Press the right key for the next slide (or swipe left)

also ...

Press the left key to go backwards (or swipe right)

Press n to toggle whether notes are shown (no equivalent if you don't have a keyboard)

Press m or double tap to see a menu of slides

 

There Does Not Exist

Something is not dead:
\hspace{3mm} ∃x ¬Dead(x)
Nothing is dead:
\hspace{3mm} ¬∃x Dead(x)
Everything is not broken:
\hspace{3mm} ∀x ¬Broken(x)
Not everything is broken:
\hspace{3mm} ¬∀x Broken(x)

Something is dead.

∃x Dead(x)

Something is not dead.

∃x ¬Dead(x)

Nothing is dead.

¬∃x Dead(x)

​1.

​2.a=a=Intro
​3.∃x (x=x)∃Intro: 2

Everything is broken.

∀x Broken(x)

Everything is not broken.

∀x ¬Broken(x)

Not everything is broken.

¬∀x Broken(x)

​1.

¬∃x Dead(x)

​2.

Dead(a)

​3.∃x Dead(x)∃Intro: 2
​4.⊥⊥Intro: 1,3
​5.¬Dead(a)¬Intro: 2-4
​6.∃x ¬Dead(x)∃Intro: 5
​1.

∃x ¬Dead(x)

​2.¬∃x Dead(x)

Counterexample:

Domain: {Ayesha, Beatrice}

Dead : { <Beatrice> }

a : Ayesha ; b : Beatrice

... Dead(b) is true; so the conclusion, ¬∃x Dead(x), is false

... and ¬Dead(a) is true; so the premise, ∃x ¬Dead(x), is true

9.12
9.18--9.19