Conditional Statements & Implications

Definition

Most theorems in mathematics appear in the form of compound statements called conditional statements. Conditional statements are also called implications.

An implication is the compound statement of the form “if 𝑝, then 𝑞.” It is denoted 𝑝⇒𝑞, which is read as “𝑝 implies 𝑞” It is false only when 𝑝 is true and 𝑞 is false, and is true in all other situations.

p	q	𝑝⇒𝑞
T	T	T
T	F	F
F	T	T
F	F	T

 

The statement 𝑝 in an implication 𝑝⇒𝑞 is called its hypothesis, premise, or antecedent, and 𝑞 the conclusion or consequence.

Implications come in many disguised forms. There are several alternatives for saying 𝑝⇒𝑞. The most common ones are

• 𝑝 implies 𝑞,
• 𝑝 only if 𝑞,
• 𝑞 if 𝑝,
• 𝑞 provided that 𝑝.

All of them mean 𝑝⇒𝑞.

Implications play a key role in logical argument. If an implication is known to be true, then whenever the hypothesis is met, the consequence must be true as well. This is why an implication is also called a conditional statement. [1]

Who

Conditional statements help mathematicians and computer programmers make decisions based on the state of a situation. While they vary in use and complexity, professionals typically use conditional statements to test hypotheses and establish rules for programs to follow. Understanding these statements and their uses can help you build your skills and improve your qualifications. [2]

What

In a conditional statement:

1. Why does a true hypothesis and a true conclusion make the conditional true?
2. Why does a True hypothesis and a false conclusion make the conditional false?
3. Why does a false hypothesis and a true conclusion make the conditional true?
4. Why does a false hypothesis and a false conclusion make the conditional true?

Suppose that you’re given the following conditional statement.

Conditional Statement
If you score at least 85 on the exam (p), you will get an A (q).

There are four possible outcomes.

1. you score at least 85 and get an A;
2. you score at least 85 and don’t get an A;
3. you score below 85 and get an A; or
4. you score below 85 and don’t get an A.

Which of above four outcomes contradict the conditional statement. Use the Implications table above as a reference.

1. The first possible outcome DOES NOT contradict the conditional statement.
2. The second possible outcome definitely DOES contradict the conditional statement. You scored at least 85, but you did not get the promised A.

Now what about outcomes 3 and 4? The thing to realize here is that taken literally, the statement says nothing about what will happen if you don’t score at least 85; it merely promises you an A if you do score at least 85.

In everyday contexts we generally don’t take such statements literally. We understand this one to mean (or at least very strongly imply) that if you fail to score at least 85, you will not get an A. But it does not actually say this, and in a formal logical context we interpret it literally. And since it says nothing about what will happen if you score below 85, you can get any letter grade after scoring below 85 without contradicting the statement. In that case there is nothing to contradict, because no prediction or promise was made. Since neither outcome — getting an A or not getting an A — contradicts it in this case, we say that it’s true.

Conclusion
The truth values for these implications are based on the idea that the statement is false if and only if the facts actually contradict it.

In the above example, that would be the case in which you scored at least 85 but did not get the promised A. In general, for an implication p=>q, the only situation (possible outcome) that absolutely contradicts it is the one in which p is true, but q is nevertheless false. It certainly isn’t contradicted when p and q are both true, and it makes no promises at all about q when p is not true.

There is one other difference between this logical implication and ordinary English usage. Ordinarily when we say that if p is true, then q is true, we are thinking of some causal connection between p and q. In formal logic of the kind with which we are dealing here, we don’t care whether there’s any relationship between p and q at all. The truth value of the statement p => q is to be determined solely on the basis of the truth values of p and q.

Brian M. Scott (https://math.stackexchange.com/users/12042/brian-m-scott), Having extreme difficulty understanding conditional statements., URL (version: 2020-08-28): https://math.stackexchange.com/q/3806647

Examples

Example #1

“If it is not raining, then Daisy is riding her bike.” [3]

We can represent this conditional statement as P→Q, where P is the statement, “It is not raining” and Q is the statement, “Daisy is riding her bike.”

Although it is not a perfect analogy, think of the statement P→Q as being false to mean that I lied and think of the statement P→Q as being true to mean that I did not lie. We will now check the truth value of P→Q based on the truth values of P and Q.

1. Suppose that both P and Q are true. That is, it is not raining and Daisy is riding her bike. In this case, it seems reasonable to say that I told the truth and that P→Q is true.
2. Suppose that P is true and Q is false or that it is not raining and Daisy is not riding her bike. It would appear that by making the statement, “If it is not raining, then Daisy is riding her bike,” that I have not told the truth. So in this case, the statement P→Q is false.
3. Now suppose that P is false and Q is true or that it is raining and Daisy is riding her bike. Did I make a false statement by stating that if it is not raining, then Daisy is riding her bike? The key is that I did not make any statement about what would happen if it was raining, and so I did not tell a lie. So we consider the conditional statement, “If it is not raining, then Daisy is riding her bike,” to be true in the case where it is raining and Daisy is riding her bike.
4. Finally, suppose that both P and Q are false. That is, it is raining and Daisy is not riding her bike. As in the previous situation, since my statement was P→Q, I made no claim about what would happen if it was raining, and so I did not tell a lie. So the statement P→Q cannot be false in this case and so we consider it to be true.

Example #2

A friend tells you “If you upload that picture to Facebook, you’ll lose your job.” Under what conditions can you say that your friend was wrong? [4]

There are four possible outcomes:

1. You upload the picture and lose your job
2. You upload the picture and don’t lose your job
3. You don’t upload the picture and lose your job
4. You don’t upload the picture and don’t lose your job

There is only one possible case in which you can say your friend was wrong: the second outcome in which you upload the picture but still keep your job. In the last two cases, your friend didn’t say anything about what would happen if you didn’t upload the picture, so you can’t say that their statement was wrong. Even if you didn’t upload the picture and lost your job anyway, your friend never said that you were guaranteed to keep your job if you didn’t upload the picture; you might lose your job for missing a shift or punching your boss instead.

In traditional logic, a conditional is considered true as long as there are no cases in which the antecedent is true and the consequent is false.

Why

See Theoretical Knowledge Vs Practical Application.

How

Many of the References and Additional Reading websites and Videos will assist you with understanding and applying conditional statements & implications.

As some professors say: “It is intuitively obvious to even the most casual observer.“

See Also

“Difference Between Causal and Conditional Relationships.” 2024. Mathematical Mysteries. July 7. https://mathematicalmysteries.org/difference-between-causal-and-conditional-relationships/.

Nordstrom, Jennifer Firkins. 2024. “Discrete Mathematics: An Active Approach to Mathematical Reasoning.” Runestone Academy. Accessed July 7. https://runestone.academy/ns/books/published/DiscreteMathText/DiscreteMath.html.

This text is meant to provide the basics for a Discrete Math course which emphasizes mathematical reasoning and an introduction to proof writing. Each section includes activities to help the students master the definitions and techniques.

Course Goals.
To introduce the student to the basic discrete mathematical structures and the theory and techniques associated with these structures.

* To continue the student’s growth in the use of mathematical language and the use of logic.
* To improve the student’s ability to write precise, valid, mathematical arguments.
* To increase the student’s confidence in working through mathematical challenges.
* To continue the growth of the student’s abilities in the appreciation and use of abstract reasoning.
* To improve the student’s problem-solving abilities.
* To increase the student’s mathematical maturity.

References

[1] Kwong, Harris. Libretexts. 2021. “2.3: Implications.” Mathematics LibreTexts. Libretexts. July 7. https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/02%3A_Logic/2.03%3A_Implications.

[2] “Conditional Statements: Examples in Math and Programming”. 2024. Accessed June 30. https://www.indeed.com/career-advice/career-development/conditional-statements.

[3] “1.1: Statements and Conditional Statements.” 2021. Mathematics LibreTexts. Libretexts. October 20. https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/01%3A_Introduction_to_Writing_Proofs_in_Mathematics/1.01%3A_Statements_and_Conditional_Statements.

[4] “17.6: Truth Tables: Conditional, Biconditional.” 2022. Mathematics LibreTexts. Libretexts. July 18. https://math.libretexts.org/Bookshelves/Applied_Mathematics/Math_in_Society_(Lippman)/17%3A_Logic/17.06%3A_Section_6-.

Additional Reading

“Arguments in Discrete Mathematics.” 2023. GeeksforGeeks. GeeksforGeeks. https://www.geeksforgeeks.org/arguments-in-discrete-mathematics/.

Arguments are an important part of logical reasoning and philosophy. It also plays a vital role in mathematical proofs. In this article, we will throw some light on arguments in logical reasoning. Logical proofs can be proven by mathematical logic. The proof is a valid argument that determines the truth values of mathematical statements. The argument is a set of statements or propositions which contains premises and conclusion. The end or last statement is called a conclusion and the rest statements are called premises.

The premises of the argument are those statements or propositions which are used to provide the support or evidence while the conclusions of an argument is that statement or proposition that simply defines that premises are intended to provide support.

“Conditional Statement.” 2024. CUEMATH. Accessed June 30. https://www.cuemath.com/data/conditional-statement/.

A conditional statement is a part of mathematical reasoning which is a critical skill that enables students to analyze a given hypothesis without any reference to a particular context or meaning. In layman words, when a scientific inquiry or statement is examined, the reasoning is not based on an individual’s opinion. Derivations and proofs need a factual and scientific basis.  Mathematical critical thinking and logical reasoning are important skills that are required to solve maths reasoning questions.

“Conditional Statements & Implications – Mathematical Reasoning: Class 11 Maths.” 2021. GeeksforGeeks. GeeksforGeeks. https://www.geeksforgeeks.org/conditional-statements-implications-mathematical-reasoning-class-11-maths/.

Generally, conditional statements are the if-then statement in which p is called a hypothesis (or antecedent or premise) and q is called a conclusion (or consequence). Conditional statements symbolized by p, q. A conditional statement p -> q is false when p is true and q is false, and true otherwise. 

“Conditional Statements: Use Of If-Then Statement: Material Implication.” 2020. BYJUS. BYJU’S. July 28. https://byjus.com/maths/use-of-if-then-statements-in-mathematical-reasoning/.

“Conditional Statements Study Guide”. 2024. Accessed June 30. https://www.ck12.org/studyguides/geometry/conditional-statements-study-guide.html.

Easdown, David. “Notes on Mathematical Implication”. February 2013. https://www.maths.usyd.edu.au/u/UG/SM/MATH3065/r/ImplicationNotes.pdf.

“Rules of Inference: Definitions, Argument Structure & Examples.” 2024. GeeksforGeeks. May 28. https://www.geeksforgeeks.org/mathematical-logic-rules-inference/.

Rules of Inference: Every Theorem in Mathematics, or any subject for that matter, is supported by underlying proofs. These proofs are nothing but a set of arguments that are conclusive evidence of the validity of the theory. The arguments are chained together using Rules of Inferences to deduce new statements and ultimately prove that the theorem is valid. 

Kirk, Donna. 2024. “2.2 Compound Statements – Contemporary Mathematics.” OpenStax. OpenStax. Accessed June 30. https://openstax.org/books/contemporary-mathematics/pages/2-2-compound-statements.

Understanding the following logical connectives, along with their properties, symbols, and names, will be key to applying the topics presented in this chapter. The chapter will discuss each connective introduced here in more detail.

“Logical Implication.” 2023. CalcWorkshop. April 1. https://calcworkshop.com/logic/logical-implication/.

A conditional statement represents an if…then statement where p is the hypothesis (antecedent), and q is the conclusion (consequent). In essence, it is a statement that claims that if one thing is true, then something else is true also.

Videos

Please get a good grasp of conditional statements (implications) before watching these videos. I suggest going over the excellent explanation by Brian M. Scott.

 

Discrete Mathematics: Logical Operators − Implication (Part 1)
1. Logical Implication.
2. Definition of Logical Implication.
3. Examples of Logical Implication.
4. Homework problems on Logical Implication.

 

 

Discrete Mathematics: Logical Operators − Implication (Part 2)
1. Different ways to represent conditional statements.
2. p implies q.
3. q when p.
4. q whenever p.
5. q follows from p.
6. p only if q.
7. Why p only if q is equivalent to if p then q.
8. Why p only if q is not equivalent to if q then p.

 

 

Discrete Mathematics: Logical Operators − Implication (Part 3)
1. Meaning of necessary condition.
2. Meaning of sufficient condition.
3. Why p is sufficient for q?
4. Why q is necessary for p?
5. Why p is not necessary for q?
6. Why q is not sufficient for p?

 

 

Learning Objectives:
1) Interpret sentences as being conditional statements
2) Write the truth table for a conditional in its implication form
3) Use truth tables to see the disjunctive form of a conditional statement as logically equivalent

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⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.