Deductive Arguments

Christopher L. Holland

Saint Louis University

January 30, 2025

Deductive Arguments

Deductive Arguments Overview

The premises are intended to guarantee the conclusion.
Every deductive argument is either valid or invalid (but not both).
Every deductive argument is either sound or unsound (but not both).

We say that a valid deductive argument preserves truth. It does so in the same way as a good refrigerator preserves food. If the food is good, a good refrigerator will preserve it; but if the food is spoiled, the refrigerator will not make it good. If the statements are true and the form is correct, the conclusion will be true; but if the premises are not true, a valid argument will not guarantee a true conclusion.

— Louis Pojman (2006, 14)

Valid or Invalid

A deductive argument is valid if it preserves truth.

If the premises are true, then the conclusion must be true.
If the premises are true, then it is impossible for the conclusion to be false.

A deductive argument is invalid if …

it does not preserve truth.
in other words, if it is not valid.

Sound or Unsound

A deductive argument is sound if it is valid and all its premises are true.
A deductive argument is unsound if it is invalid or if it has at least one false premise.

Validity

A deductive argument is valid if it preserves truth.

If the premises are true, then the conclusion must be true
If the premises are true, then it is impossible for the conclusion to be false

An argument is valid b/c of its form

Examples A & B use valid forms

Example A

1.: All humans are mortal.
2.: Socrates is a human.
∴ 3.: Socrates is mortal.

Form

1.: All \(F\) are \(G\)
2.: \(o\) is \(F\)
∴ 3.: \(o\) is \(G\)

An argument is valid b/c of its form

Examples A & B use valid forms

Example B

1.: Socrates believes in daimons.
2.: If Socrates believes in daimons,
then Socrates believes in the gods.
∴ 3.: Socrates believes in the gods.

Form

1.: \(P\)
2.: If \(P\), then \(Q\)
∴ 3.: \(Q\)

Invalidity

A deductive argument is invalid …

if it does NOT preserve truth
(in other words, if it is not valid)
if there is a counterexample
(in other words, we can give an example using the same form where the premises are true but the conclusion is false)

An argument is invalid b/c of its form

Examples C & D use invalid forms

Example C

1.: All humans are mortal.
2.: Socrates is a lemur.
∴ 3.: Socrates is mortal.

Form

1.: All \(F\) are \(G\).
2.: \(o\) is \(H\).
∴ 3.: \(o\) is \(G\).

Example D

1.: All cats are mammals.
2.: Hank is a mammal.
∴ 3.: Hank is a cat.

Form

1.: All \(F\) are \(G\).
2.: \(o\) is \(G\).
∴ 3.: \(o\) is \(F\).

Validity is about form, not content.

important

INVALID arguments can have true premises and a true conclusion.

Example

1.: Some cartoons are Martians. true
2.: Marvin is a cartoon. true
∴ 3.: Marvin is a Martian. true

Invalid Form

1.: Some \(F\) are \(G\)
2.: \(o\) is \(F\)
∴ 3.: \(o\) is \(G\)

VALID arguments can have false premises and a true conclusion.

Example

1.: All cartoons are Martians. false
2.: Marvin is a cartoon. true
∴ 3.: Marvin is a Martian. true

Valid Form

1.: All \(F\) are \(G\)
2.: \(o\) is \(F\)
∴ 3.: \(o\) is \(G\)

Example

1.: All humans are cartoons. false
2.: Marvin is not a cartoon. false
∴ 3.: Marvin is not human. true

Valid Form

1.: All \(F\) are \(G\)
2.: \(o\) is not-\(G\)
∴ 3.: \(o\) is not-\(F\)

VALID arguments can have false premises and a false conclusion.

Example

1.: All Martians are cartoons. false
2.: Marvin is not a cartoon. false
∴ 3.: Marvin is not a Martian. false

Valid Form

1.: All \(F\) are \(G\)
2.: \(o\) is not-\(G\)
∴ 3.: \(o\) is not-\(F\)

VALID arguments can have false premises and a false conclusion.

Example

1.: All mortals are humans. false
2.: Socrates is not a human. false
∴ 3.: Socrates is not mortal. false

Valid Form

1.: All \(F\) are \(G\)
2.: \(o\) is not-\(G\)
∴ 3.: \(o\) is not-\(F\)

Common Deductive Forms

Modus Ponens (MP)

valid AKA Affirming the Antecedent

Form

1.: \(P\)
2.: If \(P\), then \(Q\)
∴ 3.: \(Q\)

Example

1.: Mary is a mother.
2.: If Mary is a mother,
then Mary is a woman.
∴ 3.: Mary is a woman.

Example

1.: Socrates believes in daimons.
2.: If Socrates believes in daimons,
then Socrates believes in the gods.
∴ 3.: Socrates believes in the gods.

Denying the Antecedent

invalid

Form

1.: If \(P\), then \(Q\)
2.: Not-\(P\)
∴ 3.: Not-\(Q\)

Example

1.: If Mary is a mother, then Mary is a woman.
2.: Mary is not a mother.
∴ 3.: Mary is not a woman.

Modus Tollens (MT)

valid AKA Denying the Consequent

Form

1.: If \(P\), then \(Q\)
2.: Not-\(Q\)
∴ 3.: Not-\(P\)

Example

1.: If Leslie is a mother, then Leslie is a woman.
2.: Leslie is not a woman.
∴ 3.: Leslie is not a mother.

Example

1.: If Socrates deliberately harms the youth, then Socrates deliberately harms himself.
2.: Socrates does not deliberately harm himself.
∴ 3.: Socrates does not deliberately harm the youth.

Affirming the Consequent

invalid

Form

1.: \(Q\)
2.: If \(P\), then \(Q\)
∴ 3.: \(P\)

Example

1.: Mary is a woman.
2.: If Mary is a mother, then Mary is a woman.
∴ 3.: Mary is a mother.

Hypothetical Syllogism (HS)

valid AKA Chained Conditionals

Form

1.: If \(P\), then \(Q\)
2.: If \(Q\), then \(R\)
∴ 3.: If \(P\), then \(R\)

Example

1.: If Fido is a dog,
then Fido is a mammal.
2.: If Fido is a mammal,
then Fido is an animal.
∴ 3.: If Fido is a dog,
then Fido is an animal.

Example

1.: If death is a dreamless sleep,
then death is a blessing.
2.: If death is a blessing,
then you should not fear death.
∴ 3.: If death is a dreamless sleep,
then you should not fear death.

Disjunctive Syllogism (DS)

valid AKA Denying a Disjunct

Forms

1.: Either \(P\) or \(Q\)
2.: Not-\(P\)
∴ 3.: \(Q\)

and

1.: Either \(P\) or \(Q\)
2.: Not-\(Q\)
∴ 3.: \(P\)

Example

1.: Either Felix is a dog or Felix is a cat.
2.: Felix is not a dog.
∴ 3.: Felix is a cat.

Example

1.: Either the jury acts unjustly or Socrates acts unjustly.
2.: Socrates does not act unjustly.
∴ 3.: The jury acts unjustly.

Complex Dilemma (CD)

valid AKA Constructive Dilemma

Form

1.: Either \(P\) or \(Q\)
2.: If \(P\), then \(R\)
3.: If \(Q\), then \(S\)
∴ 4.: Either \(R\) or \(S\)

Example

1.: Either Felix is a bird or Felix is a cat.
2.: If Felix is a bird, then Felix has two legs.
3.: If Felix is a cat, then Felix has four legs.
∴ 4.: Either Felix has two legs or Felix has four.

Example

1.: Socrates corrupts the youth voluntarily or involuntarily.
2.: If he corrupts them voluntarily, then he should be punished.
3.: If he corrupts them involuntarily, then he should be instructed.
∴ 4.: Socrates should be punished or instructed.

Simple Dilemma (SD)

valid

Form

1.: Either \(P\) or \(Q\)
2.: If \(P\), then \(R\)
3.: If \(Q\), then \(R\)
∴ 4.: \(R\)

Example

1.: Either Felix is a dog or Felix is a cat.
2.: If Felix is a dog, then Felix is a mammal.
3.: If Felix is a cat, then Felix is a mammal.
∴ 4.: Felix is a mammal

Example

1.: A dead person either stops existing or goes to another place.
2.: If a dead person stops existing, then death is a blessing.
3.: If a dead person goes to another place, then death is a blessing.
∴ 4.: Death is a blessing.

Reductio ad absurdum

a common deductive argument strategy

Reductio ad absurdum (RAA)

AKA Reducing to a contradiction (or absurdity)

RAA is a common deductive argument strategy.
The basic principle: Whatever implies a contradiction (or absurdity) is false.
To give an RAA argument, assume the opposite of what you’d like to prove. Then, show that adding that assumption to the argument leads to a contradiction (or absurdity).

Proof that we do not live on Discworld

1.: Assume: We live on Discworld.
2.: If we live on Discworld, then people could fall off the edge of the world.
∴ 3.: So, people can fall off the edge of the world. (from 1&2)
4.: People cannot fall off the edge of the world.
5.: So, people both can and cannot fall off the edge of the world. (from 3&4; Contradiction!)
∴ 6.: So, we do not live on Discworld. (from 1&5)

Anselm’s Ontological Argument

1.: Assume: The greatest conceivable being (GCB) does not exist.
2.: If the GCB does not exist, then we can conceive of something greater than the GCB (a being just like that GCB that really does exist).
∴ 3.: So, we can conceive of something greater than the GCB. (from 1&2)
4.: If we can conceive of something greater than the GCB, then the GCB is not the GCB.
∴ 5.: So, the GCB is not the GCB. (from 3&4; Contradiction!)
∴ 6.: So, the GCB exists. (from 1&5)

Arguing by RAA

To show that \(A\) is false.

Assume \(A\).
Show that \(A\) leads to a contradiction (or absurdity).
Conclude not-\(A\).

Arguing by RAA

To show that \(A\) is true.

Assume not-\(A\)
Show that not-\(A\) leads to a contradiction (or absurdity).
Conclude \(A\).

Reductio ad absurdum (RAA) and modus tollens (MT)

MT: Whatever implies a falsehood is false.
RAA: Whatever implies a contradiction is false.

RAA as a special case of MT.

MT

: If \(P\), then \(Q\)
: Not-\(Q\)
∴: Not-\(P\)

RAA

: If \(P\), then (\(Q\) and not-\(Q\))
: Not-(\(Q\) and not-\(Q\))
∴: Not-\(P\)

Soundness

Sound or Unsound

A deductive argument is sound if it is valid and all its premises are true.
A deductive argument is unsound if it is invalid or if it has at least one false premise.

Soundness is about both form and content.

SOUND arguments are VALID with all true premises (and a true conclusion).

Example

1.: All Looney Tunes are cartoons. true
2.: Marvin is a Looney Tune. true
∴ 3.: Marvin is a cartoon. true

Valid Form

1.: All \(F\) are \(G\)
2.: \(o\) is \(F\)
∴ 3.: \(o\) is \(G\)

Sources

Pojman, Louis P. 2006. Philosophy: The Pursuit of Wisdom. 5th ed. Belmont, CA: Wadsworth.