# Epistemological Obstacles

### Logical Implication

Students experiencing this EO typically reason about logical implications deductively from hypothesis to conclusion (e.g. modus ponens) and struggle with reversible thinking (e.g. modus tollens). Rather than treating the implication as an invariant relationship (object conception), they may attend to reasoning about the hypothesis and conclusion separately (action conception).

###### Arnold & Norton, 2017; Norton & Arnold, 2017; Dubinsky, 1981

Related EOs:

This EO is discussed further in the **Logical Implication** Instructional Module.

Students experiencing this EO interpret logical implications as biconditional statements. For example, they may assert that the hypothesis and conclusion are either both true or both false. This conflation may arise from discrepancies between social and mathematical interpretations of everyday language.

Epp, 1999, 2000; Wagner-Egger, 2007; Rumain et al., 1983; Matarazzo & Baldasarre, 2010

Related EOs:

This EO is discussed further in the **Logical Implication** Instructional Module.

Students experiencing this EO have difficulties transforming a logical implication into its converse, contrapositive, negation, or inverse.

Arnold & Norton, 2017; Norton & Arnold, 2017; Kodroff & Roberge, 1975; Dawkins & Hub, 2017

Related EOs:

This EO is discussed further in the **Logical Implication** Instructional Module.

Students experiencing this EO interpret logical implications as only applying to the truth set of the hypothesis rather than the entire universal set. In this way, they only attend to the cases in which the hypothesis is true (implicitly treating the vacuous case as irrelevant).

This EO emerged during the project.

Related EOs:

Students experiencing this EO conflate the truth of the hypothesis with the truth of the implication itself. For example, when given a true implication, they might infer that the hypothesis must also be true.

Avital & Libeskind, 1978; Movshovitz-Hadar, 1993; Norton, Arnold, Kokushkin, & Tiraphatna, 2022

Related EOs:

This EO is discussed further in the **Mathematical Induction** Instructional Module.

Students experience this EO when they provide the "opposite statement" when asked to determine the negation of given a statement. For example, when negating "for all x, P(x)" some students might respond, "for no x, P(x)." And, when negating the logical implication P→Q, they might respond with either P→~Q or ~P→~Q.

Dawkins & Hub, 2017; Epp, 2003

Related EOs:

This EO is discussed further in the **Quantification** Instructional Module

Students experiencing this EO struggle to articulate the distinction between negating and disproving a statement. They may either see these two activities as the same thing, or they may perceive a difference but be unable to explain it.

This EO emerged during the project.

Related EOs:

This EO is not directly discussed in any Instructional Modules, but is related to Quantifiying the Negation of a Statement, which is discussed in the **Quantification** Instructional Module.

### Quantification

Students experiencing this EO treat the global quantification of a logical implication as quantifying only the hypothesis. For example, when proving a universally quantified logical implication P(x)→Q(x), they may quantify the assumption as "Assume for all x that P(x) is true." In this way, they universally quantify their assumption instead of fixing an arbitrary x and assuming P(x) is true only for this instantiated x value.

This EO emerged during the project.

Related EOs:

This EO is discussed further in the **Mathematical Induction** Instructional Module.

Students experiencing this EO may either fail to attend to the quantification of statement when it is left implicit (hidden), or they may choose the incorrect quantification when making its explicit. For example, students could intrepret the statement "If *x* > 2, then *x*^{2} > 4" as referring to a specific x. Or, they might state its negation as "*x* > 2 and *x*^{2} <= 4," omitting the existential quantification of x altogether.

Shipman, 2016; Durand-Guerrier, 2003; Ernst, 1984

Related EOs:

This EO is discussed further in the **Quantification** Instructional Module.

Students experiencing this EO will exhibit signs of cognitive overload (e.g. gesturing or verbal rehearsal) when attending to more than one quantifier in a multiply quantified statement.

Piatek-Jiminez, 2010; Dawkins & Roh, 2020b

Related EOs:

Students experiencing this EO struggle to notice or understand how the order of quantifiers in a multiply quantified statement impact mathematical logic. This may manifest as overlooking the role of the order entirely, or as an observable struggle when attempting to disambiguate the order.

Piatek-Jiminez, 2010; Dawkins & Roh, 2020b; Epp, 2003 citing Dubinsky & Yiparaki, 2000

Related EOs:

This EO is discussed further in the **Quantification** Instructional Module.

Students experiencing this EO struggle disambiguating the logical role of a variable versus its instantiated value when proving. For example, in proof by mathematical induction, we may use *n* = *k* when proving the inductive implication. Here, *k* is a fixed but arbitrary value of *n*, but it also plays the logical role of addressing all intermediate cases between the base case and the *n*th case.

Dawkins & Roh, 2020b

Related EOs:

This EO is discussed further in the **Mathematical Induction** Instructional Module.

Students experiencing this EO struggle with determining the correct quantification when finding the negation of a given statement. This issue may be exacerbated by the language present in the statement, e.g. "for any", "for some", and "for all."

Dawkins & Roh, 2016; Shipman, 2016

Related EOs:

This EO is discussed further in the **Quantification** Instructional Module.

Description

Copi, 1954; Dawkins, Roh, Eckman, 2023; Norton et al., 2022

Related EOs:

This EO is discussed further in the **Mathematical Induction** Instructional Module.

### Function

Description

Related EOs:

This EO is discussed further in the **Function** Instructional Module.

Description

Related EOs:

This EO is discussed further in the **Function** Instructional Module.

Description

Related EOs:

This EO is discussed further in the **Function** Instructional Module.