Deductive Arguments

A deductive argument is an argument from premises
to a logical consequence

Table of Contents

  1. Deductive Argument
  2. Example
  3. Aristotle’s Insight
  4. Validity
  5. Ways of Proving Validity
    1. Citing a Recognized Valid Form of Inference
    2. Deriving the Conclusion from the Premises
  6. Ways of Proving Invalidity
    1. Direct Counterexample
    2. Refutation by Logical Analogy
    3. Citing Recognized Invalid Form of Inference
  7. Evaluating Deductive Arguments
    1. Procedure
    2. Establishing the conclusion
    3. Examples of Establishing and Failing to Establish the Conclusion
  8. Exercises on Establishing the Conclusion
    1. Most US citizens were not born in the US
    2. Argument that 1 = 0.9999999…(ad infinitum)
    3. Abortion
    4. Kalam Cosmological Argument
  9. Addenda
    1. Deductive Logic
    2. Syllogisms and Venn Diagrams
    3. Technical Caveat
    4. Valid Deductive Arguments, Examples
Deductive Argument
  • A deductive argument is an argument from premises to a logical consequence, that is, an argument whose conclusion purportedly follows necessarily from its premises (purportedly allowing for bad deductive arguments).
Example
  • Argument:
    • John Oliver isn’t eligible to be president because he isn’t a natural-born U.S. citizen.
  • Argument Reconstructed:
    1. Only natural-born U.S. citizens are eligible to be president.
    2. John Oliver isn’t a natural-born U.S. citizen.
    3. Therefore, he’s not eligible to be president.
  • The argument is a valid deductive argument since the conclusion, that Oliver’s not eligible to be president, follows necessarily from the premises:
    • Only natural-born U.S. citizens are eligible to be president.
    • John Oliver isn’t a natural-born U.S. citizen.
  • Or, equivalently, it’s logically impossible (or necessarily false or a contradiction in terms) that:
    • Only natural-born U.S. citizens are eligible to be president.
    • John Oliver isn’t a natural-born U.S. citizen
    • Yet, he’s still eligible to be president.
Aristotle’s Insight
  • Compare:
    • Oliver Argument
      1. Only natural-born U.S. citizens are eligible to be president.
      2. John Oliver isn’t a natural-born U.S. citizen.
      3. Therefore, he’s not eligible to be president.
    • Damon Argument
      1. Only natural-born U.S. citizens are eligible to be president.
      2. Matt Damon isn’t a natural-born U.S. citizen.
      3. Therefore he’s not eligible to be president.
  • The Oliver Argument establishes that John Oliver is not eligible to be president. But the Damon argument fails to prove the same thing for Matt Damon, since the second premise is false.
  • Aristotle’s insight was that, even so, the arguments have the same valid logical form:
    • Only Ns are Es
    • Object O is not an N.
    • So, object O is not an E.
Validity
  • A deductive argument is valid if its conclusion follows necessarily from the premises, no matter the truth of the premises.
    • That is, if it’s logically impossible (or necessarily false or a self-contradiction) for the premises to be true and the conclusion false.
  • The John Oliver argument is valid because it’s self-contradictory that:
    1. Only natural-born U.S. citizens are eligible to be president
    2. John Oliver isn’t a natural-born U.S. citizen
    3. Yet John Oliver is eligible to be president.
  • The Damon argument is likewise valid, since it’s self-contradictory that:
    1. Only natural-born U.S. citizens are eligible to be president
    2. Matt Damon isn’t a natural-born U.S. citizen
    3. Yet Damon is eligible to be president.
  • Indeed, any argument of this form is valid:
    • Only Ns are Es
    • Object O is not an N.
    • So, object O is not an E.
  • For example:
    • Only beings with brains are conscious..
    • The spider running across the floor lacks a brain.
    • So, the spider isn’t conscious.
Ways of Proving Validity
Citing a Recognized Valid Form of Inference
  • Refutation of Newton’s Theory of Gravitation:
    • If Newton’s Theory of Gravitation were true, Mercury’s orbit would precess 5,557 seconds of arc per century.
    • Mercury’s orbit does not precess 5,557 seconds of arc per century.
    • Therefore, Newton’s Theory of Gravitation is false.
  • Valid Form of Inference
    • The argument is valid because it has the valid argument form modus tollens:
      • T → M
      • ~M
      • So, ~T
    • Where
      • T = Newton’s Theory of Gravitation is true
      • M = Mercury’s orbit precess 5,557 seconds of arc per century
      • → = if … then
      • ~ = it’s false that

View Valid Forms of Inference

Deriving the Conclusion from the Premises
  • Edwin Hubble’s argument the universe extends far beyond the Milky Way Galaxy:
    1. The universe consists either of numerous galaxies including the Milky Way or of the Milky Way Galaxy alone.
    2. If the universe consists merely of the Milky Way Galaxy, the Andromeda Nebula would not be far more distant from Earth than the stars in the Milky Way.
    3. The Andromeda Nebula (as Hubble discovered) is far more distant from Earth than the stars in the Milky Way.
    4. Therefore, the universe consists of numerous galaxies including the Milky Way.
  • Derivation
    • Let:
      • G = the universe consists of numerous galaxies
      • M = the universe consists of the Milky Way Galaxy alone
      • A = the Andromeda Nebula is far more distant from Earth than the stars in the Milky Way.
    • Argument in Symbols:
      1. G v M
        • v = or
      2. M → ~A
        • → = if … then
        • ~ = it’s false that
      3. A
      4. So, G
    • Derivation
      • ~M follows from #2 and #3 by modus tollens, the valid argument form:
        • A→B
        • ~B
        • So, ~A
      • G follows from ~M and #1 by disjunctive syllogism, the valid argument form:
        • A v B
        • ~A
        • So, B
Ways of Proving Invalidity
  • The Argument
    • Had Saddam Hussein been responsible for the 9/11 attacks, the U.S. invasion of Iraq would have been morally justified.
    • But Hussein was not responsible for the 9/11 attacks.
    • Therefore, the U.S. invasion of Iraq was not justified.
Direct Counterexample
  • A direct counterexample is a logically consistent scenario where the premises are true and the conclusion false. 
  • The Hussein argument is invalid because of the logical possibility that:
    • Hussein was not responsible for the 9/11 attacks 
    • Instead Hussein instead directly attacked the United States, justifying the US invasion.
Refutation by Logical Analogy
  • Refutation by Logical Analogy is showing an argument invalid by identifying an obviously invalid argument of the same logical form.
  • For example:
    • If Matt Damon were over seven feet tall, he would be over five feet tall.
    • But Damon is not over seven feet tall
    • Therefore, he’s not over five feet tall.
Citing Recognized Invalid Form of Inference
  • An argument can be shown invalid by identifying its invalid form of argument.
  • Fallacy of Denying the Antecedent
    • R → J
    • ~R
    • So, ~J
  • Where
    • R = Saddam Hussein was responsible for the 9/11 attacks
    • J = U.S. invasion of Iraq was morally justified
    • → = if then
    • ~ = it’s false that
Evaluating Deductive Arguments
Procedure
  1. Formulate the argument so its premises, conclusion, and logic are clear
    • Example:
      • John Oliver isn’t eligible to be president because he’s not a natural-born U.S. citizen.
        1. Only natural-born U.S. citizens are eligible to be president.
        2. John Oliver isn’t a natural-born U.S. citizen.
        3. Therefore, he’s not eligible to be president.
    • View Argument Reconstruction
  2. Determine whether premises are beyond a reasonable doubt.
  3. Determine whether the argument is valid
    1. View Validity
    2. View Ways of Proving Validity
    3. View Ways of Proving Invalidity
    4. View Deductive Fallacies
Establishing the conclusion
  • A deductive argument establishes its conclusion if it’s valid and its premises are beyond a reasonable doubt
    • Establish means to prove or make acceptable beyond a reasonable doubt
      • Merriam-Webster Unabridged
    • View Technical Caveat
Examples of Establishing and Failing to Establish the Conclusion
  • Example 1:
    • Argument
      1. If Newton’s Theory of Gravitation were true, Mercury’s orbit would precess 5,557 seconds of arc per century.
      2. Mercury’s orbit does not precess 5,557 seconds of arc per century.
      3. Therefore, Newton’s Theory of Gravitation is false.
    • Analysis
      • Premises are beyond a reasonable doubt.
      • Argument is valid.
      • Therefore the argument establishes its conclusion.
  • Example 2
    • Argument
      1. Consciousness requires a functioning brain.
      2. A person’s brain ceases to function when they die.
      3. Therefore, a person ceases to be conscious when they die.
    • Analysis
      • First premise is disputed. Second premise is true.
      • Argument is valid
      • So, the argument does not establish conclusion
  • Example 3
    • Argument
      1. Had Saddam Hussein been responsible for the 9/11 attacks, the U.S. invasion of Iraq would have been morally justified.
      2. But Hussein was not responsible for the 9/11 attacks.
      3. Therefore, the U.S. invasion of Iraq was not justified.
    • Analysis
      • First premise seems true. Second premise is true.
      • Argument is invalid
      • Therefore, the argument does not establish its conclusion.
Exercises on Establishing the Conclusion

Do these deductive arguments establish their conclusions? If not, why not?

Most US citizens were not born in the US
  • Most US citizens speak English.
  • Most people who speak English were not born in the US.
    • Britannica: As of 2020 there are 1.27 billion English speakers around the world
  • Therefore, most US citizens were not born in the US.

View Answer

Argument that 1 = 0.9999999…(ad infinitum)
  1. ⅓ = 0.3333333…(ad infinitum)
    • self-evident
  2. Therefore, 3 x ⅓  = 3 x 0.3333333…
    • multiplying both sides of line #1 by 3
  3. 1 = 3 x ⅓
    • self-evident
  4. 3 x 0.3333333… = 0.9999999…
    • self-evident
  5. Therefore, 1 = 0.9999999…
    • from lines #2, #3, and #4

View Answer

Abortion
  • Abortion should be legal because abortions will be performed whether legal or not.

View Answer

Kalam Cosmological Argument
  • Whatever begins to exist has a cause
  • The universe began to exist.
  • Therefore, the universe has a cause
  • If the universe has a cause, an uncaused, personal creator of the universe exists
  • Therefore, an uncaused, personal creator of the universe exists.

View Answer

Addenda
Deductive Logic
  • Deductive Logic is the formal (symbolic) theory of deductive reasoning.

View Deductive Logic

Syllogisms and Venn Diagrams
  • A syllogism is two-premise argument whose premises and conclusion have the forms:
    • Every A is a B
    • No A is a B
    • At least one A is a B.
    • At least one A is not a B.
  • Venn Diagrams are used to determine whether syllogisms are valid.

View Syllogisms and Venn Diagrams

Technical Caveat
  • Here’s a counterexample to the principle that an argument establishes its conclusion if it’s valid and the premises are beyond a reasonable doubt.
  • A judge conducts 10,000 trials where the defendant is guilty beyond a reasonable doubt, not just found guilty.  Consider the lengthy argument:
    • Defendant 1 is guilty.
    • Defendant 2 is guilty.
    • …..
    • Defendant 10,000 is guilty.
    • Therefore, defendants 1 through 10,000 are guilty.
  • Suppose the judge learns that DNA evidence has exonerated one of the defendants, but she doesn’t know which one. The conclusion is thus false.  Yet each premise remains true beyond a reasonable doubt, since one innocent defendant isn’t enough to undermine all 10,000 cases.  (This is a variation of the Lottery Paradox)
  • Therefore,
    • The lengthy argument is valid
    • Each premise is beyond a reasonable doubt
    • The conclusion is not beyond a reasonable doubt.
  • It’s thus false that:
    • an argument establishes its conclusion if it’s valid and the premises are beyond a reasonable doubt.
  • The counterexample can be circumvented in two ways.
  • First Way:
    • Add other things being equal to the principle:
      • An argument establishes its conclusion if, other things being equal, it’s valid and the premises are beyond a reasonable doubt.
  • Second Way:
    • The phrase the premises are beyond a reasonable doubt is ambiguous.
    • It may be understood distributively, meaning that each premise is beyond a reasonable doubt.
      • In this case the counterexample is successful.
    • But it may also be understood collectively, so that the principle becomes:
      • An argument establishes its conclusion if it’s valid and it’s beyond a reasonable doubt that all the premises are true.
Valid Deductive Arguments, Examples
  • Cheating is wrong because there’s no real difference between cheating and surreptitiously changing your grade in the instructor’s spreadsheet.
    • There’s no real difference between cheating and surreptitiously changing your grade on the instructor’s spreadsheet.
    • Changing your grade on the instructor’s spreadsheet is wrong.
    • Therefore, cheating is wrong.
  • Consciousness requires a functioning brain. But a person’s brain ceases to function when they die. Therefore, a person ceases being conscious when they die.
    • Consciousness requires a functioning brain.
    • A person’s brain ceases to function when they die.
    • Therefore, a person ceases to be conscious when they die.
  • “Death is the only final and irreversible criminal punishment.  As DNA exonerations vividly show, humans and their governments are fallible and corruptible.  Fallible governments should refrain from inflicting irreversible punishments.” (NY Times)
    • Fallible governments should refrain from inflicting irreversible punishments
    • Capital punishment is an irreversible punishment.
    • All governments are fallible.
    • Therefore, governments should refrain from inflicting capital punishment
  • Proof that the sum of two odd integers is (always) even