peterbilt and a trailer truck without having some
466
unspecified property (i.e., the item’s ligature is extended into a black region but not connected to anything).
467
The Join line rule expresses the fact that the item is either not a trailer truck or has 8 wheels. Given that the
468
item is a trailer truck, the Deiterate graph rule eliminates from further consideration the idea that it would not
469
be (i.e., the white-on-black trailer truck is erased). The Erase frame rule rephrases “not not having 8 wheels”
470
as just “having 8 wheels” (i.e., the black frame around the black-on-white “eight wheels” is erased). The Erase
471
graph rule now allows us to conclude that, given that the item is a trailer truck and has 8 wheels, the item has 8
472
wheels (i.e., the black-on-white trailer truck can be erased). (For the logic behind iconic numbers, here the
473
number 8, see Kramer, 2022. (Truck icon from thenounproject.com; for acknowledgements, see Section 6)
474
475
Compared to Figure 11, Figure 12 shows a relatively more complex proof. Figure 12A shows the
476
meanings of the icons used in the proof, Figure 12B the three premises that are the starting points of
477

ICONIC LOGIC
16
the logical argument, and Figure 12C the proof itself. Premise 1 contains, among other things, three
478
(white) subregions without content that are traversed by a line of identity, and in which this line thus
479
changes appearance. These three subregions play no role of importance in the proof, and for
480
simplicity, they are eliminated in Figure 12C. After this, Premise 1 features a black region with
481
content and a single white subregion with content. This suggests we are dealing with an implication
482
(compare with Figure 9K). Premise 1 states that tallness is transitive—that is, that if Item 1 (brownish
483
yellow line) is taller than Item 2 (dashed blue line), and if Item 2 (dashed blue line) is taller than Item
484
3 (continuous blue line), then Item 1 (brownish yellow line) is also taller than Item 3 (continuous
485
blue line). Premise 2 states that some adult happens to be taller than some toddler. Premise 3 states
486
that the toddler of Premise 2 happens to be taller than some baby. The task, now, is to prove that the
487
adult mentioned in Premise 2 must be taller than the baby mentioned in Premise 3. The proof consists
488
of the following seven steps:
489
490
Step 1: Positioning premises: Purely for convenience, Premises 2 and 3 are placed below Premise 1
491
and the graph of Premise 1 is elongated and streamlined.
492
Step 2: Erase graph rule: Of the two toddler icons, one is eliminated.
493
Step 3: Extend line rule (applied four times): Four lines are extended inward from a white area (the
494
sheet of assertion) into a black area.
495
Step 4: Join line rule (applied four times): The four lines mentioned in Step 3 are joined with existing
496
lines of identity in the black area (and given their color and line type).
497
Step 5: Deiterate graph rule (applied twice): There are two black-on-white “taller than” icons on the
498
sheet of assertion and two oppositely colored copies of them further inward. The inward ones are
499
erased.
500
Step 6: Erase frame rule: The black region has become a frame and can therefore now be erased as
501
well.
502
Step 7: Erase graph rule (applied three times) plus simplification: The icon of a toddler and two
503
“taller than” icons are erased, the dashed blue line of identity is erased, and the result is simplified to
504
improve readability. This completes the proof. Premises 1, 2, and 3 together allowed us to conclude,
505
in seven steps, that the adult pictured in Premise 2 is taller than the baby pictured in Premise 3.
506
507

ICONIC LOGIC
17
508
509
Figure 12: Relatively complex existential predicate logic. (A) Icon meanings. (B) Premises (see main text for
510
details). (C) Proof in seven steps that the adult pictured in Premise 2 must be taller than the baby pictured in
511
Premise 3. (Icons from thenounproject.com; for acknowledgements, see Section 6.)
512
513
3 Concept diagrams
514
Experiments that investigate which systems of logic are best tailored to the human mind are rare and,
515
to my knowledge, none have investigated existential or entitative graphs. Concept diagrams are
516
particularly interesting because they are quite powerful and their user-friendliness has been
517
investigated experimentally. These behavioral studies have been conducted by logicians,
518
mathematicians, and information technology experts—prominent among them John Howse and
519
colleagues (Alharbi, Howse, Stapleton, Hamie, & Touloumis, 2017; Blake, Stapleton, Rodgers,
520
Cheek, & Howse, 2014; Shams, Sato, Jamnik, & Stapleton, 2018; Shimojima & Katagiri, 2013;
521
Stapleton, Rodgers, Touloumis, & Blake, 2020; Veres & Mansson, 2004). Here, reviewing (Section
522

ICONIC LOGIC
18
3.1) and then critiquing (Section 3.2) a recent example of such studies (McGrath et al., 2022), I argue
523
that, without doing the logician’s work, behavioral scientists could have made—and still can make—
524
an important contribution to such research.
525
3.1 Using color to better identify sets and relationships
526
McGrath et al. (2022) investigated elaborations of Euler diagrams (Figure 1): concept diagrams
527
(Figure 13, 14, 15). In concept diagrams, separate Euler diagrams, each represented within a
528
rectangular area, may be interconnected and enriched with extra elements (Figures 13D-13G;
529
McGrath et al., 2022). Most of these additions are not very iconic but because the Euler diagrams are,
530
their combination is still fairly iconic.
531
532
In examples that refer to mythical creatures, McGrath et al. (2022) added solid arrows to their
533
diagrams to express only-relationships, like in “Boggarts scare only Midgets” (Figure 13A), and
534
dashed arrows to express some-relationships, like in “Boggarts scare some Midgets” (Figure 13B).
535
Mathematical symbols express, say, whether Boggarts scare at least one Midget (Figure 13B).
536
Arrows that start from the edge of a rectangle are used to express that something unspecified is
537
related to something else, like in “There is something unspecified, and it is the only thing that scares
538
Midgets” (Figure 13C). A minus sign inverts a relationship, like in “There is something unspecified,
539
and only Midgets scare it” (Figure 13D).
540
541
542
543
Figure 13: Monochrome concept diagrams. The diagrams express: (A) “Boggarts scare only Midgets”, (B)
544
“Boggarts scare at least one Midget”, (C) “There is something unspecified, and it is the only thing that scares
545
Midgets”, (D) “There is something unspecified, and only Midgets scare it”.
546
547
McGrath et al. (2022) hypothesized that concept diagrams should be easier to read, and work with, if
548
circles and arrows were colored differently (Figure 14A) or if each circle and any associated arrow
549
could be uniquely identified with a specific color (Figure 15A). Previously, in fact, a similar use of
550
color has been found to improve reasoning with Euler diagrams (Blake, Stapleton, Rodgers, &
551
Howse, 2014). To the authors’ surprise, however, the data did not corroborate their hypothesis. The
552
authors concluded that the more complex concept diagrams are, the less color helps in using them as
553
reasoning tools.
554

ICONIC LOGIC
19
555
556
557
Figure 14: Dichrome concept diagrams: (A) To distinguish them better, circles and arrows are colored
558
respectively blue and green rather than black. (B) Proposal for a monochrome alternative. (C) Proposal for a
559
monochrome alternative in which an arrow pointing to a circle is given the same color as the circle itself. (D)
560
Replacing abstract labels (here “Boggart”, “scares”, and “Midget”) with concrete icons ought to improve
561
readability. Here, the three icons represent, respectively, a boggart (a short, malicious creature), a facial
562
expression of fear, and a midget (a short but amiable person).
563
564

ICONIC LOGIC
20
565
566
Figure 15: Polychrome concept diagrams. (A) Using color to uniquely identify each circle and any associated
567
arrow. (B) Using color to make each circle and any associated arrow more distinguishable from nearby or
568
overlapping ones.
569
3.2 Using color to better distinguish sets and relationships
570
McGrath, Howse, and their colleagues should be commended for opening a new, much need research
571
field; for attempting to render logic more ergonomic; and for letting data rather than intuition be the
572
arbiter of this endeavor’s success. Their experimental studies also suggest, however, that it would
573
have been desirable to get behavioral scientists involved. To an expert in color perception, for
574
example, it is immediately apparent that McGrath et al.’s (2022) study is confounded and its
575
conclusion premature.
576
577
The dichrome and polychrome conditions featured colored circles. Unfortunately, the Ishihara
578
colorblindness test (Ishihara, 1917), although quick and easy to perform, was not used to screen out
579
colorblind participants. More importantly, in the dichrome and polychrome conditions, the colored
580
circles and arrows were more luminant (physically lighter) than the black ones in the monochrome
581
condition. As a result, they contrasted less well with their white background and were therefore less
582
visible. Moreover, the dichrome condition (Figure 15A) did not use complementary colors like blue
583
and yellow or red and green, which are the easiest to tell apart, but blue and green. These blue and
584
green colors also had rather similar luminances, which further reduced their distinguishability. In
585
simple diagrams like Figure 14A distinguishability may not be a major issue but it is in busy ones
586
like Figure 15A (Manassi & Whitney, 2018; Wolfe, Kosovicheva, & Wolfe, 2022; see also Alqadah,
587
Stapleton, Howse, & Chapman, 2016). And whereas achromatic colors like black, white, and gray
588

ICONIC LOGIC
21
can be perceived everywhere in the eye, chromatic ones like blue and green are only perceived in its
589
central part—the fovea. In sum, the visibility of the circles and arrows was bound to be much better
590
in the monochrome condition (Figure 13) than in the dichrome (Figure 14A), and especially the
591
polychrome (Figure 15A), condition.
592
593
In the dichrome condition, there is actually no need to complicate matters with chromatic colors:
594
achromatic ones will do (Figure 14B). If the arrows and circles are respectively represented in white
595
and black on a gray background, or vice versa, they will become particularly easy to distinguish
596
(Masin, 2003; see also Bressan & Kramer, 2008; Bressan & Kramer, 2017). If there is just one arrow
597
pointing to each circle, one can make visually obvious to which circle the arrow is pointing by giving
598
both the same color (Figure 14C; see also Figure 15B). Readers that are unfamiliar with ancient
599
folklore can use a dictionary to find out what, say, a boggart is but may then still have a hard time
600
picturing one in their mind. Replacing abstract labels like “Boggart” with iconic alternatives solves
601
this problem (Figure 14D) and could also bring other diagrams to life.
602
603
Even for normally sighted individuals, the cacophony of colors used in Figure 15A may perhaps be a
604
bit overwhelming. It might have been better to use only colors like blue and yellow and black and
605
white that are easy to distinguish for both normally sighted and colorblind individuals, and to use
606
these colors not to uniquely identify circles but to make nearby and overlapping ones easier to
607
distinguish from one another (Figure 15B). One could depict sets and their relationships in dark
608
colors (here black, blue, and the dark yellow known as “brown”); and sets involved in inverse
609
relationships, and these inverse relationships themselves, in light colors (here white). To depict an
610
inverse relationship, one could, in addition, use an arrow that points away from, rather than toward,
611
its associated circle. The less iconic minus sign could then be dropped.
612
613
For the design of their experiment, McGrath et al. (2022) made various other suboptimal choices.
614
Reaction time, for example, was measured without ensuring that the participants responded as fast as
615
possible while maintaining near-perfect accuracy. This goes against the tried and tested method for
616
measuring reaction times and makes the results harder to interpret. More importantly, different
617
participants were enrolled in different experimental conditions (between-subjects design), whereas it
618
would have been better if each participant had been enrolled in all conditions (within-subjects design)
619
with the order of the conditions systematically varied between participants (counterbalanced). Using
620
a between- rather than a within-subjects design is especially a concern in color perception, because
621
even such simple colors as black, white, and gray depend in complex ways on the circumstances
622
under which they are perceived (Bressan, 2006; Bressan, 2007; Bressan & Kramer, 2021b; Gilchrist,
623
2006; Kingdom, 2011; Murray, 2021; Soranzo & Gilchrist, 2019). Consider, for example, how the
624
moon looks shiny and almost white in the night’s sky but dull and dark gray under an astronaut’s
625
white boot (Bressan, 2005; Bressan & Kramer, 2021b).
626
627
Imitating the night’s sky, investigators of the perception of black, white, and gray often perform their
628
experiments in a dark laboratory and the most careful among them paint their laboratory black and
629
encase all their equipment in black as well (e.g., Bressan & Actis-Grosso, 2006; see also Gilchrist,
630
2006). Such stringent control over experimental conditions is most likely unnecessary for studies like
631
McGrath et al.’s (2022). Yet, McGrath et al. went to the other extreme and allowed viewing
632
conditions to differ from one participant to the next even though this adversely affects a between-
633
subject’s experiment more than it does a within-subjects one. That virtually no significant results
634
were obtained is not surprising. Thus, whether a judicious use of color can, or cannot, improve the
635
readability of busy concept diagrams remains an open question.
636

ICONIC LOGIC
22
4 Profiling talent and disability in logic
637
4.1 It is not truth and falsehood, but death and survival, that reign supreme
638
Expertise in perception can help us improve the ergonomic features of logic. Expertise in cognition
639
can help us understand who, from this endeavor, is likely to benefit the most. It is important to realize
640
that reasoning logically and thinking straight are two different things. For example, thinking, unlike
641
reasoning, requires picking the right premises to begin with and framing the problem in the right
642
way. In Bohr’s quantum theory of physics, a miniscule particle can be at more than one place at the
643
same time. If one reasons logically from the premise that we can travel back and forth in the three
644
dimensions of space but only forward and never backward in the dimension of time, quantum theory
645
makes little sense. To understand this theory, we need to jettison our preconceived notion of what our
646
world is like.
647
648
Our notion of what the world is like is shaped by evolution, and in evolution, it is not truth and
649
falsehood that reign supreme but death and survival (Hoffmann 2015; see also Wood, 2023). Indeed,
650
only a minority of undergraduate students perform well when tested on the Wason selection task
651
(Figure 6)—a task specifically designed to test people’s reasoning skill (Wason, 1968; Wason, 2013).
652
We are a social species, however, and most humans are keen detectors of who is violating our social
653
norms (Cosmides, Barrett, & Tooby, 2010; Cosmides & Tooby, 1992). Accuracy in the Wason
654
selection task is, in fact, significantly higher if the very same argument is presented as a moral issue
655
(e.g., about whether babies should drink alcohol or not; Figure 6) rather than as one having merely to
656
do with facts (Fiddick, Brase, Cosmides, & Tooby, 2017).
657
658
Logical reasoning and cheater detection can, however, be pitted against each other. Consider these
659
two rules:
660
661
Rule 1: If you give me your watch, I give you $20.
662
Rule 2: If I give you $20, you give me your watch.
663
664
Now, if I do not give you $20, Rule 1 implies that you do not give me your watch. Yet, from a logical
665
point of view, Rule 2 does allow me to keep my $20 and still take your watch. This violates the rules
666
of ethics but to the confusion of many socially sensitive people, it does not violate the rules of logic
667
(Cosmides & Tooby, 1992). Teaching aspiring logicians to disregard the content of an argument, and
668
simply and rigidly apply the rules of logic, may improve these people’s reasoning. The issue
669
discussed here, however, brings us to a theory of the development of social and technical skill that
670
has direct implications for how logic could become more ergonomic.
671
4.2 Symbols are for nerds, icons for everybody
672
The diametric theory of genomic imprinting suggests that there tends to be a tradeoff between social
673
skill and mathematical (and thus also logical) ability (see Bressan & Kramer, 2021a and references
674
therein, including especially Crespi & Badcock, 2008; Badcock, 2009; Badcock, 2019; see also
675
Crespi, 2020; Del Giudice, Angeleri, Brizio, & Elena, 2010; Kramer, 2022; Mokkonen, Koskela,
676
Procyshyn, & Crespi, 2018; Úbeda, 2008; Úbeda & Gardner, 2015). Tellingly, indeed, colloquial
677
language has one single term for a person who is both socially awkward and has an all-absorbing
678
attention for the minute details of some technical matter: not always derogatively, a person like that
679
is called a nerd.
680
681

ICONIC LOGIC
23
Now, some of the genes we inherit from our parents are turned on and some off. Remarkably, of so-
682
called imprinted genes, those turned on in the copy we inherit from our father are typically turned off
683
in the copy we inherit from our mother and vice versa. As a consequence, fathers and mothers push
684
offspring development in diametrically opposite directions. To some extent, male and female sex
685
chromosomes have similar opposite effects (Badcock, 2009; Badcock, 2019; Crespi & Badcock,
686
2008).
687
688
The root of the parental conflict has to do with the fact that, especially during gestation and the first
689
few years of the offspring’s life, mothers invest more in their children than fathers do (Moore &
690
Haig, 1991; Haig, 2010; Kotler & Haig, 2018). For the most part, paternal imprinting pushes for the
691
extraction of maternal resources during this early period and promotes the growth of the offspring’s
692
body and brain—in particular those parts of the brain that allow it to deal with its physical
693
environment. Maternal imprinting, instead, limits the growth and resource extraction, but does
694
stimulate the growth of those parts of the brain that allow the offspring to deal with its social
695
environment—which, later on, will enable it to take its mother’s directions.
696
697
The tug of war between parents over the expression of genes goes at their offspring’s expense, and
698
imprinted genes and sex chromosomes are implicated disproportionately often in both physical and
699
mental disease—most prominently in pairs of diseases that are due to tightly related genetic
700
mutations that come with roughly opposite symptoms (Bressan & Kramer, 2021a; Xirocostas,
701
Everingham, & Moles, 2020). Excessive paternal imprinting is associated with overgrowth of body
702
and brain before birth, which makes delivery riskier for the mother (see also Kramer & Bressan,
703
2019). In addition, it is associated with autism-spectrum disorders and strong semantic memory for
704
facts (including technical ones) but weak episodic memory for events (including social ones).
705
Conversely, excessive maternal imprinting is associated with undergrowth of body and brain before
706
birth, psychosis-spectrum disorders, and strong episodic but weak semantic memory (Bressan &
707
Kramer, 2021a). A slight tendency toward autism is associated with increased attention to detail,
708
strong semantic memory, and increased technical and mathematical ability but reduced episodic
709
memory and social skill, including a diminished ability to understand other people and to detect
710
whether they are cheating or not; a slight tendency toward psychosis is associated with the opposite
711
(Badcock, 2009; Badcock, 2019; Bressan, 2018; Crespi & Badcock, 2008; Kramer, 2022 and
712
references therein).
713
714
Malnutrition, brain damage, and problems during development may all result in memory deficits.
715
Yet, if the diametric theory of genomic imprinting is correct, then many people who are healthy and
716
have developed normally, but in whom maternal imprinting happens to be dominant, can have a
717
weak semantic memory too. These days, the less people are inclined to think analytically, the more
718
they rely on the internet as a kind of external semantic memory (Barr, Pennycook, Stolz, &
719
Fugelsang, 2015). Hotly debated is whether this may further weaken internal semantic memory and
720
its cortical substrates (Firth et al., 2019; see also Dempsey, Lyons, & McCoy, 2019; Ellis, 2019;
721
Hartanto et al., 2023). So, if we wish to improve reasoning in not just professional logicians but other
722
people too, we ought to introduce a tool that requires as little as possible from our semantic memory.
723
Arguing for a more ergonomic representation of knowledge, Wood (2023, p. 1) observes that “if
724
knowledge were simpler, we would all be wiser”. Likewise, it stands to reason that if logic burdened
725
semantic memory less, we would all argue better. And as icons are better mnemonics than symbols
726
are (Kramer, 2023), iconic logic is more ergonomic than symbolic logic is. Although strong
727
analytical thinkers may certainly benefit from iconic over symbolic logic, those who need a reasoning
728
tool the most ought to benefit the most too.
729

ICONIC LOGIC
24
4.3 Put not scribes, but readers, first
730
Given that symbolic logic burdens its readers’ semantic memory more than does iconic logic, one
731
may wonder why most logicians stick to symbolic logic anyway. Excerpts from the medical history
732
of a representative logician (Charles Sanders Peirce) may help answer this question (Pfeifer, 2014).
733
There is little doubt Peirce paid close attention to detail and was exceptionally well-versed in the
734
slow, painstaking, step-by-step kind of thinking that comes in handy in formal logic. The excerpts
735
suggest, however, that Peirce did not consider himself a good writer, was not in tune with his
736
audience during lectures, and had “too little social talent, too little art in making himself agreeable”
737
(Pfeifer, 2014, p. 29).
738
739
The combination of a poor ability to put oneself in the mind of another person, and poor social and
740
communication skills, but keen attention to detail, and a strong tendency to systematize, is typical of
741
people with an autistic tendency and frequent among mathematicians (Badcock, 2009; Badcock,
742
2019; Bressan, 2018; Bressan & Kramer, 2021a; Crespi & Badcock, 2008). Autism is partly heritable
743
(Badcock, 2009; Badcock, 2019), and Peirce came from a math-oriented family (Pfeifer, 2014). It is
744
thus tempting to suspect Peirce had an autistic tendency (for a similar view, see Pfeifer, 2013; for
745
whether Peirce may have had additional bipolar tendencies, see Brent, 1998).
746
747
If Peirce did indeed have an autistic tendency, he may not have understood how unergonomic it is to
748
represent two distinct items with one and the same line of identity while, under circumstances,
749
allowing one and the same item to be represented with two distinct lines. He may not have
750
understood that it is easier to see whether a region is shaded or not than to have to count to see
751
whether it is oddly or evenly enclosed. And he may not have understood that the meaning of labels of
752
items or properties are easier to picture and remember when they are expressed with concrete icons
753
than when they are expressed with abstract letters or words (Kramer, 2023). Peirce deemed the
754
shading of regions, and presumably also the drawing little pictures, too much of a hassle for scribes.
755
Yet with readers typically far outnumbering scribes, one would think that reducing the burden on all
756
these readers should count much more than reducing any burden on the scribes (see also Kramer,
757
2023). Today’s software lightens the scribe’s task enormously, and there is now all the more reason
758
to accommodate the reader. Of course, the ultimate arbiter of whether a system of logic is indeed
759
user-friendly or not is behavioral evidence. This means that logic, a discipline traditionally perceived
760
as standing aside from the empirical sciences, ought to become part of them instead
3
.
761
5 Acknowledgments
762
Many thanks to William Bricken for extensive correspondence regarding his iconic alternative to
763
Boolean algebra, to Ahti-Veikko Pietarinen for sending me David Pfeifer’s excerpts of Peirce’s
764
medical history, and to David Pfeifer for sending me Pfeifer (2013). And last but certainly not least,
765
many thanks, as always, to Paola Bressan for her insightful comments on the prose of various drafts
766
of this article; putting oneself in the mind of a keen reader is good, but clearly, having the comments
767
of one is much better.
768
769
Icon design in Figure 6: adult (Tamara), cognac glass (Adrien Coquet), baby (Simon Sim), baby-
770
formula (Zaenul Yahya); in Figure 9: adult (Tamara), hand (Robin Lopez), foot (Alexander
771
Skowalsky), car (Adrien Coquet), man (Guilherme Furtado), and woman (ImageCatalog); Figure 10:
772
man (Guilherme Furtado), woman (ImageCatalog); Figure 11: truck (Laxouri Studio); Figure 12:
773
3
For a similar argument for mathematics more generally, see (Kramer, 2022).

ICONIC LOGIC
25
adult (Tamara), toddler (Adrien Coquet), baby (Simon Sim), and taller-than (Adrien Coquet). Figure
774
14: fear (Michel Ouellette). All obtained via thenounproject.com.
775
6 Conflict of Interest
776
The author, Peter Kramer, declares that the research was conducted in the absence of any commercial
777
or financial relationships that could be construed as a potential conflict of interest.
778
7 Author Contributions
779
PK is solely responsible for the content of this article.
780
8 Funding
781
Not applicable.
782
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