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Logical Arguments

Apply the law of detachment to determine the conclusion of a pair of statements.

Learning Objectives

After completing this section, you should be able to:

  1. Apply the law of detachment to determine the conclusion of a pair of statements.
  2. Apply the law of denying the consequent to determine the conclusion for pairs of statements.
  3. Apply the chain rule to determine valid conclusions for pairs of true statements.

Law of Detachment

The law of detachment is a valid form of a conditional argument that asserts that if both the conditional, \(p\to q\), and the hypothesis, \(p\), are true, then the conclusion \(q\) must also be true. The law of detachment is also called affirming the hypothesis (or antecedent) and modus ponens. Symbolically, it has the form \(((p\to q)∧p)\to q\).

Law of Detachment
Premise:\(p\to q\)
Premise:\(p\)
Conclusion:\(∴\ q\)

Looking at the truth table for the conditional statement, the only time the conditional is true is when the hypothesis \(p\) is also true. The only place this happens is in the first row, where \(q\) is also true, confirming that the law of detachment is a valid argument.

\(p\)\(q\)\(p\to q\)
TTT
TFF
FTT
FFT

Another way to verify that the law of detachment is a valid argument is to construct a truth table for the argument \(((p\to q)∧p)\to q\) and verify that it is a tautology.

\(p\)\(q\)\(p\to q\)\((p\to q)∧p\)\(((p\to q)∧p)\to q\)
TTTTT
TFFFT
FTTFT
FFTFT

Venn diagrams may also be used to verify deductive arguments, which include conditional premises. Consider the statement \(p\to q:\) “If you play guitar, then you are a musician.” The set of guitarists is a subset of the set of musicians, \(p⊂q.\) To verify that an argument is valid using a Venn diagram, draw the Venn diagram representing all the premises in the argument only, as shown in . Then verify if the conclusion is also represented by the Venn diagram of the premises. If it is, the argument is valid. If it is not, the argument is not valid. The set of guitarists is drawn as a subset of the set of musicians to represent the premise, \(p\to q.\) The \(\times\) represents the premise: \(p\) is true. This completes the drawing of the premises.

Now, examine the Venn diagram to verify if the conclusion is included in the picture. The conclusion is \(q\). Because the \(\times\) is in the set \(p\), and \(p\) is a subset of \(q\), \(\times\) is also in \(q\); therefore, the law of detachment is a valid argument.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Law of Denying the Consequent

Another form of a valid conditional argument is called the law of denying the consequent, or modus tollens. Recall, that the conditional statement, \(p\to q\), is logically equivalent to the contrapositive, \(\sim q\to \ \sim p.\) So, if the conditional statement is true, then the contrapositive statement is also true. By the law of detachment, if \(\sim q\) is also true, then it follows that \(\sim p\) must also be true. Symbolically, it has the form \(((p\to q)∧\sim q)\to \ \sim p\).

Law of Denying the Consequent
Premise:\(p\to q\)
Premise:\(\sim q\)
Conclusion:\(∴\ \sim p\)

To verify if the law of denying the consequent is a valid argument, construct a truth table for the argument, \(((p\to q)∧\sim q)\to \ \sim p\), and verify that it is a tautology.

\(p\)\(q\)\(\sim p\)\(\sim q\)\(p\to q\)\((p\to q)∧\sim q\)\(((p\to q)∧\sim q)\to \sim p\)
TTFFTFT
TFFTFFT
FTTFTFT
FFTTTTT

To verify an argument of this form using a Venn diagram, again consider the premise: \(p\to q:\) “If you play guitar, then you are a musician.” We will change the second premise to \(\sim q.\) In this case, the \(\times\) represents the premise, \(\sim q.\) So, it will be placed inside the universal set of all people, but outside the set of musicians, as depicted in the Venn diagram in .

Because the \(\times\) is also outside the set of guitarists, the statement \(\sim p\) follows from the premises and the argument is valid.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Chain Rule for Conditional Arguments

The chain rule for conditional arguments is another form of a valid conditional argument. It is also called hypothetical syllogism or the transitivity of implication. Recall that the conditional statement \(p\to q\) can also be read as \(p\) implies \(q\). This is where the name transitivity of implication comes from. The transitive property for numbers states that, if \(3<4\) and \(4<5,\) then it follows that \(3<5.\) The chain rule extends this property to conditional statements. If the premises of the argument consist of two conditional statements, with the form “\(p\to q\)” and “\(q\to r,\)” then it follows that \(p\to r.\) Symbolically, it has the form \(((p\to q)∧(q\to r))\to (p\to r)\).

Chain Rule for Conditional Arguments
Premise:\(p\to q\)
Premise:\(q\to r\)
Conclusion:\(∴\ p\to r\)

To verify the chain rule for conditional arguments, construct a truth table for the argument, \(((p\to q)∧(q\to r))\to (p\to r)\), and verify that it is a tautology.

\(p\)\(q\)\(r\)\(p\to q\)\(q\to r\)\((p\to q)∧(p\to r)\)\(p\to r\)\(((p\to q)∧(q\to r))\to (p\to r)\)
TTTTTTTT
TTFTFFFT
TFTFTFTT
TFFFTFFT
FTTTTTTT
FTFTFFTT
FFTTTTTT
FFFTTTTT

To verify an argument of this form using a Venn diagram, again consider the premise \(p\to q:\) “If you play guitar, then you are a musician,” but change the second premise to \(q\to r:\) “If you are a musician, then you are an artist.” In this case, the set \(p\) of guitarists is a subset of the set \(r\) of artists, and it follows that if you are a guitarist, then you are an artist. Therefore, the conclusion \(p\to r\) follows from the premises and the chain rule for logical arguments is valid. See .

Condensed — the full section is in OpenStax Contemporary Mathematics.

Projects

Logic gates are the basis for all digital circuits.

  1. Research and document the following terms: logic gate, OR gate, AND gate, and NOT gate.
  2. Construct a diagram of a NAND gate, NOR gate, and a XOR gate by using at least two of the following gates: AND, OR, and NOT.
  3. Digital electronics use a 1 for true or on, and a 0 for false or off. Create a truth table documenting all possible cases using 0s and 1s for the NAND gate, NOR gate and XOR gate.
  4. Use a truth table to explain how XOR is related to the biconditional statement.

Fallacies are false or deceptive logical arguments.

  1. Research and document the structure of five of the following named fallacies: hasty generalization, limited choice, false cause, appeal to popularity, appeal to emotion, appeal to authority, personal attack, gamblers' ruin, slippery slope, and circular reasoning.
  2. Create a presentation highlighting one of the five fallacies researched in the previous question. The presentation must include an introductory slide with the title of the fallacy and the form or structure of the argument. The second slide must include an example of this fallacy as used in a commercial, a political cartoon or a current event or new article. The third slide must include an explanation of why the example on slide to is a representative example of the fallacy. The last slide must include citations for any materials used. No textbooks should be used as reference.

Lawyers, mathematicians, and computer programmers are a few of the careers that require knowledge of logic.

  1. What career are you interested in? Research how knowledge of logic applies to your chosen field of study. Then, write a cover letter for a position in your field you'd like to apply to. In the cover letter, include how your knowledge of logic qualifies you for the position you are applying for. If you do not think logic is important for your given career choice, find a position where logic is an essential element of the position and complete the project by pretending you are writing a cover letter for that job.

Key Concepts

  • A logical argument uses a series of facts or premises to justify a conclusion or claim. It is valid if its conclusion follows from the premises, and it is sound if it is valid, and all of its premises are true.
  • The law of detachment is a valid form of a conditional argument that asserts that if both the conditional, \(p\to q\) is true and the hypothesis, \(p\) is true, then the conclusion \(q\) must also be true.
    Law of Detachment
    Premise:\(p\to q\)
    Premise:\(p\)
    Conclusion:\(∴q\)
  • Know how to apply the law of detachment to determine the conclusion of a pair of statements.
  • The law of denying the consequent is a valid form of a conditional argument that asserts that if both the conditional, \(p\to q\) is true and the negation of the conclusion, \(\sim q\) is true, then the negation of the hypothesis \(\sim p\) must also be true.
    Law of Denying the Consequent
    Premise:\(p\to q\)
    Premise:\(\sim q\)
    Conclusion:\(∴\sim p\)
  • Know how to apply the law of denying the consequent to determine the conclusion for pairs of statements.
  • The chain rule for conditional arguments is a valid form of a conditional argument that asserts that if the premises of the argument have the form, \(p\to q\) and \(q\to r\), then it follows that \(p\to r.\)
    Chain Rule for Conditional Arguments
    Premise:\(p\to q\)
    Premise:\(q\to r\)
    Conclusion:\(∴p\to r\)
  • Know how to apply the chain rule to determine valid conclusions for pairs of true statements.

Practice (3)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Each pair of statements represents the premises in a logical argument. Based on these premises, apply the law of detachment to determine a valid conclusion.

    1. If Leonardo da Vinci was an artist, then he painted the Mona Lisa. Leonardo da Vinci was an artist.
    2. If Michael Jordan played for the Chicago Bulls, then Michael Jordan was not a soccer player. Michael Jordan played for the Chicago Bulls.
    3. If all fish have gills, then clown fish have gills. All fish have gills.
    גלה את התשובה
    1. The premises are \(p\to q:\) If Leonardo da Vinci was an artist, then he painted the Mona Lisa, and \(p:\) Leonardo da Vinci was an artist. This argument has the form of the law of detachment, so, the conclusion is \(q:\) Leonardo da Vinci painted the Mona Lisa.
    2. The premises follow the form of the law of detachment, so a valid conclusion would be \(q\). The premises are \(p\to q:\) If Michael Jordan played for the Chicago Bulls, then Michael Jordan was not a soccer player, and \(p:\) Michael Jordan played for the Chicago Bulls. The conclusion that follows from the premises is \(q:\) Michael Jordan was not a soccer player.
    3. The premises are \(p\to q:\) If all fish have gills, then clown fish have gills, and \(p:\) All fish have gills. This argument has the form of the law of detachment, so the conclusion is \(q:\) clown fish have gills.

  2. Each pair of statements represents the premises in a logical argument. Based on these premises, apply the law of denying the consequent to determine a valid conclusion.

    1. If Leonardo da Vinci was an artist, then he painted the Mona Lisa. Leonardo da Vinci did not paint the Mona Lisa.
    2. If Michael Jordan played for the Chicago Bulls, then Michael Jordan was not a soccer player. Michael Jordan was a soccer player.
    3. If all fish have gills, then clown fish have gills. Clown fish do not have gills.
    גלה את התשובה
    1. The premises are \(p\to q:\) If Leonardo da Vinci was an artist, then he painted the Mona Lisa, and \(\sim q:\) Leonardo da Vinci did not paint the Mona Lisa. This argument has the form of the law of denying the consequent, so the conclusion is \(\sim p:\) Leonardo da Vinci was not an artist.
    2. The premises follow the form of the law of denying the consequent, so a valid conclusion would be \(\sim p\). The premises are: \(p\to q:\) If Michael Jordan played for the Chicago Bulls, then Michael Jordan was not a soccer player, and \(\sim q:\) Michael Jordan was a soccer player. The conclusion that follows from the premises is \(\sim p:\) Michael Jordan did not play for the Chicago Bulls.
    3. The premises are \(p\to q:\) If all fish have gills, then clown fish have gills, and \(\sim q:\) Clown fish do not have gills. This argument has the form of the law denying the consequent, so the conclusion is \(\sim p:\) Some fish do not have gills.

  3. Each pair of statements represents true premises in a logical argument. Based on these premises, apply the chain rule for conditional arguments to determine a valid and sound conclusion.

    1. If my roommate goes to work, then my roommate will get paid. If my roommate gets paid, then my roommate will pay their bills.
    2. If robins can fly, then some birds can fly. If some birds can fly, then we will watch birds fly.
    3. If Irma is a teacher, then Irma has a college degree. If Irma has a college degree, then Irma graduated from college.
    גלה את התשובה
    1. The premises are \(p\to q:\) “If my roommate goes to work, then they will get paid,” and \(q\to r:\) “If my roommate gets paid, then my roommate will pay their bills.” This argument has the form of the chain rule for conditional arguments, so the valid conclusion will have the form “\(p\to r.\)” Because all the premises are true, the valid and sound conclusion of this argument is: “If my roommate goes to work, then my roommate will pay their bills.”
    2. The premises are \(p\to q:\) “If robins can fly, then some birds can fly,” and \(q\to r:\) “If some birds can fly, then we will watch them fly.” This argument has the form of the chain rule for conditional arguments, so, the valid conclusion will have the form “\(p\to r.\)” Because all the premises are true, the valid and sound conclusion of this argument is: “If robins can fly, then we will watch birds fly.”
    3. The premises are \(p\to q:\) (see line 1 of solution 1 and 2 above) “If Irma is a teacher, then Irma has a college degree,” and \(q\to r:\) “If Irma has a college degree, then Irma graduated from college.” This argument has the form of the chain rule for conditional arguments, so the valid conclusion will have the form “\(p\to r.\)” Because all the premises are true, the valid and sound conclusion of this argument is: “If Irma is a teacher, then Irma graduated from college.”

Symbols used here

n!
factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.
\binom{n}{k}
binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
x \in A,\ A \subseteq B
element of, subset
x belongs to A; every element of A is in B.
A \cup B,\ A \cap B,\ A \setminus B
union, intersection, difference
In either; in both; in A but not B.
\emptyset,\ |A|
empty set, cardinality
The set with no elements; the number of elements of A.
\forall,\ \exists
for all, there exists
Quantifiers: every x; at least one x.
\neg,\ \wedge,\ \vee,\ \Rightarrow,\ \Leftrightarrow
not, and, or, implies, iff
Logical connectives.
\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.
a \equiv b \pmod n
congruent modulo n
n divides a − b; a and b have the same remainder.
O(n^2),\ \Theta,\ \Omega
big-O notation
Grows no faster than n² (up to a constant), for large n.
a \bmod n
remainder
What is left after dividing a by n.

How to: Logical Arguments

  1. Apply the law of detachment to determine the conclusion of a pair of statements.
  2. Apply the law of denying the consequent to determine the conclusion for pairs of statements.
  3. Apply the chain rule to determine valid conclusions for pairs of true statements.
  4. If Leonardo da Vinci was an artist, then he painted the
  5. If Michael Jordan played for the Chicago Bulls, then Michael Jordan was not a soccer player. Michael Jordan played for the Chicago Bulls.
  6. If all fish have gills, then clown fish have gills. All fish have gills.
  7. The premises are
  8. The premises follow the form of the law of detachment, so a valid conclusion would be

Questions people ask

What makes mathematics "discrete"?

It deals with separate, countable objects — integers, graphs, statements — rather than continuous quantities. No limits, no infinitesimals; instead induction, counting and logic.

How does a proof by induction work?

Show the statement for the first case, then show that whenever it holds for n it holds for n + 1. Like dominoes: the first falls, and each knocks over the next.

נסה את שלך.

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

יותר בפנים. Discrete Math & Logic

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