Valid & Invalid Argument Forms

Gig Φ Philosophy
(at-a-glance overviews of philosophical concepts)

Published September, 2022

Deductively Valid Argument Forms

HELPFUL HINTS: Modus Ponens & Tollens

Below are two of the most standard logical syllogisms derived from a conditional statement: modus ponens (MP) and modus tollens (MT). Before reviewing them, it is worth noting a few features of these arguments that can help us differentiate them from the syllogistic forms that follow:

Conditional Statements = If P, then Q

P = Antecedent

Q = Consequent

Modus Ponens (MP) = Affirming the Antecedent

Modus Tollens (MT) = Denying the Consequent

The names tell you what structure each premise takes:

Latin [‘modus’] = P1. If P, then Q.

Anglo = P2. [Affirm P] or [Deny Q].

Although MP and MT are the most commonly used deductively valid argument forms, they have some dangerously similar invalid counterparts, covered at the end of this section.

HELPFUL HINTS: Disjunctive & Hypothetical Syllogisms

Following MP and MT, we have two other syllogisms: disjunctive syllogism (DS) and hypothetical syllogism (HS). The former deals with its namesake (i.e., disjunctive statements) while the latter form is composed entirely of conditional statements. As with our first two deductively valid argument forms, here are some features unique to DS and HS:

Disjunctive Statements = Either P, or Q

P & Q = Disjuncts

DISJUNCTIVE Syllogism (DS) = 

P1 = Either P, or Q

P2 = Deny either disjunct (P or Q)

C = Remaining disjunct (Q or P)

HYPOTHETICAL Syllogism (HS) = 

P1, P2, & C = Conditionals

Pattern : : A = B , B = C , ∴ A = C

B = consequent of P1 & antecedent of P2, should not appear in C.

HELPFUL HINTS: Destructive & Constructive Dilemmas

The final two syllogisms we will cover are variations on the same structure: constructive dilemma (CD) and destructive dilemma (DD). Each of which begins with two conjoined conditionals followed by two disjunctive statements. Here are some features unique to CD and DD:

Both Dilemmas =

P1 = [conditional] & [conditional]

P2 & C = disjunctive statements

CONSTRUCTIVE Dilemma (CD) = 

P2 = P1 antecedents = Either [1st], or [2nd]

C = P1 consequents = Either [1st], or [2nd]

DESTRUCTIVE Dilemma (CD) = 

P2 = Deny P1 consequents = Neither [1st], nor [2nd]

C = Deny P1 antecedents = Neither [1st], nor [2nd]

Invalid Argument Forms

HELPFUL HINTS IN / VALID ARGUMENTS

Having completed our overview of deductively valid argument forms, we will now turn towards two of their invalid counterparts.

The two invalid arguments we will focus on are best understood as fallacious inversions of Affirming the Antecedent (MP) and Denying the Consequent (MT). These are denying the antecedent (DA) and affirming the consequent (AC), respectively. As with their valid counterparts, here are some helpful hints:

Conditional Statements = Are only ever FALSE when the ...

Antecedent = TRUE

Consequent = FALSE

VALID INFERENCES:

AFFIRMing the Antecedent

DENYing the Consequent

INVALID INFERENCES:

DENYing Antecedent

AFFIRMing Consequent

HELPFUL HINTS INVALID ARGUMENT FORMS

Let us assess one more example of DA and AC respectively to better understand why these syllogisms are invalid. In other words, why their conclusions could still be false even if their premises are true:

EXAMPLE DENYING THE ANTECEDENT

P1. If it is raining, then I need an umbrella.

P2. It is not raining.

C. ∴ I do not need an umbrella.

Notice here that when the antecedent condition (i.e., “it is raining”) is not met, nothing can be derived from the conditional statement in the first premise. Who is to say whether or not I need my umbrella when it is not raining? There is not sufficient reason to arrive at the conclusion.

EXAMPLE AFFIRMING THE CONSEQUENT

P1. If it is raining, then I need an umbrella.

P2. I need an umbrella.

C. ∴ it is raining.

Notice here that even if the consequent condition (i.e., “needing an umbrella”) is met, this says nothing about whether or not the antecedent condition (i.e., “it is raining”) has also been met. One could possibly need an umbrella even if it were not raining. Again, leaving the conclusion unestablished.

Conditional

"if ... , then ... " statement

Antecedent

comes after if " ... "

Consequent

comes after then " ... "

Disjunctive

"either ... , or ... " statement

Modus Ponens Affirming the Antecedent

P1. If P, then Q.

P2. P.

C. Therefore, Q.

Modus Tollens  Denying the Consequent

P1. If P, then Q.

P2. Not Q.

C. Therefore, not P.

Hypothetical Syllogism

P1. If P, then Q.

P2. If Q, then R.

C. Therefore, if P, then R.

Disjunctive Syllogism

P1. Either P, or Q.

P2. Not P. [or Not Q.]

C. Therefore, Q. [∴ P.]

Constructive Dilemma

P1. Either P, or Q.

P2. If P, then R.

P3. If Q, then S.

C. Therefore, either R, or S. [CD]

Note on Validity

To ensure validity, please review the role of variables in valid argument forms. If a variable appears in multiple places, that means the phrasing should be identical. The exception to this is when the variable is negated (~), in which case, you would phrase it opposite from its original phrasing.

P  ⸬  "it is raining"

Not P  ⸬  ~ P  ⸬  "it is not raining"

Standard Form

P1. Numbered premises, each on a separate line.

P2. Line separating premises from conclusion.

C. Identified conclusion.

Example

P1. If I follow the instructions, then I will receive full credit on this assignment.

P2. I followed the instructions.

C. Therefore, I will receive full credit on this assignment.

Argument Examples

Modus Ponens (MP)

MP #1

P1. If animals have no moral standing [P], then it is morally permissible to eat animals [Q].

P2. Animals have no moral standing [P].

C. Therefore, it is morally permissible to eat animals [Q].

MP #2

P1. If there is evil in the world [P], then God does not exist [Q].

P2. There is evil in the world [P].

C. Therefore, God does not exist [Q].

Modus Tollens (MT)

MT #1

P1. If it is morally permissible to eat animals [P], then animals do not have the ability to suffer (i.e., sentience) [Q].

P2. Animals do have the ability to suffer (i.e., sentience) [~Q].

C. Therefore, it is not morally permissible to eat animals [~P].

MT #2

P1. If God did not exist [P], then there would be no similarities between major world religions [Q].

P2. There are similarities between major world religions [~Q].

C. Therefore, God does exist [~P].

Disjunctive Syllogism (DS)

DS #1

P1. Either God exists and there is a successful response to the Problem of Evil [P], or there is no successful response to the Problem of Evil and God does not exist [Q].

P2. It is not the case that God exists and there is a successful response to the Problem of Evil [~P].

C. Therefore, there is no successful response to the Problem of Evil and God does not exist [Q].

DS #2

P1. Either it is morally impermissible to eat animals because they are sentient [P], or it is morally permissible to eat animals because they do not have [sufficient] moral standing/value [Q].

P2. It is not the case that it is morally permissible to eat animals because they do not have [sufficient] moral standing/value [~Q].

C. Therefore, it is morally impermissible to eat animals because they are sentient [P].

Hypothetical Syllogism (HS)

HS #1

P1. If God does not exist [P], then belief in God is irrational [Q].

P2. If belief in God is irrational [Q], then it and other religious beliefs should not be used as justification for other beliefs or claims [R].

C. Therefore, if God does not exist [P], then it and other religious beliefs should not be used as justification for other beliefs or claims [R].

HS #2

P1. If it is morally permissible to eat animals [P], then it is morally permissible to eat other sentient creatures [Q].

P2. If it is morally permissible to eat other sentient creatures [Q], then it is morally permissible to eat humans [R].

C. Therefore, if it is morally permissible to eat animals [P], then it is morally permissible to eat humans [R].

Constructive Dilemma (CD)

CD #1

P1. Either it is morally permissible to eat animals [P], or it is morally impermissible to eat animals [Q].

P2. If it is morally permissible to eat animals [P], then people who consume meat and dairy need not change their eating habits [R].

P3. If it is morally impermissible to eat animals [Q], then people who consume meat and dairy should change their eating habits [S].

C. Therefore, either people who consume meat and dairy need not change their eating habits [R], or people who consume meat and dairy should change their eating habits [S].

AT-A-GLANCE: Argument Forms

Argument Forms.pdf

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