How can I define Propositional Expansion precisely?
Suppose that our universe of discourse is $X={ x_1, x_2, cdots , x_n }$
Then, intuitively, the following seems to be truth.
$$(exists xin X) (P(x)) := (P(x_1) lor P(x_2) lor cdots lor P(x_n))$$
$$(forall xin X) (P(x)) := (P(x_1) wedge P(x_2) wedge cdots wedge P(x_n))$$
I want to make this definition clearer.
And from this definition, I want to prove the properties about quantifiers only using the inference rules of propositional logic. (I have heard about infinitary logic. What I want is just a case where the universe of discourse is finite.)
If you have an idea, or if you know the references, please help me.
logic propositional-calculus first-order-logic
asked Dec 2 at 10:03
amoogae
125
|
show 1 more comment
Suppose that our universe of discourse is $X={ x_1, x_2, cdots , x_n }$
Then, intuitively, the following seems to be truth.
$$(exists xin X) (P(x)) := (P(x_1) lor P(x_2) lor cdots lor P(x_n))$$
$$(forall xin X) (P(x)) := (P(x_1) wedge P(x_2) wedge cdots wedge P(x_n))$$
I want to make this definition clearer.
And from this definition, I want to prove the properties about quantifiers only using the inference rules of propositional logic. (I have heard about infinitary logic. What I want is just a case where the universe of discourse is finite.)
If you have an idea, or if you know the references, please help me.
logic propositional-calculus first-order-logic
asked Dec 2 at 10:03
amoogae
125
It is quite hard... Propositional formulas have finite lenght, while fitst-order logic deals also with infinite domains.
– Mauro ALLEGRANZA
Dec 2 at 10:07
See Infinitary logic.
– Mauro ALLEGRANZA
Dec 2 at 10:08
I have heard about infinitary logic. What I want is just a case where the universe of discourse is finite.
– amoogae
Dec 2 at 10:10
Ok; if the universe $X$ is finite, then the above definitions are correct.
– Mauro ALLEGRANZA
Dec 2 at 10:13
With the above def, Universal instantiation is simply Conjunction elimination.
– Mauro ALLEGRANZA
Dec 2 at 10:15
|
show 1 more comment
Suppose that our universe of discourse is $X={ x_1, x_2, cdots , x_n }$
Then, intuitively, the following seems to be truth.
$$(exists xin X) (P(x)) := (P(x_1) lor P(x_2) lor cdots lor P(x_n))$$
$$(forall xin X) (P(x)) := (P(x_1) wedge P(x_2) wedge cdots wedge P(x_n))$$
I want to make this definition clearer.
And from this definition, I want to prove the properties about quantifiers only using the inference rules of propositional logic. (I have heard about infinitary logic. What I want is just a case where the universe of discourse is finite.)
If you have an idea, or if you know the references, please help me.
logic propositional-calculus first-order-logic
asked Dec 2 at 10:03
amoogae
125
Suppose that our universe of discourse is $X={ x_1, x_2, cdots , x_n }$
Then, intuitively, the following seems to be truth.
$$(exists xin X) (P(x)) := (P(x_1) lor P(x_2) lor cdots lor P(x_n))$$
$$(forall xin X) (P(x)) := (P(x_1) wedge P(x_2) wedge cdots wedge P(x_n))$$
I want to make this definition clearer.
And from this definition, I want to prove the properties about quantifiers only using the inference rules of propositional logic. (I have heard about infinitary logic. What I want is just a case where the universe of discourse is finite.)
If you have an idea, or if you know the references, please help me.
logic propositional-calculus first-order-logic
logic propositional-calculus first-order-logic
asked Dec 2 at 10:03
amoogae
125
asked Dec 2 at 10:03
amoogae
125
edited Dec 2 at 10:24
asked Dec 2 at 10:03
amoogae
125
asked Dec 2 at 10:03
amoogae
125
asked Dec 2 at 10:03
amoogae
125
125
It is quite hard... Propositional formulas have finite lenght, while fitst-order logic deals also with infinite domains.
– Mauro ALLEGRANZA
Dec 2 at 10:07
See Infinitary logic.
– Mauro ALLEGRANZA
Dec 2 at 10:08
I have heard about infinitary logic. What I want is just a case where the universe of discourse is finite.
– amoogae
Dec 2 at 10:10
Ok; if the universe $X$ is finite, then the above definitions are correct.
– Mauro ALLEGRANZA
Dec 2 at 10:13
With the above def, Universal instantiation is simply Conjunction elimination.
– Mauro ALLEGRANZA
Dec 2 at 10:15
|
show 1 more comment
It is quite hard... Propositional formulas have finite lenght, while fitst-order logic deals also with infinite domains.
– Mauro ALLEGRANZA
Dec 2 at 10:07
See Infinitary logic.
– Mauro ALLEGRANZA
Dec 2 at 10:08
I have heard about infinitary logic. What I want is just a case where the universe of discourse is finite.
– amoogae
Dec 2 at 10:10
Ok; if the universe $X$ is finite, then the above definitions are correct.
– Mauro ALLEGRANZA
Dec 2 at 10:13
With the above def, Universal instantiation is simply Conjunction elimination.
– Mauro ALLEGRANZA
Dec 2 at 10:15
It is quite hard... Propositional formulas have finite lenght, while fitst-order logic deals also with infinite domains.
– Mauro ALLEGRANZA
Dec 2 at 10:07
It is quite hard... Propositional formulas have finite lenght, while fitst-order logic deals also with infinite domains.
– Mauro ALLEGRANZA
Dec 2 at 10:07
See Infinitary logic.
– Mauro ALLEGRANZA
Dec 2 at 10:08
See Infinitary logic.
– Mauro ALLEGRANZA
Dec 2 at 10:08
I have heard about infinitary logic. What I want is just a case where the universe of discourse is finite.
– amoogae
Dec 2 at 10:10
I have heard about infinitary logic. What I want is just a case where the universe of discourse is finite.
– amoogae
Dec 2 at 10:10
Ok; if the universe $X$ is finite, then the above definitions are correct.
– Mauro ALLEGRANZA
Dec 2 at 10:13
Ok; if the universe $X$ is finite, then the above definitions are correct.
– Mauro ALLEGRANZA
Dec 2 at 10:13
With the above def, Universal instantiation is simply Conjunction elimination.
– Mauro ALLEGRANZA
Dec 2 at 10:15
With the above def, Universal instantiation is simply Conjunction elimination.
– Mauro ALLEGRANZA
Dec 2 at 10:15
|
show 1 more comment
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It is quite hard... Propositional formulas have finite lenght, while fitst-order logic deals also with infinite domains.
– Mauro ALLEGRANZA
Dec 2 at 10:07
See Infinitary logic.
– Mauro ALLEGRANZA
Dec 2 at 10:08
I have heard about infinitary logic. What I want is just a case where the universe of discourse is finite.
– amoogae
Dec 2 at 10:10
Ok; if the universe $X$ is finite, then the above definitions are correct.
– Mauro ALLEGRANZA
Dec 2 at 10:13
With the above def, Universal instantiation is simply Conjunction elimination.
– Mauro ALLEGRANZA
Dec 2 at 10:15