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Introduction to Philosophy

5.2 Logical Statements

Introduction to Philosophy5.2 Logical Statements

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Table of contents
  1. Preface
  2. 1 Introduction to Philosophy
    1. Introduction
    2. 1.1 What Is Philosophy?
    3. 1.2 How Do Philosophers Arrive at Truth?
    4. 1.3 Socrates as a Paradigmatic Historical Philosopher
    5. 1.4 An Overview of Contemporary Philosophy
    6. Summary
    7. Key Terms
    8. References
    9. Review Questions
    10. Further Reading
  3. 2 Critical Thinking, Research, Reading, and Writing
    1. Introduction
    2. 2.1 The Brain Is an Inference Machine
    3. 2.2 Overcoming Cognitive Biases and Engaging in Critical Reflection
    4. 2.3 Developing Good Habits of Mind
    5. 2.4 Gathering Information, Evaluating Sources, and Understanding Evidence
    6. 2.5 Reading Philosophy
    7. 2.6 Writing Philosophy Papers
    8. Summary
    9. Key Terms
    10. References
    11. Review Questions
    12. Further Reading
  4. 3 The Early History of Philosophy around the World
    1. Introduction
    2. 3.1 Indigenous Philosophy
    3. 3.2 Classical Indian Philosophy
    4. 3.3 Classical Chinese Philosophy
    5. Summary
    6. Key Terms
    7. References
    8. Review Questions
    9. Further Reading
  5. 4 The Emergence of Classical Philosophy
    1. Introduction
    2. 4.1 Historiography and the History of Philosophy
    3. 4.2 Classical Philosophy
    4. 4.3 Jewish, Christian, and Islamic Philosophy
    5. Summary
    6. Key Terms
    7. References
    8. Review Questions
    9. Further Reading
  6. 5 Logic and Reasoning
    1. Introduction
    2. 5.1 Philosophical Methods for Discovering Truth
    3. 5.2 Logical Statements
    4. 5.3 Arguments
    5. 5.4 Types of Inferences
    6. 5.5 Informal Fallacies
    7. Summary
    8. Key Terms
    9. References
    10. Review Questions
    11. Further Reading
  7. 6 Metaphysics
    1. Introduction
    2. 6.1 Substance
    3. 6.2 Self and Identity
    4. 6.3 Cosmology and the Existence of God
    5. 6.4 Free Will
    6. Summary
    7. Key Terms
    8. References
    9. Review Questions
    10. Further Reading
  8. 7 Epistemology
    1. Introduction
    2. 7.1 What Epistemology Studies
    3. 7.2 Knowledge
    4. 7.3 Justification
    5. 7.4 Skepticism
    6. 7.5 Applied Epistemology
    7. Summary
    8. Key Terms
    9. References
    10. Review Questions
    11. Further Reading
  9. 8 Value Theory
    1. Introduction
    2. 8.1 The Fact-Value Distinction
    3. 8.2 Basic Questions about Values
    4. 8.3 Metaethics
    5. 8.4 Well-Being
    6. 8.5 Aesthetics
    7. Summary
    8. Key Terms
    9. References
    10. Review Questions
    11. Further Reading
  10. 9 Normative Moral Theory
    1. Introduction
    2. 9.1 Requirements of a Normative Moral Theory
    3. 9.2 Consequentialism
    4. 9.3 Deontology
    5. 9.4 Virtue Ethics
    6. 9.5 Daoism
    7. 9.6 Feminist Theories of Ethics
    8. Summary
    9. Key Terms
    10. References
    11. Review Questions
    12. Further Reading
  11. 10 Applied Ethics
    1. Introduction
    2. 10.1 The Challenge of Bioethics
    3. 10.2 Environmental Ethics
    4. 10.3 Business Ethics and Emerging Technology
    5. Summary
    6. Key Terms
    7. References
    8. Review Questions
    9. Further Reading
  12. 11 Political Philosophy
    1. Introduction
    2. 11.1 Historical Perspectives on Government
    3. 11.2 Forms of Government
    4. 11.3 Political Legitimacy and Duty
    5. 11.4 Political Ideologies
    6. Summary
    7. Key Terms
    8. References
    9. Review Questions
    10. Further Reading
  13. 12 Contemporary Philosophies and Social Theories
    1. Introduction
    2. 12.1 Enlightenment Social Theory
    3. 12.2 The Marxist Solution
    4. 12.3 Continental Philosophy’s Challenge to Enlightenment Theories
    5. 12.4 The Frankfurt School
    6. 12.5 Postmodernism
    7. Summary
    8. Key Terms
    9. References
    10. Review Questions
  14. Index

Learning Objectives

By the end of this section, you will be able to:

  • Identify the necessary and sufficient conditions in conditionals and universal affirmative statements.
  • Describe counterexamples for statements.
  • Assess the truth of conditionals and universal statements using counterexamples.

Specific types of statements have a particular meaning in logic, and such statements are frequently used by philosophers in their arguments. Of particular importance is the conditional, which expresses the logical relations between two propositions. Conditional statements are used to accurately describe the world or construct a theory. Counterexamples are statements used to disprove a conditional. Universal statements are statements that assert something about every member of a set of things and are an alternative way to describe a conditional.

Conditionals

A conditional is most commonly expressed as an if–then statement, similar to the examples we discussed earlier when considering hypotheses. Additional examples of if–then statements are “If you eat your meat, then you can have some pudding” and “If that animal is a dog, then it is a mammal.” But there are other ways to express conditionals, such as “You can have pudding only if you eat your meat” or “All dogs are mammals.” While these sentences are different, their logical meaning is the same as their correlative if–then sentences above.

All conditionals include two components—that which follows the “if” and that which follows the “then.” Any conditional can be rephrased in this format. Here is an example:

Statement 1: You must complete 120 credit hours to earn a bachelor’s degree.

Statement 2: If you expect to graduate, then you must complete 120 credit hours.

Whatever follows “if” is called the antecedent; whatever follows “then” is called the consequent. Ante means “before,” as in the word “antebellum,” which in the United States refers to anything that occurred or was produced before the American Civil War. The antecedent is the first part of the conditional, occurring before the consequent. A consequent is a result, and in a conditional statement, it is the result of the antecedent (if the antecedent is true).

Necessary and Sufficient Conditions

All conditionals express two relations, or conditions: those that are necessary and those that are sufficient. A relation is a relationship/property that exists between at least two things. If something is sufficient, it is always sufficient for something else. And if something is necessary, it is always necessary for something else. In the conditional examples offered above, one part of the relation is required for the other. For example, 120 credit hours are required for graduation, so 120 credit hours is necessary if you expect to graduate. Whatever is the consequent—that is, whatever is in the second place of a conditional—is necessary for that particular antecedent. This is the relation/condition of necessity. Put formally, Y is a necessary condition for X if and only if X cannot be true without Y being true. In other words, X cannot happen or exist without Y. Here are a few more examples:

  • Being unmarried is a necessary condition for being a bachelor. If you are a bachelor, then you are unmarried.
  • Being a mammal is a necessary condition for being a dog. If a creature is a dog, then it is a mammal.

But notice that the necessary relation of a conditional does not automatically occur in the other direction. Just because something is a mammal does not mean that it must be a dog. Being a bachelor is not a necessary feature of being unmarried because you can be unmarried and be an unmarried woman. Thus, the relationship between X and Y in the statement “if X, then Y” is not always symmetrical (it does not automatically hold in both directions). Y is always necessary for X, but X is not necessary for Y. On the other hand, X is always sufficient for Y.

Take the example of “If you are a bachelor, then you are unmarried.” If you know that Eric is a bachelor, then you automatically know that Eric is unmarried. As you can see, the antecedent/first part is the sufficient condition, while the consequent/second part of the conditional is the necessary condition. X is a sufficient condition for Y if and only if the truth of X guarantees the truth of Y. Thus, if X is a sufficient condition for Y, then X automatically implies Y. But the reverse is not true. Oftentimes X is not the only way for something to be Y. Returning to our example, being a bachelor is not the only way to be unmarried. Being a dog is a sufficient condition for being a mammal, but it is not necessary to be a dog to be a mammal since there are many other types of mammals.

Dog in a field with a person and two sheep.
Figure 5.4 All dogs are mammals, but not all mammals are dogs. Being a dog is a sufficient condition for being a mammal but it is not necessary to be a dog to be a mammal. (credit: “Sheepdog Trials in California” by SheltieBoy/Flickr, CC BY 2.0)

The ability to understand and use conditionals increases the clarity of philosophical thinking and the ability to craft effective arguments. For example, some concepts, such as “innocent” or “good,” must be rigorously defined when discussing ethics or political philosophy. The standard practice in philosophy is to state the meaning of words and concepts before using them in arguments. And oftentimes, the best way to create clarity is by articulating the necessary or sufficient conditions for a term. For example, philosophers may use a conditional to clarify for their audience what they mean by “innocent”: “If a person has not committed the crime for which they have been accused, then that person is innocent.”

Counterexamples

Sometimes people disagree with conditionals. Imagine a mother saying, “If you spend all day in the sun, you’ll get sunburnt.” Mom is claiming that getting sunburnt is a necessary condition for spending all day in the sun. To argue against Mom, a teenager who wants to go to the beach might offer a counterexample, or an opposing statement that proves the first statement wrong. The teenager must point out a case in which the claimed necessary condition does not occur alongside the sufficient one. Regular application of an effective sunblock with an SPF 30 or above will allow the teenager to avoid sunburn. Thus, getting sunburned is not a necessary condition for being in the sun all day.

Counterexamples are important for testing the truth of propositions. Often people want to test the truth of statements to effectively argue against someone else, but it is also important to get into the critical thinking habit of attempting to come up with counterexamples for our own statements and propositions. Philosophy teaches us to constantly question the world around us and invites us to test and revise our beliefs. And generating creative counterexamples is a good method for testing our beliefs.

Universal Statements

Another important type of statement is the universal affirmative statement. Aristotle included universal affirmative statements in his system of logic, believing they were one of only a few types of meaningful logical statements (On Interpretation). Universal affirmative statements take two groups of things and claim all members of the first group are also members of the second group: “All A are B.” These statements are called universal and affirmative because they assert something about all members of group A. This type of statement is used when classifying objects and/or the relationships. Universal affirmative statements are, in fact, an alternative expression of a conditional.

Universal Statements as Conditionals

Universal statements are logically equivalent to conditionals, which means that any conditional can be translated into a universal statement and vice versa. Notice that universal statements also express the logical relations of necessity and sufficiency. Because universal affirmative statements can always be rephrased as conditionals (and vice versa), the ability to translate ordinary language statements into conditionals or universal statements is helpful for understanding logical meaning. Doing so can also help you identify necessary and sufficient conditions. Not all statements can be translated into these forms, but many can.

Counterexamples to Universal Statements

Universal affirmative statements also can be disproven using counterexamples. Take the belief that “All living things deserve moral consideration.” If you wanted to prove this statement false, you would need to find just one example of a living thing that you believe does not deserve moral consideration. Just one will suffice because the categorical claim is quite strong—that all living things deserve moral consideration. And someone might argue that some parasites, like the protozoa that causes malaria, do not deserve moral consideration.

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