# predicate calculus

• Dec 11th 2009, 03:04 AM
ruleworld
predicate calculus

Consider the following statements:

Every file in MyDocuments is encrypted or hidden. Every encrypted
file in MyDocuments is secure. Every secure file in MyDocuments
contains secrets. There is a file in MyDocuments which does not contain
secrets. Therefore there is a file in MyDocuments which is hidden.

Choose an appropriate universal set and collection of predicates, and express this argument in the language of predicate calculus.
• Dec 11th 2009, 12:59 PM
emakarov
Let the universal set be the set of all files in MyDocuments. What about choosing the collection of predicates, e.g., E(x) means "x is envrypted"? Can you do that?
• Dec 11th 2009, 04:08 PM
ruleworld
i am trying the problem. please check if its ok or not.

Let
U, the set of all files in MyDocuments

we define the following functions

E(x), indicates if file x is encrypted
H(x), indicates if file x is hidden
Secure(x), indicates if file x is secure
Secret(x), indicates if file x contains secrets

we can now write the given statements using predicate and logical connectives

x ∈ U: E(x) ∨ H(x)
x ∈ U: E(x) ⇒Secure(x)
x ∈ U: Secure(x) Secret(x)
∃x ∈ U: ¬( Secret(x) ) ⇒ ∃x ∈ U: H(x)
• Dec 12th 2009, 07:59 AM
emakarov
I agree with what you wrote, except that the last line looks like one formula, i.e., one assumption. The claim to be proved should be clearly separated from all the assumptions.

Let us know if you need help with constructing a formal or informal derivation, etc.
• Dec 12th 2009, 03:19 PM
ruleworld
Quote:

Originally Posted by emakarov
I agree with what you wrote, except that the last line looks like one formula, i.e., one assumption. The claim to be proved should be clearly separated from all the assumptions.

Let us know if you need help with constructing a formal or informal derivation, etc.

I don't know what to do with the last line. would it be something like joining all the lines together like this? please help

x ∈ U:(E(x) ∨ H(x)) ∧ ∀x ∈ U: (E(x) ⇒Secure(x)) ∧ x ∈ U:(Secure(x) Secret(x))∧ ∃x ∈ U: ¬( Secret(x) ) ⇒ ∃x ∈ U: H(x)
• Dec 13th 2009, 05:06 AM
ruleworld
• Dec 13th 2009, 06:38 AM
Seppel
Hello ruleworld!

Quote:

Originally Posted by ruleworld
i am trying the problem. please check if its ok or not.

Let
U, the set of all files in MyDocuments

we define the following functions

E(x), indicates if file x is encrypted
H(x), indicates if file x is hidden
Secure(x), indicates if file x is secure
Secret(x), indicates if file x contains secrets

we can now write the given statements using predicate and logical connectives

x ∈ U: E(x) ∨ H(x)
x ∈ U: E(x) ⇒Secure(x)
x ∈ U: Secure(x) Secret(x)
∃x ∈ U: ¬( Secret(x) ) ⇒ ∃x ∈ U: H(x)

In the last line of the quote you write $\exists x\in U:\neg (Secret(x))\Longrightarrow \exists x\in U: H(x)$, so it looks like this statement is given. Actually $\exists x\in U:\neg (Secret(x))$ is one of the givens and $\exists x\in U: H(x)$ is the conclusion.

Emakarov meant that this should be clear for everybody who reads your text. Thus I suggest you write it for example like this

given
given
given
.
.
.
given
-------
conclusion

In this form the readers can see which are the givens and what the conlcusion is.

Best wishes,
Sebastian
• Dec 13th 2009, 07:38 AM
emakarov
Yes, that was my idea.

If one wants to express the whole problem as one formula (for example, in order to feed it to an automatic theorem prover and see if it is derivable), then the long implication given by OP is also correct. I would only put extra parentheses to avoid ambiguity and possibly add some indentation, something like this:

(∀x ∈ U: (E(x) ∨ H(x))) ∧
(∀x ∈ U: (E(x) ⇒ Secure(x))) ∧
(∀x ∈ U: (Secure(x) ⇒ Secret(x))) ∧
(∃x ∈ U: ¬(Secret(x))) ⇒
∃x ∈ U: H(x)

Bloody editor: I don't know how to indent the last line by a couple of spaces.

• Dec 13th 2009, 12:01 PM
ruleworld
Thanks very much for ur help Emakarov and Seppel.