Lecture #19, Nov 20, 2020
0.0.1 Example. Consider the wff
x = y → (∀x)x = y (1)
Here are a few interpretations: 1.D={3},xD =3,yD =3.
Since D contains one element only the above “choice” was made for us, being unique.
Thus (1) translates as
3 = 3 → (∀x ∈ D)x = 3 (2)
Incidentally, (2) is TRUE.
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Basic Logic© by George Tourlakis
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2. This time I take
D={3,5},andagainxD =3andyD =3. Thus (1) translates as:
3 = 3 → (∀x ∈ D)x = 3 (3) This time (3) is FALSE since “3 = 3” is TRUE as before,
BUT
Basic Logic© by George Tourlakis
“(∀x ∈ D)x = 3” is FALSE.
0.0.2 Example. Let’s interpret the following a few different ways:
(∀x)(x ∈ y ≡ x ∈ z) → y = z (1)
1. First this is true if we really are talking about sets as “∈” compels us to think, being THE predicate of set theory that says “is a member of”.
Incidentally, (1) if interpreted in Set Theory, says that any two sets y and z are equal if they happen to have the same elements (x is in y iff x is in z). Hence is true, as I noted.
2. Let us now interpret in number theory (of N).
Take D = N and ∈D=<, where “<” is the relation “less
than” on N.
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Basic Logic© by George Tourlakis
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Wait a minute! Can I do that?! Can I interpret “∈” as something OTHER than “is a member of”?
Of course you can!
Only “=, (, ), ¬, ∨, ∧, →, ≡” translate as themselves! EVERYTHING ELSE is fair game to translate as you please!
So (1) translates as:
(∀x∈N)(x