# More predicate logic problems

• lazycritic
In summary: From 1 and 2 you can infer that ~Faa and Fa are the same. From 3 and 4 you can infer that if Fa and ~Faa are the same, then ~Faa is also true.

#### lazycritic

Hey,

If anyone here likes working on these problems, I'm working on a couple more. I finished the first two. These go:

1. Vx Vy (Fxy ---> ~Fyx) |- -ExFxx
(now my work - maybe wrong
2. Vy (Fay ---> ~Fya) 1,VE
3. (Faa ---> ~Faa) 2,VE
4. | Faa H(for -->E)
5. | ~Faa 3,4, CON
6. Faa
7. ~~Faa 3,6, MT
(I find this hard to articulate in the logical format, but basically, if ~Faa is proved wrong by Faa in line six(~(~Faa)), then MT says that ~Faa can be infered...)
8. ~Faa
9. -ExFxx 8, EI(can I do this?)

Actually ended up working it out as I posted it. Is it right, though? Btw, does anyone know the tex way to do this?

Test: $$VxEy(Mxy -> Fxy)$$

Didn't work.

Last edited:
I think I got the next problem, a theorom, but if anyone can look over it, I'd appreciate it.

4) |- ~Ex(Fx & ~Fx)

1. | ~[~Ex(Fx & ~Fx)] H(for con) - that how would I say I'm hypothesizing for a contradition?

2. | Ex(Fx & ~Fx) 1, DN
3. | Fa & ~Fa 2, E Elim.(tex would be nice here)
4. ~Ex(Fx & ~Fx) 1-3 Con?

Bump. Anyone mind checking these?

$$\forall \exists \neg \vee \wedge \rightarrow \leftrightarrow \vdash$$
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lazycritic said:
1. Vx Vy (Fxy ---> ~Fyx) |- -ExFxx
(now my work - maybe wrong
2. Vy (Fay ---> ~Fya) 1,VE
3. (Faa ---> ~Faa) 2,VE
4. | Faa H(for -->E)
5. | ~Faa 3,4, CON
From 4 and 5 you can infer Faa -> ~Faa (which you already have). Conditional proof gives you a conditional- a conditional which says that your hypothesis implies whatever you can derive while assuming it.
Remember I don't know much about quantifier elimination and introduction. Maybe you could post your rules for these.
If you can get
3. Faa -> ~Faa
Then
4. ~Faa v ~Faa [3, Implication]
5. ~Faa [4, Tautology]
Of course, those are what my rules are called, you may use different names or different rules. But if you can see that (P -> Q) <=> (~P v Q) and (~P v ~P) <=> ~P, you can probably find the rules you need.

Patience, grasshopper :tongue2:

lazycritic said:
how would I say I'm hypothesizing for a contradition?
That seems fine. There are two rules that use provisional hypotheses: Conditional proof and Reductio ad absurdum. You can find both rules here. I highly recommend you read those rules!
4) |- ~Ex(Fx & ~Fx)

1. | ~[~Ex(Fx & ~Fx)] H(for con) - that

2. | Ex(Fx & ~Fx) 1, DN
3. | Fa & ~Fa 2, E Elim.(tex would be nice here)
4. ~Ex(Fx & ~Fx) 1-3 Con?
Not by any rules I know. However, you can do:
1. | Fa & ~Fa [Reductio]
2. | Fa [1, Simplification]
3. | ~Fa [1, Simplification]
Now what can you infer from that?

## 1. What is predicate logic?

Predicate logic is a type of formal logic that deals with propositions, statements, and their relationships. It uses symbols and rules to represent and manipulate logical statements, and is often used in mathematics, computer science, and philosophy.

## 2. How is predicate logic different from propositional logic?

Predicate logic allows for the use of variables and quantifiers, while propositional logic only deals with simple statements and logical connectives. This makes predicate logic more powerful and flexible in representing complex statements and arguments.

## 3. What are the main components of a predicate logic statement?

A predicate logic statement consists of a subject, a predicate, and possibly one or more quantifiers. The subject is the object or individual being described, while the predicate is the property or relation being attributed to the subject. Quantifiers specify the scope and quantity of the statement.

## 4. What are some common quantifiers used in predicate logic?

Some common quantifiers include "for all" (∀) and "there exists" (∃). These quantifiers indicate the scope of the statement, such as whether it applies to all or some objects in a given domain. Other quantifiers, such as "for every" (∀) and "for each" (∀), can also be used.

## 5. How can predicate logic be used in real-world applications?

Predicate logic has many practical applications, including in computer programming, artificial intelligence, and database management. It can also be used in philosophical and mathematical proofs, as well as in linguistics to analyze the structure of natural language statements.