Deductive Arguments

A deductive argument is an argument whose conclusion follows necessarily from its premises

Contents

Deductive Arguments

A deductive argument is an argument whose conclusion (purportedly) follows necessarily from its premises in the sense that it’s logically impossible that its premises are true and conclusion false.
- “Logically impossible” is used in the sense of contradiction, in the sense, for example, that it’s impossible that all swans are white though some are not.
- “Purportedly” allows for bad deductive arguments.

A simple deductive argument:
- Only citizens can vote.
- He’s not a citizen.
- Therefore, he can’t vote.
The conclusion follows necessarily from the premises in the sense that the negation of the conclusion contradicts the premises:
- Only citizens can vote.
- He’s not a citizen.
- He can vote.

Two Forms

There are two forms of deductive argument.
- The basic form consists of premises and conclusion.
- The enhanced form consists of premises, conclusion, and a derivation of the conclusion from the premises.

Basic Form, Examples

From the third-century Greek philosopher Sextus Empiricus:
- Everything human is an animal.
- Socrates is human.
- Therefore Socrates is an animal.
Why John Oliver, an American citizen, can’t be president:
- Only natural-born U.S. citizens are eligible to be president
- John Oliver isn’t a natural-born U.S. citizen
- Therefore, John Oliver is not eligible to be president.
Argument that Newton’s Theory of Gravitation is wrong:
- If Newton’s Theory of Gravitation were correct, 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.
  - The observed precession is 5599.7.
- Therefore, Newton’s Theory is false.
The Greeks discovered that the Morning Star is the Evening Star:
- The Morning Star is the planet Venus.
- The Evening Star is the planet Venus.
- So the Morning Star is the Evening Star.
An argument there’s no afterlife:
- 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.
Edwin Hubble inferred that the universe is more than just the Milky Way from his observation of a star in the Andromeda Nebula.
- The universe is either a myriad of galaxies (including the Milky Way) or the Milky Way Galaxy alone.
- If the universe is the Milky Way Galaxy alone, the Andromeda Nebula would be no further from Earth than the stars in the Milky Way.
- The Andromeda Nebula is much further from Earth than the stars in the Milky Way.
- Therefore, the universe is a myriad of galaxies.

Enhanced Form, Example

Proof that the sum of two odd integers is even.

Premises
- An integer is even if it’s two times some integer.
- An integer is odd if it’s two times some integer + 1.
Derivation
1. Let x and y be two odd integers.
2. Then, x = 2a + 1, for some integer a.
  - Definition of odd integer
3. And y = 2b + 1, for some integer b.
  - Definition odd integer
4. So x + y = 2a + 1 + 2b + 1.
5. Thus, x + y = 2a + 2b + 2.
  - Rearranging line 4
6. And so x + y = 2(a + b + 1).
  - Factoring line 5
7. Thus x + y = two times some integer (a + b + 1) and is therefore even.
  - Definition of even integer.

View Additional Proofs.

Validity and Soundness

A deductive argument is valid if it’s logically impossible that its premises are true and conclusion false.
A deductive argument is sound if it’s valid and all its premises are true (and so therefore is its conclusion).
The John Oliver argument above is valid and sound.
- Only natural-born U.S. citizens are eligible to be president
- John Oliver isn’t a natural-born U.S. citizen
- Therefore, John Oliver is not eligible to be president.
This argument that Matt Damon can’t be president is valid and unsound.
- Only natural-born U.S. citizens are eligible to be president
- Matt Damon isn’t a natural-born U.S. citizen
- Therefore, Matt Damon is not eligible to be president.
The following argument is invalid (and therefore automatically unsound).
- All natural-born U.S. citizens are eligible to be president
- John Oliver isn’t a natural-born U.S. citizen
- Therefore, John Oliver is not eligible to be president.
All six basic arguments above are valid. All but one are sound; whether consciousness requires a functioning brain is up for debate.

Three Ways of Defining Validity

Modality
- An argument is valid if it’s logically impossible that its premises are true and conclusion false. Or, alternatively, if the conclusion necessarily follows from the premises.
Form
- An argument is valid if the form of the argument is such that no argument of that form has true premises and a false conclusion.
Derivation
- An argument is valid if the conclusion can be derived from the premises using basic valid forms of inference.
- View Basic Valid Argument Forms.

Example:
- If he robbed the bank he wasn’t in jail at the time.
- He was in jail at the time of the robbery.
- So he didn’t rob the bank.
Modal Validity
- The argument is valid because the following propositions are logically incompatible:
  - If he robbed the bank he wasn’t in jail at the time.
  - He was in jail at the time of the robbery.
  - He robbed the bank.
Formal Validity
- The argument is valid since no argument of this form has true premises and a false conclusion:
  - If A then B
  - It’s false that B
  - So it’s false that A
Derivational Validity
- The argument is valid because the conclusion can be derived from the premises:
  1. If he robbed the bank he wasn’t in jail at the time.
    - Premise
  2. He was in jail at the time of the robbery.
    - Premise
  3. He didn’t rob the bank.
    - Follows from lines 1 and 2 by modus tollens:
      - If A then B
      - It’s false that B
      - So it’s false that A

Evaluating Deductive Arguments

Argument evaluation is the three step process of argument reconstruction, determining whether the premises are true, and assessing how strongly the premises support the conclusion.
Assessing the evidential support of a deductive argument is a binary affair. A deductive argument is either valid (and the conclusion follows from the premises) or invalid (and the conclusion doesn’t follow from the premises).

Proving an Argument Valid or Invalid

An argument can be proved valid by
1. citing a recognized valid form of Inference,
2. deriving the conclusion from the premises.
An argument can be proved invalid by
1. a direct counterexample,
2. refutation by logical analogy,
3. citing a recognized invalid form of inference.

Proving Validity

1. Citing a Recognized Valid Form of Inference

Argument that Newton’s Theory of Gravitation is wrong:
- 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.
The argument has the valid argument form modus tollens:
- If A then B
- It’s false that B
- So, it’s false that A.
View Basic Valid Argument Forms.

2. Deriving the Conclusion from the Premises

Hubble’s argument that the universe extends beyond the Milky Way.
1. The universe is either a myriad of galaxies (including the Milky Way) or the Milky Way Galaxy alone.
  - Premise
2. If the universe is the Milky Way Galaxy alone, the Andromeda Nebula would be no further from Earth than the stars in the Milky Way.
  - Premise
3. The Andromeda Nebula is much further from Earth than the stars in the Milky Way.
  - Premise
4. The universe is not the Milky Way Galaxy alone
  - From lines 2 and 3 by modus tollens, the argument form:
    - If A then B
    - It’s false that B
    - So, it’s false that A.
  - View Basic Valid Argument Forms.
5. Therefore, the universe is a myriad of galaxies.
  - From lines 1 and 4 by disjunctive syllogism, the argument form:
    - A or B
    - It’s false that A
    - So, B.
  - View Basic Valid Argument Forms.

Proving Invalidity

Consider 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.
Three ways of proving the argument invalid.

1. Direct Counterexample

A direct counterexample is a logically consistent scenario where the premises are true and the conclusion false.
Suppose the following are true:
- Had Saddam Hussein been responsible for the 9/11 attacks, the U.S. invasion of Iraq would have been morally justified.
- Hussein was not responsible for the 9/11 attacks.
- But Hussein directly attacked the United States.
Then the U.S. invasion of Iraq would have been justified, contradicting the conclusion.

2. Refutation by Logical Analogy

Refutation by Logical Analogy is proving that an argument is invalid by identifying an obviously invalid argument of the same form.
The Hussein argument has the same form as this absurd argument:
- If Matt Damon were over seven feet tall, he would be over two feet tall.
- Damon is not over seven feet tall.
- Therefore, he’s not over two feet tall.

3. Citing a Recognized Invalid Form of Inference

An argument can be shown invalid by identifying its invalid form of argument.
The Hussein argument is invalid because it’s an instance of the fallacy of Denying the Antecedent, the argument form:
- If A then B
- It’s false that A
- So, it’s false that B.
View Deductive Fallacies

Validity and Epistemic Probability

The conclusion of a valid argument follows necessarily from its premises.
But what does this mean for a person using the argument to establish that something is true?
It depends on the person’s epistemic relation to the the premises.
Suppose, based on the person’s evidence, that it’s certain that the premises true. It seems reasonable, then, to infer that it’s certain that the conclusion is true.
But what if, based on the evidence, it’s beyond a reasonable doubt that the premises are true but not certain that they are?
- For example, it might be beyond a reasonable doubt that my car is where I parked it an hour ago though not absolutely certain.
- View Epistemic Probability
You might think that if the premises of a valid argument are beyond a reasonable doubt so too is the conclusion.
Suppose, however, that an argument P₁, P₂, … P_n therefore C is valid. And suppose that it’s beyond a reasonable doubt that P₁ is true, that P₂ is true and so on through P_n. It’s possible, nonetheless, that it’s not beyond a reasonable doubt that C is true. That the premises of a valid argument are all beyond a reasonable doubt does not logically entail the conclusion is beyond a reasonable doubt.

A Counterexample:

A judge conducts 10,000 trials where the defendant is (actually) guilty beyond a reasonable doubt (and not just found guilty). Consider the lengthy argument:
- Defendant 1 is guilty.
- Defendant 2 is guilty.
- …..
- Defendant 10,000 is guilty.
- Therefore, all the defendants, 1 through 10,000, are guilty.
The argument is valid.
Suppose the judge learns that DNA evidence has exonerated one of the defendants, but she doesn’t know which one. She reviews every trial in light of the 1/10,000 possibility that the defendant is innocent and, in each case, confirms her opinion that the defendant is guilty beyond a reasonable doubt. Therefore:
- The conclusion is false.
- At least one premise is false.
- The argument is valid.
- Based on what the judge knows, each premise is beyond a reasonable doubt.
- Based on what the judge knows, the conclusion is not beyond a reasonable doubt.
Thus it’s not logically necessary that:
- If the premises of a valid deductive argument are beyond a reasonable doubt, the conclusion is beyond a reasonable doubt.

Logical Modalities

Deductive validity belongs to a family of modal notions that characterize propositions:
- the properties of possibility, necessity, impossibility, and contingency;
- the relations of entailment, incompatibility, equivalence, and subcontrariety.

Necessity, Possibility, Impossibility, and Contingency

A proposition is logically necessary if its negation implies a contradiction, e.g. everything is what it is and not something else.
A proposition is logically impossible if it’s a self-contradiction, e.g. some people are taller than themselves.
A proposition is contingently true if it’s true and not logically necessary, e.g. some molecules have three atoms.
A proposition is logically possible if it’s not logically impossible, e.g. some human beings are over 15 feet tall.
A proposition is contingently false if it’s false and logically possible.
A proposition is contingent if it’s neither logically necessary nor logically impossible.

Entailment, Incompatibility, Equivalence, and Subcontrariety

A set S of propositions logically entails a proposition Q (or Q is a logical consequence of S) if it’s logically impossible that the propositions of S are all true and Q false.
- An argument is valid if its conclusion is a logical consequence of its premises.
Two propositions are logically equivalent if each logically entails the other.
- View Basic Logical Equivalences.
Two propositions are logically contradictory if it’s logically impossible that they are both true and logically impossible that they are both false.
A set of propositions is logically incompatible if it’s logically impossible that they are all true.
A set of propositions is logically subcontrary if it’s logically impossible that they are all false.

Addendum

Basic Valid Argument Forms

Basic Logical Equivalences

Syllogisms

A syllogism is a deductive argument that has two premises and conclusion, each of which has one of the forms:
- All S are P
- No S are P
- Some S are P
- Some S is not P
For example;
- All M are A.
- All D are M.
- So, all D are A.

Valid Syllogisms

Venn Diagrams can be used to determine whether a syllogism is valid or not.
- View Syllogisms and Venn Diagrams

Formalization of Validity in Deductive Logic

Deductive Logic, the formal theory of deductive reasoning, uses both derivational and formal validity to “formalize” the concept of validity.
Take Propositional Logic, for example, which uses (i) capital letters to stand for statements and (ii) the symbols &, ~, v, and → to mean “and”, “not”, “or”, and “if then” respectively.
Derivational validity:
- An argument in propositional logic is valid if the conclusion can be derived from the premises using a specific list of rules of inference such as:
  - Modus ponens: Infer Q from P → Q and P.
  - Disjunctive syllogism: infer Q from P v Q and ~P.
  - View Basic Valid Argument Forms.
Formal validity:
- An argument in propositional logic is valid if there is no way of assigning the truth values T and F to its sentence letters that makes the premises true and the conclusion false.
- For example there is no way of assigning T and F to P and Q that makes
  - P v Q true
  - ~P true
  - Q false.
Not surprisingly, the formally valid arguments are the same as the derivationally valid arguments.

Are these arguments valid?

Most US citizens speak English.
Most people who speak English were not born in the US.
Therefore, most US citizens were not born in the US.

View Answer

⅓ = 0.3333333…(ad infinitum)
Therefore, 3 x ⅓ = 3 x 0.3333333…(ad infinitum)
1 = 3 x ⅓
3 x 0.3333333… = 0.9999999…(ad infinitum)
Therefore, 1 = 0.9999999…(ad infinitum)

View Answer

Love is blind.
God is love.
Ray Charles is blind.
Therefore, Ray Charles is God.

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Everything ceases to exist at some time or other.
Therefore, there is a time when nothing exists.

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No integer is greater than itself.
Therefore, no integer is greater than all integers.

View Answer

Total pacifism is a good principle if every country practices it.
But not every country does.
So it isn’t.

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Living human beings have brains.
Human zygotes lack brains.
Therefore, human zygotes aren’t living human beings.

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Mark is eligible for paid vacation only if he has been employed for at least 90 days.
Mark has been employed for 90 days.
He’s therefore eligible for paid vacation.

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The defendant’s statement and the plaintiff’s testimony cannot both be true.
The defendant’s statement is false.
Therefore the plaintiff’s testimony is true.

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Some people speak both French and German.
Some people speak both German and Italian.
Therefore some people speak both French and Italian.

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Sam speaks French or German.
Sam speaks German or Italian.
Therefore, Sam speaks French or Italian.

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There will be no bonus unless the company makes a profit.
The company will not make a profit.
Therefore, there will be no bonus.

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Without both a driver’s license and a credit card, they won’t cash your check.
You have a driver’s license but you don’t have a credit card.
Therefore, they won’t cash your check.

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Not everyone who’s wealthy is conservative.
Not everyone who’s conservative is a Republican.
Therefore, not everyone who’s wealthy is a Republican.

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All and only conscious beings have souls.
Dogs do not have souls.
Therefore, dogs are not conscious beings.

View Answer

The Theory of Evolution and Genesis cannot both be true.
If the Bible is the inerrant word of God, Genesis is true.
There’s strong empirical evidence that the Theory of Evolution is true.
Therefore, there’s strong empirical evidence that the Bible is not the inerrant word of God.

View Answer

If human beings have free will, conscious thoughts can alter the course of events in the brain.
If conscious thoughts can alter the course of events in the brain, telekinesis is real. But telekinesis isn’t real.
Therefore, human beings don’t have free will.

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Most college graduates have well-paying jobs.
Therefore, most people who have well-paying jobs are college graduates.

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If the Bible is the word of God, then every statement in the Bible is true.
If every statement in the Bible is true, Jesus of Nazareth was crucified about 9:00 AM on Passover (Mark 15:25).
If every statement in the Bible is true, Jesus of Nazareth was crucified about noon the day before Passover (John 19:14).
It can’t both be true that Jesus of Nazareth was crucified about 9:00 AM on Passover and about noon the day before Passover.
Therefore the Bible is not the word of God.

View Answer

If the creation story in Genesis is a true literal description, then for the first three days of the earth’s existence there was no sun.
There are no days and nights if there is no sun.
Therefore, the creation story in Genesis is not a true literal description.

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If the theory of evolution were true, there would be no gaps in the fossil record.
There are gaps in the fossil record.
Therefore, the theory of evolution is false.

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If there’s free will, people can control some of their physical movements.
A person can control a physical movement only if they can control the events that initiated it.
Physical movements are initiated by neurons firing in the motor cortex of the brain.
People can’t control neurons firing in the motor cortex of their brains.
Hence, there’s no free will.

View Answer

If Earth rotated on its axis, there would be constant hurricane-force winds at the equator.
No such winds exist.
Therefore, Earth does not rotate on its axis.

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Any being that knows beforehand that innocent children will suffer and die and who could have easily prevented the suffering and dying but lets the suffering and dying happen anyway is morally reprehensible.
Natural disasters sometimes happen that result in the suffering and death of innocent children.
If there is a supreme being, that being knows beforehand that such disasters will happen (since that being is omniscient).
If there is a supreme being, that being can easily prevent such disasters from happening (since that being is omnipotent).
If there is a supreme being, that being is not morally reprehensible (since that being benevolent).
Therefore, there is no supreme being.

View Answer

For everything that begins to exist there is something else that caused it to begin to exist.
The universe began to exist.
Therefore, there is something other than the universe that caused it to exist.

View Answer

Answers

Most US citizens speak English
- Invalid
- Refutation by Logical Analogy
  - Most Texans are US citizens (true)
  - Most US citizens do not live in Texas (true)
  - Therefore, most Texans do not live in Texas (false

Go Back

1 = 0.9999999
- Valid
- The derivation is a mathematical proof that 1 = 0.9999999…
- Statements #1, #3, and #4 are self-evident premises.
- Statement #2 is derived from #1 using the principle that
  - if a = b and both a and b are multiplied by the same number n, then n x a = n x b.
- Statement #4 is derivied from #2, #3, and #4 by the transitivity of identity:
  - a = b
  - b = c
  - Therefore a = c

Go Back

Ray Charles is blind
- Invalid
- There are multiple senses of the verb to be:
  - View Multiple Senses of To Be
- The Ray Charles argument is valid only if the is of identity is used throughout the argument. But the first two premises use the is of predication:
- But not all premises of the Ray Charles argument use the is of identity:
  - Ray Charles is blind, i.e. has the condition of being blind
  - Love is blind, i.e. has the condition of being blind
  - God = Love.
  - Therefore, Ray Charles = God.
- Asserting that Ray Charles and love are blind ascribes to each the condition of blindness; it doesn’t say they’re identical to that condition. By contrast, the final premise and conclusion use the is of identity, asserting God is the same thing as Love and that Ray Charles is the same thing as God. The argument is thus invalid, having same form as the argument:
  - Ernest Hemingway was American-born.
  - Mark Twain was American-born.
  - Samuel Clemens = Mark Twain.
  - Therefore Ernest Hemingway = Samuel Clemens

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Everything ceases to exist at some time or other
- Invalid
- The argument can be rephrased:
  - Everything is such that there is a time when it no longer exists.
  - Therefore, there is a time such that everything no longer exist at that time.
- Refutation by logical analogy:
  - Every integer is such that there is an integer greater than it. (true)
  - There is an integer such that every integer is greater than it. (false)
- The argument, translated into predicate logic, is invalid:
  - (x)(Ey)Gyx
  - Therefore, (Ey)(x)Gyx
    - where Gyx = integer y is greater than integer x.

Go Back

No integer is greater than itself
- Valid
- If there’s an integer greater than all integers, then it’s greater itself.
- But no integer is greater than itself, per the first premise.
- Therefore, my modus tollens, there’s no integer greater than all integers.
- The argument is valid in predicate logic, where G means greater than:
  - ~(Ex)Gxx
  - ~(Ex)(y)Gxy

Go Back

Total pacifism is a good principle
- Invalid
- Fallacy of Affirming the Consequence
- Invalid argument form
  - A → B
  - B
  - Therefore, A.

Go Back

Living human beings have brains
- Valid
- If human zygotes were living human beings they would both have brains (because they’re human beings) and not have brains (because they’re human zygotes). But that’s impossible. So human zygotes aren’t living human beings.
- The argument in predicate logic:
  - (x)(Hx → Bx)
  - (x)(Zx → ~Bx)
  - So, (x)(Zx → ~Hx)

Go Back

Mark is eligible for paid vacation
- Invalid
- To say that A only if B is to say that B is a necessary condition of A.
- Thus to say that a gun fires only if there’s a live round in the chamber is to say that a live round in the chamber is a necessary condition for the gun firing, i.e. that the gun won’t fire without a live round in the chamber.
- To say that Mark is eligible for paid vacation only if he has been employed for at least 90 days is to say that being employed for at least 90 days is a necessary condition for being eligible for paid vacation, i.e. he’s not eligible if hasn’t been employed for 90 days (~E → ~90)
- What it doesn’t say is that being employed for 90 days is a sufficient condition for being eligible, i.e. if he’s been employed for 90 days then he’s eligible (90 → E).

Go Back

The defendant’s statement
- Invalid
- Refutation by Logical Analogy
  - The statements that all swans are white and that no swans are white cannot both be true (assuming there are swans).
  - The statement that all swans are white is false. (Black swans exist.)
  - Therefore, the statement that no swans are white is true. (false)

Go Back

Some people speak both
- Invalid
- Refutation by Logical Analogy:
  - Some integers are both odd and greater than 10, e.g. 11
  - Some integers are both greater than 10 and even, e.g. 12
  - Therefore some integers are both odd and even. (false)

Go Back

Sam speaks French or German
- Invalid
- Counterexample: Sam speaks only German.
  - In which case the premises are true and the conclusion false

Go Back

There will be no bonus
- Valid
- “A unless B” means “A if not B”.
- Thus to say that there will be no bonus unless the company makes a profit is to say that there will be no bonus if the company does not make a profit.
- The argument is thus valid per modus tollens:
  - P → Q and ~Q therefore ~P.

Go Back

Without both
- Valid
- The first premise says they won’t cash your check if you don’t have both a driver’s license and a credit card.
  - ~(D & C) → ~M
- That you don’t have a credit card logically implies you don’t have a driver’s license and a credit card.
  - ~C entails ~(D & C)
- So by modus ponens they won’t cash your check.

Go Back

Not everyone who’s wealthy
- Invalid
- Refutation by Logical Analogy
  - Not every integer greater than 10 is less than 7, e.g 11.
    - True
  - Not every integer less than 7 is greater than 5, e.g. 4.
    - True
  - Not every integer greater than 10 is greater than 5
    - False

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All and only conscious beings
- Valid
- That all and only conscious beings have souls implies that every conscious being has a soul.
- If dogs were conscious beings, then, they would have souls.
- But they don’t.
- So they’re not conscious beings.

Go Back

The Theory of Evolution and Genesis
- Valid
- There’s strong evidence that that theory of Evolution is true (premise 3).
- The Theory of Evolution and Genesis cannot both be true (premise 1).
- If (i) A is evidence for B and (ii) B and C are incompatible, then A is evidence that C is false.
- Hence, there’s strong evidence that Genesis is false. (~G)
- If the Bible is the inerrant word of God, Genesis is true. (B → G)
- B → G and ~G logically entail ~B, by modus tollens.
- If (i) A is evidence for B and (ii) B logically entails C, then A is evidence for C.
- Therefore, there’s strong evidence that the Bible is not the inerrant word of God.

Go Back

If human beings have free will
- Valid
- Form of Argument
  - F → C
  - C → T
  - ~T
  - So, ~F
- ~C follows from C → T and ~T by modus tollens
- ~F follows from F → C and ~C by modus tollens

Go Back

Most college graduates
- Invalid
- Refutation by Logical Analogy
  - Most integers among the integers 11, 12, and 13 are odd numbers. (true)
  - So, most odd numbers are among the integers 11, 12, and 13. (false)

Go Back

If the Bible is the word of God
- Valid
- Form of Argument
  - B → T
  - T → P
  - T → D
  - ~(P & D)
  - So, ~B
- The first three premises imply B → (P&D)
- ~B follows from B → (P&D) and ~(P & D) by modus tollens

Go Back

If the creation story in Genesis
- Valid if the argument is this:
  - Gen → No sun for first three days
  - No sun for first three days → Days
  - No sun for first three days → ~Sun
  - ~Sun → (~Days &~Nights)
  - So, ~Gen
- The argument is valid because, according to the argument, if there is no sun for the first three days then, during that time, there were days (three of them) and there were no days (no sun), which is impossible. Therefore:
  - Gen → No sun for first three days
  - No sun for first three days → (Days & ~Days)
  - Therefore, Gen → (Days & ~Days).
- ~Gen follows by modus tollens since it can’t be true that Days & ~Days.

Go Back

If the theory of evolution were true
- Valid by modus tollens:
  - E → G
  - ~G
  - So, ~E

Go Back

If there’s free will
- Valid
- The argument has a valid form in Predicate Logic:
  - F → ∀x(Px → ∃y(My & Cxy))
  - ∀x∀y(Px & My & Cxy → ∀z(Izy → Cxz))
  - ∀x(Mx → ∃y(Ny & Iyx))
  - ∀x∀y(Px & Ny → ~Cxy)
  - So, ~F
- where F: there is free will, P: is a person, M: is a physical movement, C: controls, I: initiates, N: is neurons firing in the motor cortex

Go Back

If Earth rotated on its axis
- Valid by modus tollens
  - R → H
  - ~H
  - So, ~R

Go Back

Any being that knows beforehand
- Valid
- Argument in Predicate Logic
  - ∀x∀y(Kxy & Dy & Pxy → Rx)
  - ∃xDx
  - ∀x∀y(Sx & Dy → Kxy))
  - ∀x∀y(Sx & Dy → Pxy))
  - ∀x(Sx → ~Rx)
  - So, ~∃xSx
- where
  - D: is a disaster that results in the suffering and death of innocent children.
  - K: x knows beforehand that y will occur
  - P: x could have prevented y
  - R: is morally reprehensible
- Informal proof
  - Suppose a and b are any entities such that Sa and Db
  - Then
    - Kab by premise 3
    - Pab by premise 4
    - Kab & Db & Pab → Ra by premise 1
    - Therefore Ra by modus ponens
    - Sa → ~Ra by premise 5
    - ~Sa by modus tollens
    - ∀x~Sx, since a was any entity
    - Therefore ~∃xSx

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For everything that begins to exist
- Valid per Predicate Logic (universal specification and modus ponens)
  - ∀x(Bx → ∃y(Cyx & y≠x))
  - Bu
  - So, ∃y(Cyu & y≠u)
- where B: begins to exist, C: 1 caused 2 to exist, and u = the universe.

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