2.7: Counterexamples

Not all deductions are valid. To show that a particular deduction is not valid, you need to show that it is possible for its conclusion to be false at the same time that all of its hypotheses are true. To do this, you should find an assignment to the variables that makes all of the hypotheses true, but makes the conclusion false.

Exercise (PageIndex{1})

Show that the deduction [A lor B, quad A Rightarrow B, quad herefore A] is not valid.


To make the conclusion false, we let (A) be false. Then, to make the first hypothesis true, we must let (B) be true. Fortunately, this also makes the second hypothesis true.


Let (A) be false, and let (B) be true. Then

[A lor B = mathsf{F} lor mathsf{T} = mathsf{T} ]

and [A Rightarrow B = mathsf{F} Rightarrow mathsf{T} = mathsf{T},]

so both hypotheses of the deduction are true. However, the conclusion of the deduction (namely, (A)) is false.

Since we have a situation in which both hypotheses of the deduction are true, but the conclusion of the deduction is false, the deduction is not valid.

Any situation in which all of the hypotheses of a deduction are true, but the conclusion is false, is called a counter example to the deduction.

[ ext{To show that a deduction is not valid, find a counterexample.}]

Exercise (PageIndex{2})

Show that each of these deductions is invalid, by finding a counterexample.

  1. (A lor B), (A Rightarrow B)
  2. (P lor Q), (P & Q)
  3. (A Rightarrow (B & C)), (lnot A Rightarrow (B lor C)), (C)
  4. (P Rightarrow Q), (lnot P Rightarrow R), (Q & (P lor R))

Do Genesis 1 and 2 Contradict Each Other?

A concerned reader wonders how to respond to an issue that has created doubt in her husband's faith. Tim Chaffey, AiG-US, addresses his concerns.

My husband recently had a discussion with one of his professors regarding Hebrew scriptures in Genesis. My husband has concluded that because there seems to be [contradiction] between the order of creation in Genesis chapters 1 and 2 that the Bible is not flawless. I do not share his conclusion and would like to have an answer for him. The scriptures in question are Genesis 1:1–2:3 vs. Genesis 2:4-22 . He has claimed to be a believer for 10 years, but now believes that man has mucked up the Word of God and that the Bible is not completely accurate and has flaws. Could you help me?

–R.H., USA

Thank you for contacting Answers in Genesis.

This is a common argument used against the traditional understanding of Genesis (i.e., God created everything in six normal-length days approximately 6,000 years ago). This argument attempts to show that inconsistencies exist between the first two chapters in the Bible. Critics and skeptics use it in their efforts to show the Bible cannot be trusted. Some Christians who believe in billions of years use it in trying to show that these chapters should not be understood in their plain sense. However, the argument is based on a misunderstanding of Genesis 2 .

Genesis 1:1–2:3 provides us with a chronological account of what God did on each of the days during Creation Week. Genesis 2:4–25 zooms in on Day Six and shows some of the events of that day.1 Let’s take a look at what happened on Day Six, according to Genesis 2 , and we’ll see there is no discrepancy here.

  • Adam is created ( Genesis 2:7 )
  • Garden of Eden created ( Genesis 2:8–9 )
  • Description of river system in Eden ( Genesis 2:10–14 )
  • Adam put in Garden and given instructions ( Genesis 2:15–17 )
  • Adam names some of the kinds of animals ( Genesis 2:18–20 )
  • God creates Eve ( Genesis 2:21–22 )
  • Description of Adam, Eve, and marriage ( Genesis 2:23–25 )

The particular issue that people have with Genesis 2 is that the order of the creation of man, animals, and trees seems to be contrary to the order stated in Genesis 1 .

Genesis 2:7 describes the creation of man.

Following the creation of man, Genesis 2:9 mentions that God created trees, including the tree of life and the tree of the knowledge of good and evil.

Then Genesis 2:19 mentions the creation of certain land animals.

At first glance this seems to be a contradiction because Genesis 1 has the animals and trees created prior to the creation of man however, both issues can be resolved by an understanding of the original language and the translation process.2 The Hebrew word for formed in both passages is yatsar. The New King James Version (quoted above) translates the verb in its perfect form.

However, this Hebrew word may also be translated in its pluperfect form. In this case, it would read that God “had formed” these creatures, as some other translations have it (e.g. ESV, NIV, etc.) For example, Genesis 2:19 in the NIV states:

This rendering eliminates any problem with the chronology because it refers to what God had already done earlier in Creation Week. This would mean that the plants ( Genesis 2:9 ) and the animals ( Genesis 2:19 ) had already been formed by God earlier in Creation Week. William Tyndale was the first to translate an English Bible directly from the original languages,3 and He also translated the verb in its pluperfect form.

(For more information on this topic, please see “Two Creation Accounts?”)

This seems to stem from a misunderstanding of the doctrine of biblical inerrancy, which is clearly spelled out in our statement of faith.

It is important to notice that inerrancy only applies to the original autographs (manuscripts). It does not extend to every copy and translation. As a result of this misunderstanding, people have sometimes come across an error in one of the translations and mistakenly assumed that the Bible must contain errors. In truth, the error was made by either a translation committee or a scribe responsible for copying the manuscript.

I would recommend a book entitled Nothing But the Truth by Brian Edwards. It explains the issues of translation and inerrancy in good detail, and would address your husband’s questions (also see “Why 66?”).

To automatically assume that this is a contradiction portrays the author of Genesis in a pretty dim light. Was he so inept that he couldn’t keep from contradicting himself in the first two chapters or were these chapters written with two different focuses? Rather than immediately assuming that the writer could not get his facts straight in the first two chapters, one should dig a little deeper (as you have done by asking us) to see if there is a better explanation.

While man and the devil often do attempt to muck up God’s Word, we can have confidence that God’s Word is true and accurate from the very beginning.

On counterexamples to the Hughes conjecture

In 1957 D.R. Hughes published the following problem in group theory. Let G be a group and p a prime. Define H p ( G ) to be the subgroup of G generated by all the elements of G which do not have order p. Is the following conjecture true: either H p ( G ) = 1 , H p ( G ) = G , or [ G : H p ( G ) ] = p ? After various classes of groups were shown to satisfy the conjecture, G.E. Wall and E.I. Khukhro described counterexamples for p = 5 , 7 and 11. Finite groups which do not satisfy the conjecture, anti-Hughes groups, have interesting properties. We give explicit constructions of a number of anti-Hughes groups via power-commutator presentations, including relatively small examples with orders 5 46 and 7 66 . It is expected that the conjecture is false for all primes larger than 3. We show that it is false for p = 13 , 17 and 19.

Transforming Your Self

Getting Started

1.4 Transforming Your Self - PDF Book

1.5 List of Videos corresponding to the Calls


2 Introduction To Transforming Your Self

Basic Understandings

3.1 Self-concept, Values, & Self-esteem

3.2 The Power of Self-Concept

3.3 Elements of a Healthy Self-concept

Strengthening The Self

4.1 Changing Structure - Part 1

4.2 Changing Structure - Part 2

4.3 Changing Structure - Part 3

4.4 Changing Structure - Part 4

4.5 Changing Structure Demonstration

5.4 Changing Time Demonstration

5.5 Changing Content - Part 1

5.6 Changing Content - Part 2

5.7 Changing Content - Part 3

5.8 Changing Content - Part 4

5.9 Changing Content Demonstration

Expanding The Self

6.1 Utilizing Mistakes - Part 1

6.2 Utilizing Mistakes - Part 2

6.3 Utilizing Mistakes - Part 3

6.4 Eliciting Counterexamples Demonstration

6.5 Integrating Counterexamples

6.6 Transforming Mistakes - Part 1

6.7 Transforming Mistakes - Part 2

6.8 Transforming Mistakes - Part 3

6.9 Transforming Counterexamples Demonstration

6.10 Transforming Mistakes - Part 4

6.11 Transforming Mistakes - Part 5

6.12 Grouping Counterexamples Demonstration

6.14 Eliciting Values Demonstration - Part 1

6.15 Eliciting Values Demonstration - Part 2

6.16 Building A New Quality Of Self-concept

6.17 Building A New Quality Of Self-concept Demonstration - Part 1

6.18 Building A New Quality Of Self-concept Demonstration - Part 2

Transforming The Self

7.1 Transforming An Uncertain Quality - Part 1

7.2 Transforming An Uncertain Quality - Part 2

7.3 Transforming An Uncertain Quality Demonstration - Part 1

7.4 Transforming An Uncertain Quality Demonstration - Part 2

7.5 Changing The Not Self - Part 1

7.6 Changing The Not Self - Part 2

7.7 Transforming An Unwanted Quality

7.8 Transforming An Unwanted Quality Exercise

7.9 Transforming An Unwanted Quality Demonstration

Boundaries Of The Self

8.1 Discovering & Changing Boundaries - External Boundaries

8.2 Discovering & Changing Boundaries - Changing External Boundaries

8.3 Discovering & Changing Boundaries - Internal Boundaries

8.4 Discovering & Changing Boundaries - Changing Internal Boundaries

8.5 Discovering & Changing Boundaries - Mind-Body Split

8.6 Connecting With Others - Connection & Disconnection

8.7 Connecting With Others - Mapping Across From Disconnection To Connection

Exponents and Their Properties

This lesson teaches students the rules for combining exponents when multiplying or dividing powers with the same base. Students will:

  • discover the rule for combining exponents when multiplying powers with the same base.
  • practice the rule for combining exponents when multiplying powers with the same base.
  • discover the rule for combining exponents when dividing powers with the same base.
  • practice the rule for combining exponents when dividing powers with the same base.

Essential Questions


  • Exponent: A numeral that tells how many times a number or variable is used as a factor. For example, in 2 7 , 2 is the base and 7 is the exponent this means 2 is multiplied by itself 7 times.


Prerequisite Skills


  • Multiplying Powers worksheet (M-8-4-1_Multiplying Powers and KEY.docx) for each student.
  • Dividing Powers worksheet (M-8-4-1_Dividing Powers and KEY.docx) for each student.
  • Powers in Expressions packet (14 sheets) (M-8-4-1_Powers in Expressions.docx) one copy or additional copies as needed.

Related Unit and Lesson Plans

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

  • Multiplying Powers worksheet (M-8-4-1_Multiplying Powers and KEY.docx) for each student.
  • Dividing Powers worksheet (M-8-4-1_Dividing Powers and KEY.docx) for each student.
  • Powers in Expressions packet (14 sheets) (M-8-4-1_Powers in Expressions.docx) one copy or additional copies as needed.

Formative Assessment

  • Students may be assessed on their completion of the Multiplying Powers worksheet.
  • Students&rsquo understanding may be evaluated based on their completion of the Dividing Powers worksheet .
  • Student performance and teacher observation during Activity 3 using Powers in Expressions (M-8-4-1_Powers in Expressions.docx) will aid in determining level of student mastery.

Suggested Instructional Supports

Scaffolding, Active Engagement, Modeling, Formative Assessment
W: Students will learn how to combine and simplify powers with the same base whether they involve multiplication or division. Students will practice these skills with several varied sample problems.
H: Students will be hooked by beginning the lesson with simple questions (evaluating 3 2 and 3 5 ) by allowing them to hypothesize and discover the rule for simplifying expressions when multiplying powers with the same base.
E: Students will examine several examples and explore simplifying them independently with the Multiplying Powers worksheet as well as the Dividing Powers worksheet, both completed after teacher-guided instruction. Students will further explore key ideas in the class-based Activity 3.
R: Students will revise and refine their understanding of multiplying and dividing powers with like bases, then simplifying results as they go over correct and incorrect assumptions and conclusions on the practice worksheets. Practice and evaluation of results will serve as a review of lesson concepts.
E: Students may be evaluated based on their performance on the practice worksheet problems. Correct and reteach as needed to ensure that students have a clear understanding of the procedures required to multiply and divide powers with like bases. Comparison of Activity 3 answers between classmates will also help students
self-evaluate their progress.
T: Use the Extension section to tailor the lesson to meet the needs of students. The Routine section provides ideas for reviewing lesson concepts throughout the year. The Small Group section includes activities for students who may benefit from additional practice. The Expansion section is designed for students who are prepared for a challenge that goes beyond the requirements of the standard.
O: The two individual concepts (multiplying and dividing powers with the same base) are introduced sequentially, and students are given a chance to practice both of these skills. Finally, after practicing these skills, students get a chance to combine the skills and simplify more complex expressions.

Instructional Procedures

Ask students to evaluate 3 2 . (9) Ask students to evaluate 3 5 . (243) Then, write on the board:

&ldquoWhat is the value of 3 2 times 3 5 ?&rdquo (2,187) &ldquoIs 2,187 a power of 3? Can it be written as 3 to some exponent?&rdquo Give students time to experiment or work it out. They should arrive at the conclusion that 2,187 = 3 7 . After they draw this conclusion, replace the question mark in the equation above with a 7. Then, write:

&ldquoWhat is the value of 2 3 times 2 7 ?&rdquo (1,024) &ldquoIs 1,024 a power of 2? Can it be written as 2 to some exponent?&rdquo Give students some time to think about it and work on it. Students will discover the general rule for adding exponents at different times, so ask students to not shout out their answers to allow other students to work it out and discover the pattern on their own. Once students determine that 2 3 × 2 7 = 2 10 , replace the question mark with a 10.

&ldquoIs there a relationship between the exponents on the left-hand side of the equal sign and the right-hand side of the equal sign?&rdquo (One is a sum of the other two.)

Ask students a variety of questions such as: &ldquo4 5 times 4 8 equals?&rdquo (4 13 ). It should only take a few questions before every student has mastered the rule for exponents.

&ldquoIt&rsquos important when you use this rule that the base, the number we&rsquore raising to the exponent, is the same.&rdquo Illustrate this with a couple of counterexamples such as:

Explain that this is a case in which the rule does not apply because 5 and 2 are not the same number.

Ask students how, for example, they could find the value of 4 3 . They should note that they can find it by representing the result as 4 × 4 × 4. Likewise, ask them how they could find the value of 4 5 . (4 × 4 × 4 × 4 × 4) &ldquoSo, we can write 4 3 × 4 5 as (4 × 4 × 4) × (4 × 4 × 4 × 4 × 4), which is the same as 4 8 .&rdquo

Have students complete the Multiplying Powers worksheet (M-8-4-1_Multiplying Powers and KEY.docx).

&ldquoWhen we multiplied powers with the same base, we found that our rule was that we should add the exponents together. Does anyone have any guesses as to what we should do to the exponents when we divide powers with the same base?&rdquo Students will most likely guess, at some point, to subtract them from one another.

&ldquoLet&rsquos take a look. Let&rsquos try 6 5 ÷ 6 2 . What is 6 5 ?&rdquo (7,776) &ldquoAnd what is 6 2 ?&rdquo (36) &ldquoAnd what is 7,776 ÷ 36?&rdquo (216) &ldquoSo, according to our hypothesis, we should subtract the exponents and we should end up with 6 3 . Is 6 3 equal to 216?&rdquo (Yes.) &ldquoSo it looks like subtraction is a good rule. When dividing powers with the same base, subtract the exponents.&rdquo

Write on the board. &ldquoFor this problem, in which order should we subtract the exponents? Should we do 9 &ndash 3 or 3 &ndash 9?&rdquo Guide students toward understanding that we subtract the exponent of the divisor from the exponent of the dividend, or the exponent of the denominator from the exponent of the numerator. If students say that it must be 9 &ndash 3 because otherwise you get a negative number and/or you can&rsquot have negative exponents, make sure to point out that you can have negative exponents so you can&rsquot use that as a general rule.

&ldquoNow, let&rsquos look at how we can represent a problem like . We can write it as: .

&ldquoThen, we can cancel some 7s from the numerator and denominator. How many 7s can we cancel?&rdquo (three) &ldquoSo, we&rsquoll cancel those out.&rdquo Cross out three 7s from both the numerator and denominator. &ldquoWhat&rsquos left in the numerator?&rdquo (7 × 7 × 7 × 7 × 7 × 7 or 7 6 ). &ldquoWhat&rsquos left in the denominator?&rdquo Here, make students recognize that even though we&rsquove canceled everything out, the denominator is still 1.

&ldquoSo we have 7 6 divided by 1, but that&rsquos just 7 6 . So it looks like the algebra supports our rule.&rdquo

Have students complete the Dividing Powers worksheet (M-8-4-1_Dividing Powers and KEY.docx).

&ldquoNow, we&rsquore going to combine it all together and look at expressions that multiply and divide powers with the same base.&rdquo

Give 14 students each a single page from the Powers in Expressions packet (M-8-4-1_Powers in Expressions.docx). Note: Use fewer pages or create additional pages as necessary there&rsquos really no limit on the number of pages to use as long as there are sufficient × and ÷ pages.

Separate students based on whether they have a page with a number (a power of 7) or an operation (either × or ÷). Have each group (students with numbers and students with operations) form a single-file line. Then, have the first student with a number come up and show the number, followed by the first student with an operation, followed by another student with a number etc. They should stand next to one another to form an expression like 7 8 × 7 &minus4 . Have students respond with the simplified value of the expression (expressed as a power of 7, in this case, 7 4 ).

This activity can be continued with simple expressions (those involving a single operation) but can be expanded by continuing to have students from the front of each line come up and continue the expression, as long as the expression alternates between number and operation. A sample longer expression might be:

Remind students that multiplication and division are performed from left to right. The above expression, for example, can be simplified:

Depending on the class, students can interact with those making the expressions in several different ways:

  • Students can write down their answers for assessment later (in which case you should record the answers as well).
  • Students can simply respond verbally out of turn.
  • Students can respond verbally by raising their hand.
  • Teams can be formed, and students from each team must raise their hands and respond verbally.

After the activity has gone on awhile, students with numbers and operations can exchange places with those that don&rsquot have numbers and operations. (Therefore, having about half the class with numbers and operations might be suitable so every student can have a chance at both parts of the activity.)

Use the following strategies to tailor the lesson to meet the needs of students throughout the year.

4. Prominent Faults of an Old Theory

4.1 Introduction

Below are eight biblical texts, which, according to IWS, must report the eventualities in the order they occurred. Conversely, according to this theory, the eventualities occurred in the order they are reported. But, as I demonstrate, read that way, these texts become non-sequiturs. It creates farcical and irrational scenarios inside the narratives. But if we do not assume that the order of eventualities and verbs are always the same, the texts make perfect sense.

4.2 Biblical examples of exceptions to IWS

[Please note: I have translated all the verses I discuss in this article. Also, the wayiqtols are boldface in the Hebrew text and translation, and are marked with superscripted boldface letters. Where I refer to these letters in my analysis, it is usually to the whole verb phrase, not just its verb. For example, in Mike x ran through the park, (x) often refers to “ran through the park,” not just “ran.” In addition, if qatals18 are pertinent to the analysis, they are underlined. The layout of the examples is text, followed by translation, then analysis.]

4.2.1 The crossing of the Jordan (Joshua 3:14–17 4:10–12, 18)

(In the following I have only provided the portions of text necessary to discern the order of the eventualities portrayed.)

וַיְהִ֗י בִּנְסֹ֤עַ הָעָם֙ מֵאָ֣הֳלֵיהֶ֔ם לַעֲבֹ֖ר אֶת־הַיַּרְדֵּ֑ן וְהַכֹּהֲנִ֗ים נֹֽשְׂאֵ֛י הָאָר֥וֹן הַבְּרִ֖ית לִפְנֵ֥י הָעָֽם׃ 15 וּכְב֞וֹא נֹשְׂאֵ֤י הָֽאָרוֹן֙ עַד־הַיַּרְדֵּ֔ן וְרַגְלֵ֤י הַכֹּֽהֲנִים֙ נֹשְׂאֵ֣י הָֽאָר֔וֹן נִטְבְּל֖וּ בִּקְצֵ֣ה הַמָּ֑יִם וְהַיַּרְדֵּ֗ן מָלֵא֙ עַל־כָּל־גְּדוֹתָ֔יו כֹּ֖ל יְמֵ֥י קָצִֽיר׃ 16 וַיַּעַמְד֡וּ הַמַּיִם֩ הַיֹּרְדִ֙ים מִלְמַ֜עְלָה קָ֣מוּ נֵד־אֶחָ֗ד הַרְחֵ֙ק מְאֹ֜ד . . .

17 וַיַּעַמְד֣וּ הַכֹּהֲנִ֡ים נֹ֠שְׂאֵי הָאָר֙וֹן בְּרִית־יְהוָ֜ה בֶּחָֽרָבָ֛ה בְּת֥וֹךְ הַיַּרְדֵּ֖ן הָכֵ֑ן וְכָל־יִשְׂרָאֵ֗ל עֹֽבְרִים֙ בֶּחָ֣רָבָ֔ה עַ֤ד אֲשֶׁר־תַּ֙מּוּ֙ כָּל־הַגּ֔וֹי לַעֲבֹ֖ר אֶת־הַיַּרְדֵּֽן׃ . . . .

4:10–12, 18

10 וְהַכֹּהֲנִ֞ים נֹשְׂאֵ֣י הָאָר֗וֹן עֹמְדִים֘ בְּת֣וֹךְ הַיַּרְדֵּן֒ עַ֣ד תֹּ֣ם כָּֽל־הַ֠דָּבָר אֲשֶׁר־צִוָּ֙ה יְהוָ֤ה אֶת־יְהוֹשֻׁ֙עַ֙ לְדַבֵּ֣ר אֶל־הָעָ֔ם כְּכֹ֛ל אֲשֶׁר־צִוָּ֥ה מֹשֶׁ֖ה אֶת־יְהוֹשֻׁ֑עַ וַיְמַהֲר֥וּ הָעָ֖ם וַֽיַּעֲבֹֽרוּ ׃ 11 וַיְהִ֛י כַּֽאֲשֶׁר־תַּ֥ם כָּל־הָעָ֖ם לַֽעֲב֑וֹר וַיַּעֲבֹ֧ר אֲרוֹן־יְהוָ֛ה וְהַכֹּהֲנִ֖ים לִפְנֵ֥י הָעָֽם׃ 12 וַ֠יַּעַבְרוּ בְּנֵי־רְאוּבֵ֙ן וּבְנֵי־גָ֜ד וַחֲצִ֙י שֵׁ֤בֶט הַֽמְנַשֶּׁה֙ חֲמֻשִׁ֔ים לִפְנֵ֖י בְּנֵ֣י יִשְׂרָאֵ֑ל כַּאֲשֶׁ֛ר דִּבֶּ֥ר אֲלֵיהֶ֖ם מֹשֶֽׁה׃ . . . .

18 וַ֠יְהִי ) בַּעֲלוֹת] (כַּעֲל֙וֹת[ הַכֹּהֲנִ֜ים נֹשְׂאֵ֙י אֲר֤וֹן בְּרִית־יְהוָה֙ מִתּ֣וֹךְ הַיַּרְדֵּ֔ן נִתְּק֗וּ כַּפּוֹת֙ רַגְלֵ֣י הַכֹּהֲנִ֔ים אֶ֖ל הֶחָרָבָ֑ה וַיָּשֻׁ֤בוּ מֵֽי־הַיַּרְדֵּן֙ לִמְקוֹמָ֔ם וַיֵּלְכ֥וּ כִתְמוֹל־שִׁלְשׁ֖וֹם עַל־כָּל־גְּדוֹתָֽיו׃

And it a was when the people pulled up [i.e. pulled up stakes] from their tents to cross the Jordan, and the priests carrying the ark of the covenant being in front of/in the presence of/before the people, |15| when those carrying the ark came as far as the Jordan, the feet of the priests carrying the ark dipped into the edge of the water. (Now the Jordan was full—over all its banks—all the days of harvest.) |16| The water which came down from upstream b stopped: it rose up in a heap a great distance away [[cities named, specifying how far]] (The people crossed opposite Jericho.)|17|The priests carrying the ark of the covenant of YHWH c stood on dry ground in the middle of the Jordan while all Israel were crossing on dry ground, until all the nation had finished crossing the Jordan.

|4:10| The priests carrying the ark were standing in the middle of the Jordan until every word which YHWH had commanded Joshua to speak to the people according to what Moses commanded Joshua was complete. The people d hurriedly e crossed. |11| And it f was, as soon as all the people had finished crossing, the ark of YHWH and the priests g crossed in front of/in the presence of the people. |12| And the sons of Reuben, and the sons of Gad, and half of the tribe of Manasseh h crossed over armed in front of/in the presence of the Sons of Israel, just as Moses had spoken to them. About forty thousand military units crossed in the presence of YHWH for [i.e. ready for] battle, to the Aravah of Jericho. . . .

|18| And it i was, when the priests carrying the covenant of YHWH came up from the middle of the Jordan, the soles of the feet of the priests withdrew onto the dry ground, the water of the Jordan j returned to its place, and it k went [i.e. flowed] as formerly [lit. “yesterday and three days before’]—over all its banks. [[date stamp]].

The passage above is in many ways a hinge text.19 Our concern, however, is with the order of the crossings of the Jordan reported here. According to IWS, the military vanguard of Reuben, Gad, and the half tribe of Manasseh crossed (h) after the priests carrying the Ark crossed (g), because the crossing of the three tribes is reported after the crossing of the priests. The order of eventualities would then be as follows. While the priests were standing in the middle of the dry riverbed, all the people crossed (e).Then the priests carrying the ark crossed (g), and finally the three tribes crossed (h). But do the details of the text (and world knowledge) allow for this order? First of all, what does the text mean by crossing the river? The text reports that at flood stage, the river is wider than its banks, so “crossing” cannot mean to go from bank to bank. It must mean going from dry ground on one side to dry ground on the other side (and here, miraculously, dry ground in between). Thus, when the text reports that the priests crossed the river, it entails that they came up from where the river would normally be at flood stage. Second, the text reports that as soon as the priests came up from the river, the river returned to flood stage (4:18). Consequently, anyone who crossed after the priests would have to swim across the swollen torrent. According to IWS, then, the three tribes would have been swimming—in full military gear—since it makes them cross after the priests! Hence, IWS has reduced the text to absurdity. Moreover, it has caused it to contradict itself, because “all” (mentioned twice in 3:17 and again in 4:11) would include the vanguard in question, and all were supposed to have crossed on dry ground! So, clearly, the armed men of Reuben, Gad, and half of the tribe of Manasseh did not cross after the priests carrying the Ark. Hence, in this text the order of the eventualities cannot be that of the wayyiqtols representing them. Since the text must make sense, IWS is obviously invalid here. Thus, plainly this text is non-iconic.

4.2.2 Jezebel writes letters (1 Kings 21:8–9)

וַתִּכְתֹּ֤ב סְפָרִים֙ בְּשֵׁ֣ם אַחְאָ֔ב וַתַּחְתֹּ֖ם בְּחֹתָמ֑וֹ וַתִּשְׁלַ֣ח
( הַסְפָרִים) [סְפָרִ֗ים] אֶל־הַזְקֵנִ֤ים וְאֶל־הַֽחֹרִים֙ אֲשֶׁ֣ר בְּעִיר֔וֹ הַיֹּשְׁבִ֖ים אֶת־נָבֽוֹת׃

9 וַתִּכְתֹּ֥ב בַּסְּפָרִ֖ים לֵאמֹ֑ר קִֽרְאוּ־צ֔וֹם וְהוֹשִׁ֥יבוּ אֶת־נָב֖וֹת בְּרֹ֥אשׁ הָעָֽם׃

She [Jezebel] a wrote letters in the name of Ahab, b sealed them with his seal, and c sent the letters to the elders and to the nobles who were in his [Naboth’s] city, who lived with Naboth.

|9| She d wrote in the letters, “call for a fast and seat Naboth at the head of the people.”

The chronological order of the eventualities represented in this brief, chilling text is easily seen. With the first three wayyiqtols, “wrote” (a), “sealed” (b), and “sent” (c), the order of the eventualities and the verbs which recount them are identical. Jezebel would not have sent unsealed scrolls without the king’s seal her nefarious scheme would not have worked. Nor would she have sealed blank scrolls. But—consider closely the fourth wayyiqtol. If IWS were applied, verb (d), “wrote,” would refer to a subsequent writing of the letters after they were sealed (b) and sent (c)—as if Jezebel had run after the messengers, retrieved the letters she had written and sealed, broken open the seals, and hastily scribbled in them again—preposterous! But, if we understand that this is a clear example of flashback, then the text may be read as common sense dictates: that “wrote” (d) refers to the one-and-only writing of the letters, which chronologically preceded their sealing (b) andsending (c). Thus, the first mention of her writing is repeated in the second mention, after which we are told the content of the letters—a coherent reading. Such a reading makes perfect sense of this solemn record of one of Jezebel’s crimes. Thus, the order of the eventualities represented by the verbs is a/d b c. Hence IWS cannot be applied to this text either. It is non-iconic.

4.2.3 Abraham journeys to the land of Canaan with his household (Genesis 12:4–5)

וַיֵּ֣לֶךְ אַבְרָ֗ם כַּאֲשֶׁ֙ר דִּבֶּ֤ר אֵלָיו֙ יְהוָ֔ה וַיֵּ֥לֶךְ אִתּ֖וֹ ל֑וֹט וְאַבְרָ֗ם בֶּן־חָמֵ֤שׁ שָׁנִים֙ וְשִׁבְעִ֣ים שָׁנָ֔ה בְּצֵאת֖וֹ מֵחָרָֽן׃ 5 וַיִּקַּ֣ח אַבְרָם֩ אֶת־שָׂרַ֙י אִשְׁתּ֜וֹ וְאֶת־ל֣וֹט בֶּן־אָחִ֗יו וְאֶת־כָּל־רְכוּשָׁם֙ אֲשֶׁ֣ר רָכָ֔שׁוּ וְאֶת־הַנֶּ֖פֶשׁ אֲשֶׁר־עָשׂ֣וּ בְחָרָ֑ן וַיֵּצְא֗וּ לָלֶ֙כֶת֙ אַ֣רְצָה כְּנַ֔עַן וַיָּבֹ֖אוּ אַ֥רְצָה כְּנָֽעַן׃

Abram a went just as YHWH had spoken to him. And Lot b went with him. (Now Abram was seventy-five years old when he went out of Haran.) Abram c took Sarai, his wife, Lot, the son of his brother, and all their possessions, which they had acquired, and every person, whom they had acquired in Haran. And they d went out to go to the land of Canaan. And they e entered the land of Canaan.

Clearly (b) “And Lot went with him,” is reprised in (c) “And Abram took Sarai, his wife, and Lot, the son of his brother” [emphasis mine]. We cannot explore the reason for this repetition here, but the eventuality of Lot having been taken by Abram is plainly the very same eventuality as Lot having gone with him. There is no temporal progression here. Moreover, the eventuality is further examined in the text in the fourth main clause: (d) is plural, because Abram did not go out of his country by himself he took his whole household (including Lot). But it is still looking at the same eventuality. Again, therefore, time does not advance. Wayyiqtols (b), (c),and (d) all refer to the same eventuality IWS would erroneously have these three verbs refer to three sequential eventualities. Hence, this text cannot be iconic, either.

4.2.4 Assessment of Esau’s actions (Genesis 25:34)20

וְיַעֲקֹ֞ב נָתַ֣ן לְעֵשָׂ֗ו לֶ֚חֶם וּנְזִ֣יד עֲדָשִׁ֔ים וַיֹּ֣אכַל וַיֵּ֔שְׁתְּ וַיָּ֖קָם וַיֵּלַ֑ךְ וַיִּ֥בֶז עֵשָׂ֖ו אֶת־הַבְּכֹרָֽה׃

As for Jacob, he gave Esau bread and lentil stew. And he [Esau] a ate and b drank, c arose and d went. So, Esau e despised his birthright.

Esau probably did not wait until he had eaten all the stew before he had anything to drink. Yet IWS would have it so. Indeed, rather, most likely he alternated between eating and drinking as we do, given that the two actions represented by (a) and (b) are compatible. On the other hand, (c) and (d) are most likely not compatible with the first two, and thus must occur after them in time. (e) is altogether different from the rest. It is a summary assessment of what Esau has done. Thus, time does not advance. IWS wrongly insists that it does. Again, the text is non-iconic.21

4.2.5 Moses’ instructions to the spies (Numbers 13:17ff)

וַיִּשְׁלַ֤ח אֹתָם֙ מֹשֶׁ֔ה לָת֖וּר אֶת־אֶ֣רֶץ כְּנָ֑עַן וַיֹּ֣אמֶר אֲלֵהֶ֗ם עֲל֥וּ זֶה֙ בַּנֶּ֔גֶב וַעֲלִיתֶ֖ם אֶת־הָהָֽר׃ 18 וּרְאִיתֶ֥ם אֶת־הָאָ֖רֶץ מַה־הִ֑וא וְאֶת־הָעָם֙ הַיֹּשֵׁ֣ב עָלֶ֔יהָ הֶחָזָ֥ק הוּא֙ הֲרָפֶ֔ה הַמְעַ֥ט ה֖וּא אִם־רָֽב׃ 19 וּמָ֣ה הָאָ֗רֶץ אֲשֶׁר־הוּא֙ יֹשֵׁ֣ב בָּ֔הּ הֲטוֹבָ֥ה הִ֖וא אִם־רָעָ֑ה וּמָ֣ה הֶֽעָרִ֗ים אֲשֶׁר־הוּא֙ יוֹשֵׁ֣ב בָּהֵ֔נָּה הַבְּמַֽחֲנִ֖ים אִ֥ם בְּמִבְצָרִֽים׃ 20 וּמָ֣ה הָ֠אָרֶץ הַשְּׁמֵנָ֙ה הִ֜וא אִם־רָזָ֗ה הֲיֵֽשׁ־בָּ֥הּ עֵץ֙ אִם־אַ֔יִן וְהִ֙תְחַזַּקְתֶּ֔ם וּלְקַחְתֶּ֖ם מִפְּרִ֣י הָאָ֑רֶץ וְהַ֙יָּמִ֔ים יְמֵ֖י בִּכּוּרֵ֥י עֲנָבִֽים׃

Moses sent them to spy out the land of Canaan. And he said to them, “Go up here into the Negev, then go up into the hill country. See the land, what it is, and what the people who dwell in it are like. Are they strong or are they weak? Whether they are few or many. And what is the land in which they dwell: is it good or bad? And what are the cities like in which they dwell? Are they in camps or in fortifications? And what of the soil: is it rich or poor? Are there any trees in it or not? Strengthen yourselves and take some of the fruit of the land.” (Now the days were the days of the first fruits of the grapes).

Clearly, Moses gave the spies this long charge concerning their mission as he sent them out, or before he sent them out, not afterwards. They would not have been there after he sent them. If sending is a process, the text elaborates on this process. Part of the process is the charge. However, if it is an instantaneous event, it must follow the charge. In addition, for the former way of understanding, although sent and said are compatible, and thus, not constrained to happen at different times, the linearity of texts requires this verbal sequence for the latter way, the verbs are in reverse temporal order. Either understanding yields non-iconicity, and therefore, IWS is yet again found to be in error.

4.2.6 Joshua orders an ambush to be set against Ai (Joshua 8:3–4)

יָּ֧קָם יְהוֹשֻׁ֛עַ וְכָל־עַ֥ם הַמִּלְחָמָ֖ה לַעֲל֣וֹת הָעָ֑י וַיִּבְחַ֣ר יְ֠הוֹשֻׁעַ שְׁלֹשִׁ֙ים אֶ֤לֶף אִישׁ֙ גִּבּוֹרֵ֣י הַחַ֔יִל וַיִּשְׁלָחֵ֖ם לָֽיְלָה׃ 4 וַיְצַ֙ו אֹתָ֜ם לֵאמֹ֗ר רְ֠אוּ אַתֶּ֞ם אֹרְבִ֤ים לָעִיר֙ מֵאַחֲרֵ֣י הָעִ֔יר אַל־תַּרְחִ֥יקוּ מִן־הָעִ֖יר מְאֹ֑ד וִהְיִיתֶ֥ם כֻּלְּכֶ֖ם נְכֹנִֽים׃

Joshua and all the men of war a arose to go up to Ai. Joshua b chose thirty thousand men, the best warriors, and c sent them at night. |4| He d commanded them, “Look, you are going to set an ambush for the city. Do not be very far from the city. And all of you be ready.”

The pertinent issue for us in these verses is the temporal sequence—or lack thereof—between (c) and (d). Here Joshua is deploying men for an ambush, which, as world knowledge instructs us, requires utmost secrecy for it to succeed. The idea that Joshua shouted the orders to the ambushers after they left to position themselves—which would have been the case if IWS were true—and thus compromise the mission, is ludicrous. Thus, this is another obvious exception to IWS. Clearly this text cannot be iconic.

4.2.7 The Philistines gather for battle (1 Samuel 17:1)

וַיַּאַסְפ֙וּ פְלִשְׁתִּ֤ים אֶת־מַֽחֲנֵיהֶם֙ לַמִּלְחָמָ֔ה וַיֵּאָ֣סְפ֔וּ שֹׂכֹ֖ה אֲשֶׁ֣ר לִיהוּדָ֑ה וַֽיַּחֲנ֛וּ בֵּין־שׂוֹכֹ֥ה וּבֵין־עֲזֵקָ֖ה בְּאֶ֥פֶס דַּמִּֽים׃

The Philistines a gathered their camp for battle. They b amassed at Sokoh, which belongs to Judah, and c camped between Sokoh and “Azekah in Ephes Dammim.

It is clear from both the immediate and extended context what this text describes: the staging of the Philistines in the Valley of Elah to fight against the forces of Saul. (a) gives us a general Introductory Encapsulation: the Philistines gathered together their forces to engage in battle. (b) and (c) give us the particulars of the location of their camp, with (c) further specifying the place beyond what (b) does. The result is general, followed by specific, followed by even more specific. The elaboration is spatial: it concerns the circumstances of the event it does not break down the eventuality into sub-events. In this case (b) obviously occurred within the same time interval in which (a) happened. And, (c) happened within this interval as well. Consequently, there is no temporal progression represented by the textual sequence—yet another example of a non-iconic text. Again IWS does not apply.

4.2.8 The account of Uriah’s death (2 Samuel 11:17)

וַיֵּ֙צְא֜וּ אַנְשֵׁ֤י הָעִיר֙ וַיִּלָּחֲמ֣וּ אֶת־יוֹאָ֔ב וַיִּפֹּ֥ל מִן־הָעָ֖ם מֵעַבְדֵ֣י דָוִ֑ד וַיָּ֕מָת גַּ֖ם אוּרִיָּ֥ה הַחִתִּֽי׃

The men of the city a came out and b fought with Joab. Some of the people from the servants of David c fell. Also, Uriah the Hittite d died.

The text above, although short, is extremely poignant and deserves more than the brief attention I can give it here.22 But to the matter at hand. (a) and (b) give us the circumstances that resulted in the army of Israel suffering casualties. This brings us to (c) and (d). (c) recounts the casualties sustained in the battle: “some of the servants of David.” (d) zooms in on one of the loyal servants who gave their lives fighting for their king, namely, Uriah, in a classic movement from general to specific, with the curt (only four Hebrew words) grim report: “Also, Uriah the Hittite died.” As to the temporal profile of this text, Uriah’s death is part of the death of the rest, and occurred therefore within the same time span as theirs. Hence, there is no temporal progression between (c) and (d)—a final parade example of an exception to IWS. It too is non-iconic.

4.3 Discussion

In the foregoing examples, I engaged in “temporal reasoning’,23 a methodology for carefully analyzing the temporal relationships between the eventualities represented by the verbs in any discourse (written or spoken). Applying this technique to Scripture invariably leads to a coherent reading. And may I be so bold to say, it yields a better reading of the text than mindlessly assuming IWS, which in the eight examples above resulted in nonsensical readings. And these eight are merely a small subset of a great many more.24

What means these clear-cut exceptions to IWS, then? Is it not the same as the significance of obvious exceptions to any other theory?

According to the scientific method, for a hypothesis to reach the status of a theory, it should be repeatedly tested. And according to Karl Popper’s refinement of the scientific method, “Every genuine test of a theory is an attempt to falsify it, or to refute it”(Popper 1963, pp. 33–39).25 If a theory fails such tests, it should be rejected. Considering the multitude of exceptions to IWS, a strict adherence to the scientific method dictates that it should never have attained the status of a theory.

Notwithstanding, what should never be not infrequently obtains regardless. It is possible, and not uncommon, for a hypothesis to reach the status of a theory solely because of the fame of those who first proposed it, without it being properly tested. One of the most famous examples is the geocentric model, which stood almost unchallenged 1500 years, because Aristotle and Ptolemy had proposed it.

I believe that this is what happened with IWS. This understanding of how the wayyiqtol functions—as a verb which always indicates a sequence—was adopted because of the stature of and respect for Ewald and Driver. Also, because of their towering reputation, very few have challenged their ideas.26

In addition, although even one exception should topple a theory,27 theories tend to become entrenched. But whenever the exceptions begin to multiply and explanations for them become increasingly wild and unlikely, such theories are eventually—even if reluctantly—rejected.28 Famously, the Aristotle-Ptolemy model was unable to fully explain the motions of some of the planets (particularly Mars). Yet it persisted. But, as observational astronomy refined its art, the deficiencies of the epicycles’ model became more and more apparent and impossible to ignore. And eventually, Copernicus, then Galileo, then Kepler and Newton, were able to overthrow it and replace it with a new science.

And so should it be in this case: the many exceptions to IWS must cause us to conclude that this particular implication of perspectival aspect is wholly invalid, and perhaps calls into question the wisdom of applying perspectival aspect to the BH verbal system altogether. To recapitulate: the eventuality sequence does not always match the wayyiqtol sequence which represents it. This requires that how the temporal sequence is discerned be rethought. To that end, I propose a new model.

Because Driver’s claim that the order of Creation contradicts the order in Genesis 1 is solely based on his false understanding of the wayyiqtol, there exists no contradiction—which obviates resorting to allegory or some halting explanation “against idiom,”29 in order to face the charge of contradiction.30 Thus, this branch of the two-pronged attack has been successfully repulsed.

Now to face the second prong of the assault by the enemy forces: that the text is incoherent. To do so, I apply temporal reasoning to Genesis 2–3. To understand the contours of the technique requires a rudimentary grasp of how texts, eventualities, and time interrelate. A comprehensive model of how they interact (fully developed in Grappling with the Chronology of the Genesis Flood (2014)) is outlined here to demonstrate how to “navigate the flow of time in biblical narrative.”31

4. Discontinuity of Differential for sc-Diffeomorphisms

The purpose of this section is to show that sc-diffeomorphisms—in contrast to the basic germs in §3—can have a discontinuous differential, viewed as a map to the space of bounded linear operators as in Proposition 3.2.

Theorem 4.1. There exists a sc-smooth diffeomorphism s : F → F on a sc-Banach space F = ( F i ) i ∈ N 0 , whose differential d s : F i + 1 → L ( F i , F i ) is discontinuous for any scale i ∈ N 0 .

The construction of this map s : F → F is also an example of a sc-Fredholm map with discontinuous differential, since s is equivalent, via the sc-diffeomorphism s, to the identity map i d F , which is a basic germ (as it satisfies Definition 3.1 with W = F , k = N = 0 , and B ≡ 0 ).

Remark 4.2. A sc-diffeomorphism is defined (ref. 8, p. 12) to be a homeomorphism f : U → V between open subsets U ⊂ E , V ⊂ F of sc-Banach spaces, such that both f and f − 1 are sc-smooth. It then follows that the differential d u f ≔ d f ( u ) : E k → F k is an isomorphism on scale k ∈ N 0 at base points u ∈ U ∩ E k + 1 . In particular d f ( u ) : E → F is a sc-isomorphism for u ∈ U ∩ E ∞ .

Indeed, the chain rule (ref. 3, theorem 1.1) applied to the identities g ○ f = i d U and f ○ g = i d V for g ≔ f − 1 yields d f ( u ) g ○ d u f = i d E k for u ∈ E k + 1 and d u f ○ d f ( u ) g = i d E k for f ( u ) ∈ F k + 1 . Here f ( u ) ∈ F k + 1 follows by sc-continuity of f from u ∈ E k + 1 . □

To construct the example in Theorem 4.1, we work with an abstract model for the sc-Banach space E = ( H 3 i ( S 1 ) ) i ∈ N 0 . For that purpose we start with an infinite dimensional vector space E ≔ ∑ n = 1 N x n e n | N ∈ N , x 1 , … , x N ∈ R generated by a sequence of formal variables ( e n ) n ∈ N . We obtain norms ‖ x ‖ i ≔ ⟨ x , x ⟩ i on E by defining inner products with ⟨ e n , e m ⟩ i ≔ ( n m ) 3 i δ n , m . Then each completion of E in a norm ‖ ⋅ ‖ i defines a Banach space E i ≔ E ¯ ‖ ⋅ ‖ i , and the embeddings E i + 1 ⊂ E i are compact so that E ≔ ( E i ) i ∈ N 0 is a sc-Banach space. (This follows from the compact Sobolev embeddings H 3 i ( S 1 ) ↪ H 3 j ( S 1 ) for i > j .) Here an explicit sc-isomorphism E 0 ≃ H 0 ( S 1 ) mapping E i to H 3 i ( S 1 ) can be obtained by taking real and imaginary parts of the complex orthogonal basis ( e − 1 k θ ) k ∈ N 0 of L 2 ( S 1 ) = H 0 ( S 1 ) and normalizing these real valued functions to obtain a collection of smooth functions ( e n ) n ∈ N ⊂ C ∞ ( S 1 ) = ⋂ i ∈ N 0 H 3 i ( S 1 ) that have inner products ⟨ e n , e m ⟩ H 3 i ≔ n 6 i δ n , m . Thus they form an orthonormal basis of H 0 ( S 1 ) and the ‖ ⋅ ‖ i closure of the finite span E ↪ H 0 ( S 1 ) exactly corresponds to the subspace H 3 i ( S 1 ) ⊂ H 0 ( S 1 ) .)

Proof of Theorem 4.1. We construct a map s : F → F on F ≔ R × E by s : ( t , x ) ↦ ( t , s t ( x ) ) , s t ∑ n = 0 ∞ x n e n ≔ ∑ n = 0 ∞ f n ( t ) x n e n for a sequence of smooth functions f n : R → [ 1 2 , 1 ] , t ↦ f 1 2 ( n ( n + 1 ) t + 1 − n ) obtained by reparameterizing a smooth function f : R → [ 1 2 , 1 ] chosen with f ( − ∞ , 1 2 ] ≡ 1 , f [ 1 , ∞ ) ≡ 1 2 , and s u p p f ′ ⊂ ( 1 2 , 1 ) . First note that by construction we have f n ( − ∞ , 1 n + 1 ] ≡ 1 and f n [ 1 n , ∞ ) ≡ 1 2 . So, the family of linear maps s t restricts to s t = i d E for t ≤ 0 and s t E N = 1 2 i d E N on E N ≔ s p a n < e n | n ≥ N >for t ≥ 1 N . Thus, d s : R × E i + 1 → L ( R × E i , R × E i ) cannot be continuous for any i ∈ N 0 since d s ( t , x ) < 0 >× E i : ( 0 , X ) ↦ ( 0 , s t ( X ) ) is discontinuous at t = 0 in L ( E i , E i ) by ‖ s 1 / n − s 0 ‖ L ( E i , E i ) ≥ ‖ s 1 / n ( e n ) − s 0 ( e n ) ‖ i ‖ e n ‖ i − 1 = ‖ 1 2 e n − e n ‖ i ‖ e n ‖ i − 1 = 1 2 . On the other hand, since f n ( t ) ≠ 0 , the map s has an evident inverse given by s − 1 : t , ∑ n = 0 ∞ y n e n ↦ t , ∑ n = 0 ∞ y n f n ( t ) e n . To prove the theorem it remains to show that s and s − 1 are well defined and sc-smooth. For that purpose note that s − 1 is of the same form as s, with the function f replaced by 1 f . So it suffices to consider the map s, as long as we only use common properties of the functions f n in both cases. Since s u p p f 1 ′ ⊂ ( 1 2 , 1 ) and the derivatives of f 1 = f and f 1 = f − 1 are uniformly bounded, we have for all n ∈ N s u p p f n ( k ) ⊂ 1 n + 1 , 1 n ∀ k ≥ 1 , f n ( k ) ∞ = n ( n + 1 ) 2 k f 1 ( k ) ∞ ≤ n 2 k C k ∀ k ≥ 0 . [8] Next, we write s ( t , x ) = ( t , ρ 0 ( t , x ) ) and—to prove that ρ 0 : R × E → E and thus s is well defined and sc-smooth—we more generally study the maps arising from the derivatives f n ( k ) = d k d t k f n on shifted sc-spaces E k ≔ ( E k + i ) i ∈ N 0 for k ∈ N 0 , ρ k : R × E k → E , t , ∑ n = 0 ∞ x n e n ↦ ∑ n = 0 ∞ f n ( k ) ( t ) x n e n . We can rewrite this as ρ k ( t , ⋅ ) = ∑ n = 0 ∞ f n ( k ) ( t ) p n in terms of the orthogonal projections to R e n ⊂ E 0 , p n : E → E , x ↦ ⟨ x , e n ⟩ 0 e n . Then for k ≥ 1 the supports of f n ( k ) are disjoint, so we have ρ k ( t , ⋅ ) = f N t ( k ) ( t ) p N t with N t ≔ ⌊ t − 1 ⌋ for t > 0 and ρ k ( t , ⋅ ) ≡ 0 for t ≤ 0 as well as in a small neighborhood t ∼ 1 n for each n ∈ N . Note also for future purposes the estimates for x ∈ E i + k and k ≥ 0 ‖ p n ( x ) ‖ i = ⟨ x , e n ⟩ 0 ‖ e n ‖ i ‖ e n ‖ i + k ‖ e n ‖ i + k = n − 3 k ‖ ⟨ x , e n ⟩ 0 e n ‖ i + k = n − 3 k ‖ p n ( x ) ‖ i + k , [9] ∑ n = N ∞ p n ( x ) i = ∑ n = N ∞ ‖ p n ( x ) ‖ i 2 1 / 2 = ∑ n = N ∞ n − 6 k ‖ p n ( x ) ‖ i + k 2 1 / 2 ≤ N − 3 k ∑ n = 0 ∞ ‖ p n ( x ) ‖ i + k 2 1 / 2 = N − 3 k ∑ n = 0 ∞ p n ( x ) i + k = N − 3 k ‖ x ‖ i + k . [10] We will show for all k ∈ N 0 that ρ k : R × E k → E is well defined, s c 0 , and sc-differentiable with tangent map T ρ k = ( ρ k , D ρ k ) : R × E k + 1 × R × E k → E 1 × E given by D ρ k : ( t , x , T , X ) ↦ ρ k ( t , X ) + T ⋅ ρ k + 1 ( t , x ) . [11] Once this is established, T ρ k is s c 0 by scale-continuity of ρ k , ρ k + 1 . In fact, T ρ k , as a sum and product of s c 1 maps, is s c 1 , and further induction proves that ρ k and thus also s and s − 1 are all sc ∞ .

The above claims and Eq. 11 for t ≠ 0 follow from the maps ρ k : E k + i → E i all being classically differentiable with differential D ρ k ( t , x , T , X ) = d d s s = 0 ρ k ( t + s T , x + s X ) = d d s s = 0 ∑ n = 0 ∞ f n ( k ) ( t + s T ) p n ( x + s X ) = ∑ n = 0 ∞ T f n ( k + 1 ) ( t ) p n ( x ) + f n ( k ) ( t ) p n ( X ) = T ⋅ ρ k + 1 ( t , x ) + ρ k ( t , X ) . To see that ρ 0 is well defined note that ( e n ) n ∈ N 0 ⊂ E i is orthogonal on each scale i ∈ N 0 , so ρ 0 ( t , x ) i = ∑ f n ( t ) p n ( x ) i = ∑ f n ( t ) 2 ‖ p n ( x ) ‖ i 2 1 / 2 ≤ sup n ‖ f n ‖ ∞ 2 ∑ ‖ p n ( x ) ‖ i 2 1 / 2 = sup n ‖ f n ‖ ∞ ⋅ ∑ p n ( x ) i = ‖ f 1 ‖ ∞ ‖ x ‖ i ≤ 2 ‖ x ‖ i , where ‖ f 1 ‖ ∞ = ‖ f ‖ ∞ = 1 or ‖ f 1 ‖ ∞ = ‖ 1 f ‖ ∞ = 2 if we choose f : R → R with values in [ 1 2 , 1 ] .

To check sc-continuity of ρ 0 at t = 0 we fix a level i ∈ N 0 and x ∈ E i and estimate for R × E i ∋ ( t , h ) → 0 with N t ≔ ⌊ t − 1 ⌋ for t > 0 and N t ≔ ∞ for t ≤ 0 ‖ ρ 0 ( t , x + h ) − ρ 0 ( 0 , x ) ‖ i = ‖ ρ 0 ( t , h ) + ρ 0 ( t , x ) − x ‖ i ≤ ‖ ρ 0 ( t , h ) ‖ i + ‖ ∑ ( f n ( t ) − 1 ) p n ( x ) ‖ i ≤ 2 ‖ h ‖ i + ∑ n = N t ∞ ( f n ( t ) − 1 ) p n ( x ) i ≤ 2 ‖ h ‖ i + sup n ‖ f n − 1 ‖ ∞ ∑ N t ∞ p n ( x ) i → | t | + ‖ h ‖ i → 0 0 . Here we used the facts that f n ( t ) = 1 for n ≤ t − 1 − 1 , and that x = lim N → ∞ ∑ n = 0 N p n ( x ) ∈ E i converges, hence as N t = ⌊ t − 1 ⌋ → ∞ with t → 0 we have ∑ n = N t ∞ p n ( x ) i → 0 .

Differentiability of ρ 0 with D ρ 0 ( 0 , x , T , X ) = ρ 0 ( 0 , X ) + T ρ 1 ( 0 , x ) = X as claimed in Eq. 11 amounts to estimating for x ∈ E i + 1 and t > 0 , using Eqs. 8 and 10 ρ 0 ( t , x + X ) − ρ 0 ( 0 , x ) − ρ 0 ( 0 , X ) i = ∑ f n ( t ) p n ( x + X ) − x − X i = ∑ n = N t ∞ ( f n ( t ) − 1 ) p n ( x + X ) i ≤ sup n ‖ f n − 1 ‖ ∞ ‖ ∑ n = N t ∞ p n ( x + X ) ‖ i ≤ N t − 3 ‖ x + X ‖ i + 1 , whereas for t ≤ 0 we have ρ 0 ( t , x + X ) − ρ 0 ( 0 , x ) − ρ 0 ( 0 , X ) i = x + X − x − X i = 0 . So together we obtain the required convergence of difference quotients, ‖ ρ 0 ( t , x + X ) − ρ 0 ( 0 , x ) − ρ 0 ( 0 , X ) ‖ i | t | + ‖ X ‖ i + 1 ≤ max 0 , ⌊ t − 1 ⌋ − 3 ‖ x + X ‖ i + 1 | t | + ‖ X ‖ i + 1 → | t | + ‖ X ‖ i + 1 → 0 0 . For k ≥ 1 recall that ρ k ( t , ⋅ ) = f N t ( k ) ( t ) p N t with N t = ⌊ t − 1 ⌋ for t > 0 and ρ k ( t , ⋅ ) ≡ 0 for t ≤ 0 as well as in a small neighborhood t ∼ 1 n for each n ∈ N . Thus the maps ρ k ( t , ⋅ ) are evidently well defined and linear on each scale in E i , and continuous (in fact classically smooth) with respect to t ∈ R < 0 >. To check continuity at t = 0 we fix a level i ∈ N 0 and x ∈ E k + i and estimate for h ∈ E k + i and t > 0 ‖ ρ k ( t , x + h ) ‖ i = f N t ( k ) ( t ) p N t ( x + h ) i ≤ ‖ f N t ( k ) ‖ ∞ ‖ p N t ( x + h ) ‖ i ≤ N t 2 k C k N t − 3 k ‖ x + h ‖ i + k ≤ N t − k C k ‖ x + h ‖ k + i , where we used Eqs. 8 and 9. Since ρ k ( t , x ) = 0 for t ≤ 0 this proves continuity ‖ ρ k ( t , x + h ) − ρ k ( 0 , x ) ‖ i ≤ max 0 , ⌊ t − 1 ⌋ − k C k ‖ x + h ‖ k + i → | t | + ‖ h ‖ k + i → 0 0 . Finally, differentiability for k ≥ 1 with D ρ k ( 0 , x , T , X ) = ρ k ( 0 , X ) + T ρ k + 1 ( 0 , x ) = 0 as claimed in Eq. 11 follows from the analogous estimate for x ∈ E k + i + 1 and t > 0 ρ k ( t , x + X ) − ρ k ( 0 , x ) − ρ k ( 0 , X ) i = ρ k ( t , x + X ) i ≤ N t − k − 3 C k ‖ x + X ‖ k + i + 1 , while for t ≤ 0 we have ρ k ( t , x + X ) − ρ k ( 0 , x ) − ρ k ( 0 , X ) i = 0 . So together we obtain the required convergence of difference quotients, ‖ ρ k ( t , x + X ) − ρ k ( 0 , x ) − ρ k ( 0 , X ) ‖ i | t | + ‖ X ‖ k + i + 1 ≤ max 0 , ⌊ t − 1 ⌋ − k − 3 C k ‖ x + X ‖ k + i + 1 | t | + ‖ X ‖ k + i + 1 → | t | + ‖ X ‖ k + i + 1 → 0 0 . This proves for all k ∈ N 0 that ρ k is s c 0 and sc-differentiable with Eq. 11, and thus finishes the proof of sc-smoothness of s and s − 1 .

Logic and Mathematical Reasoning an introduction to proof writing

For propositions (P) and (Q ext<,>) the conditional sentence (P implies Q) is the proposition “If (P ext<,>) then (Q ext<.>)” The proposition (P) is called the antecedent, (Q) the consequent. The conditional sentence (P implies Q) is true if and only if (P) is false or (Q) is true. In other words, (P implies Q) is equivalent to (( eg P) vee Q ext<.>)

A conditional is meant to make precise the standard language construct “If …, then …”, but it is has some seemingly counterintuitive properties. For example, do you think the statement “if the moon is made of green cheese, then it is tasty,” is true or false? What about the statement, “if the moon is made of green cheese, then the Red Sox will win the world series,” is it true or false? In fact, both statement are true because the antecedent, “the moon is made of green cheese”, is false. Note, in each case, we are not asking about the truth of the atomic propositions, but rather the statement as a whole. Moreover, there is no reason the antecedent and consequent need to be logically connected, which violates our intuition.

Exercise 1.2.2

Suppose you are a waiter in a restaurant and you want to make sure that everyone at the table is obeying the law: the drinking age is 21. You know some information about who ordered what to drink and their ages which is indicated in the table below. What is the minimal additional information you need to determine if the law is obeyed?

Person Age Drink
A 33
B Beer
C 15
D Coke
Table 1.2.3

B's age and C's drink. You can think of obeying the law as making “If under 21, then no alcohol,” a true statement. Then the statement is true whenever each person is either 21 and up or did not order alcohol. A is above 21, so he is obeying the law no matter what he ordered. B ordered alcohol, so we must check how old he is to determine if the law is obeyed. C is under 21, so we must check what he ordered to determine if the law is obeyed. D order coke, so he is obeying the law regardless of his age.

Definition 1.2.4 Converse, Contrapositive

Let (P) and (Q) be propositions and consider the conditional (P implies Q ext<.>) Then the

converse is (Q implies P ext<.>) contrapositive is (( eg Q) implies ( eg P) ext<.>)

Theorem 1.2.5 Contrapositive Equivalence
  1. A conditional sentence and its contrapositive are equivalent.
  2. A conditional sentence and its converse are not equivalent.
Definition 1.2.6 Biconditional

For propositions (P) and (Q ext<,>) the biconditional sentence (P iff Q) is the proposition “(P) if and only if (Q ext<.>)” (P iff Q) is true exactly when (P) and (Q) have the same truth value.

Theorem 1.2.7 De Morgan's Laws

For propositions (P) and (Q ext<,>)

  1. ( eg (P wedge Q)) is equivalent to (( eg P) vee ( eg Q) ext<>)
  2. ( eg (P vee Q)) is equivalent to (( eg P) wedge ( eg Q) ext<.>)

These can be read in English as “the negation of a conjunction is the disjunction of the negations,” and “the negation of a disjunction is the conjunction of the negations.”

Theorem 1.2.8 Commutativity of Conjunction and Disjunction

For propositions (P) and (Q ext<,>)

So there is no ambiguity when we say “the conjunction of (P) and (Q ext<,>)” or “the disjunction of (P) and (Q ext<,>)”

Theorem 1.2.9 Associativity of Conjunction and Disjunction

For propositions (P ext<,>) (Q) and (R ext<,>)

  1. (P wedge (Q wedge R)) is equivalent to ((P wedge Q) wedge R ext<>)
  2. (P vee (Q vee R)) is equivalent to ((P vee Q) vee R ext<.>)

So there is no ambiguity in the propositions (P wedge Q wedge R) or (P vee Q vee R ext<.>)

Theorem 1.2.10 Distributivity of Conjunction and Disjunction

For propositions (P ext<,>) (Q) and (R ext<,>)

  1. (P wedge (Q vee R)) is equivalent to ((P wedge Q) vee (P wedge R) ext<>)
  2. (P vee (Q wedge R)) is equivalent to ((P vee Q) wedge (P vee R) ext<.>)

You should interpret this as indicating that conjunction and disjunction distribute over each other.

Theorem 1.2.11 Conditional Equivalences

For propositions (P) and (Q ext<,>)

  1. (P implies Q) is equivalent to (( eg P) vee Q ext<>)
  2. ( eg(P implies Q)) is equivalent to (P wedge ( eg Q) ext<>)
  3. (P iff Q) is equivalent to ((P implies Q) wedge (Q implies P) ext<.>)
Remark 1.2.12 Dictionary of implication

The logical conditional (P implies Q) has several translations into English which include:

  • If (P ext<,>) then (Q ext<.>)
  • (P) implies (Q ext<.>)
  • (P) is sufficient for (Q ext<.>)
  • (P) only if (Q ext<.>)
  • (Q ext<,>) if (P ext<.>)
  • (Q) whenever (P ext<.>)
  • (Q) is necessary for (P ext<.>)
  • (Q ext<,>) when (P ext<.>)

Similarly, the biconditional (P iff Q) translates into a few of English phrases including:

Linguistic rule writing: An iterative process

In writing linguistic rules, I generally follow these six steps:

  1. Identify categories of interest (e.g., “wh-question”, “polar question”, “not a question”).
  2. Come up with one or two big linguistic generalizations about each category.
  3. Come up with some counterexamples to the generalizations and revise/expand them as necessary.
  4. Write rules capturing the new generalizations.
  5. Test the rules on a bunch of examples.
  6. Fine-tune the rules by addressing any false positives and testing new examples.

Often these steps will not be perfectly sequential. For example, identifying the categories you want to target (step 1) basically amounts to stating some generalizations about them (step 2), so you may do these steps at the same time. Before I illustrate how this flow can guide our rule writing, let’s dig into spaCy a bit.

Section 3.10

The following solutions use information found in the Reading Section.

Exercise 3.10.1: Definitions

Fill in the blanks to complete the following sentences.

State Theorem 3.10.1 from the Reading Section.
If x0 and y0 are one solution to the linear Diophantine equation ax + by = c, where a, b, c Z, and d = gcd(a, b), then the general solution to the
linear Diophantine equation ax + by = c is given by

where t Z.

Exercise 3.10.2: Examples

You should attempt all these exercises yourself, using the textbook as an aid. Once you have attempted each question, check your answers by following the appropriate links. If you are stuck on a question, choose the link that gives you a hint and then try the question again.

In the previous section you found one solution to each of the following linear Diophantine equations. For each equation, use the solution you found in the previous section to find the general solution. Then list all solutions which involve only positive integers.

1. Find a formula for all integers x and y, given that x and y satisfy the linear Diophantine equation 91x + 221y = 676.
List all the solutions which involve positive values for x and y.

2. Find a formula for all integers m and n, given that m and n satisfy the linear Diophantine equation 105m + 56n = - 14.
List all the solutions which involve positive values for m and n.

Extra Practice Problems
In the previous section you found one solution to each of the following linear Diophantine equations. For each equation, use the solution you found in the previous section to find the general solution. Then list all solutions which involve only positive integers. Once you have finished both of these problems, click here to check your answers.

Watch the video: DCS WORLD. LAUNCH VIDEO (October 2021).