nazmi

Logic — 2026-10-16

lecture 7

logic RAA counterexample

Indirect Arguments

indirect argument (reductio ad absurdum) proof by contradiction

• establish the conclusion not-A
• by supposing that A
• and showing that this leads to something absurd

modus tollens (MT): if $A$ then $B,$ not $B$ therefore not $A$
modus ponens (MP): if $A$ then $B$, $A$, therefore $B$.

the validiy of MT follows from that of MP and that of reductio ad absurdum

1. if A then B	premise
2. not B	premise
3. A	supposition
4. B	from 1, 3, by MP
5. Contradiction!	from 2, 4
6. not A	from 3-5, by RAA

absurdity: we cannot have both $B$ and $notB$

• the importance of bookkeeping in our derivations:
  • in step (3), a supposition is made without any inputs required
  • in step (5), an absurdity is found and the supposition is discharged
  • the supposition is active in steps (3-5), cf. the indentation to the right
  • the input for RAA in step (6) is not one or two separate lines, but the entire subderivation of steps (3-5)

why does RAA work?

1. $A_1$	premise
. .	.
$n$ $A_n$	…
$(n+1)$ $S$	supposition
$(k)$ Contradiction!	…
$(k+1)$ not $S$	from $n+1-k$, by RAA

$A_1$, …, $A_n$, together with $S$, entail a contradiction
if $A_1$, …, $A_n$, and $S$ can all be true together,
then that contradiction can be true as well
but contradictions can never be true!
so $A_1$, …, $A_n$, and $S$ cannot all be true together.
more specifically: if $A_1$, …, $A_n$, are all true, then S has to be false
in other words $A_1$, …, $A_n$, entail not S

the counterexample method

ackshually!!!

how to show that an argument is invalid?

• show that there exists a possible situation in which all premises are true. yet the conclusion is false.
  • if you don’t find such a situation…
  • … is it because it really doesn’t exist?
  • … or because you haven’t looked hard enough?
• show that there does not exist a derivation that starts with the premises and ends with the conclusion
  • if you don’t find such a derivation…
  • … is it because it really doesn’t exist?
  • … or because you haven’t looked hard enough?

some logicians are Belgians, Lorenz is a logician, so Lorenz is Belgian

• true premises, true conlcusion, but the argument is invalid.
  • I can imagine a situation where some logicians are Belgians, Lorenz is a logician, but Lorenz is Dominican

you might as well argue:

some animals are zebras, Garfield is an animal, so Garfield is a zebra

this is a counterexample

• its premises are true

• its conclusion is false

• hence, the argument is invalid

• the original argument has the same form as this one

• so the original argument is invalid

scientists:

disadvantage:

• coming up with a possible situation requires imagination
• coming up with a counterexample also requires imagination

advantage:

• possible situations are murky, what can be coherently imagined?

• de facto truth is completely clear

• shared argument form: some $F$ are $G$, $n$ is $F$, so $n$ is $G$

• this argument form is invalid.

but invalid argument forms may have valid instances.

some logicians are logicians, Lorenz is a logician, so Lorenz is a logician

a very incorrect application of counterexample can be found.

• searching for a counterexample to some argument $A$

1. find an argument form $F$ such that
   • $A$ is an instance of $F$
   • $F$ needs to be valid in order for $A$ to be valid
2. find an argument $C$ such that
   • $C$ is an instance of $F$ (so $A$ and $C$ have the same form, namely $F$)
   • $C$ has true premises and a false conlcusion (the counterexample)

A

$A$: some logicians are Belgian, Lorenz is a logician, so Lorenz is Belgian

• we can take:
  • $F$: some $F$ are $G$, $n$ is $F$, so $n$ is $G$
  • $C$: some animals are zebras, Garfield is an animal, so Garfield is a zebra

---

$A$: some logicians are logicians, Lorenz is a logician, so Lorenz is a logician

• we cannot take:
  • $F$: some $F$ are $G$, $n$ is $F$, so $n$ is $G$
  • whatever

Logical validity

7/10 exams had an example from this chapter
validity exam

new notion of validity → logical validity (narrows down deductive validity)

no dogs are chickens, so no chickens are dogs
the game
no logicians are irrational, so no irrational people are logicians

• deductively valid
• argument form: no $F$ are $G$, so no $G$ are $F$

Ann is a widow, so Ann is a woman

Sophie is a widow, so Sophie is a woman

abcdefghijklmnopqrstuvxyz0123456789

• deductively valid

• argument form 1: $n$ is $F$, so $n$ is $G$

  • this argument form is invalid
  • invalid instance: Lorenz is a logician, so Lorenz is silly

• argument form 2: $n$ is a widow, so $n$ is a woman

  • this argument form is valid its not really a form anymore
  • all its instances are valid

tbc 2026-10-19