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rockysheedy

Member

Registered: 2006-06-11

Posts: 12

What is rule P and T in Predicate Calculus?

Hello Guys,

I don't require a stepwize solution for this problem? I just want to know what is rule P and T in Predicate Calculus.

The Question is as follows. I am even writing in the Answer. Please explain me what it is.

Topic:- Discrete Mathematics.

Question:- In Theory of Inference for the Predicate Calculus the Method of Derivation of Predicate forumlas uses the rules P and T. What are these Rules P and T?

Problem: - Show that (x) (H(x)->M(x)) n H(s) <-> M(s). This Problem is a symbolic Translation of a well-known argument known as "socrates argument" which is given by:

All men are mortal.

Socrates is a man.

Therefore Socrates is a mortal.

If we deonte H(x): x is a man, M(x): x is a mortal, and s: Socrates, we can put the argument in the form.

Solution:-

{1} (1) (x) (H(x)->M(x)) P------- What is P in this step? What is {1} in this step?
{1} (2) H(s)->M(s) US, (1) ------- What is US, (1) in this step?
{3} (3) H(s) P ------- Again what is P in this step? Why {3} is used in this step and not {1}?
{1,3} (4) M(s) T, (2), (3) ------- why {1,3} is used in this step and what does T, (2), (3) means?

Is {1}, {3} and {1,3} the Universe of Discourse?

regards,
rocky sheedy.

Last edited by rockysheedy (2007-06-28 03:17:41)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: What is rule P and T in Predicate Calculus?

Well, it’s clear that P means “premise”. Thus, ∀x[H(x)→M(x)] and H(s) are your premises. Premises are the statements that are given from which a conclusion is to be deduced.

US, I would say, is “universal substitution”. In a predicate statement introduced by the universal quantifier, say ∀xF(x), if s is in the universe of discourse, you may deduce the propositional statement F(s) from the predicate one. This is called a universal substitution.

I’m not sure what T stands for, but it should clear from your first three statements that the conclusion M(s) follows. From the rules of the propositional calculus, the statements P→Q and P imply Q; in your question, P is H(s) and Q is M(s).

Last edited by JaneFairfax (2007-06-28 04:28:55)

rockysheedy

Member

Registered: 2006-06-11

Posts: 12

Re: What is rule P and T in Predicate Calculus?

Thank you for the explanation.

Why are these {1}, {3} and  {1,3} used in the above problem?

Does T means Tautology? I know that CP means Conditional Proof.

If the question is itself half? Please let me know.

Does T, (2), (3) in last step means that Rule T is applied to Step number 2 and 3?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: What is rule P and T in Predicate Calculus?

The numbers are just to label the steps in the deduction.

I don’t think T stands for “tautology”. The rule used to derive the last step is that from P and P→Q, the statement Q may be logically inferred. I don’t know the actual name for this rule though. But it’s definitely not a tautology. (A tautology is a statement that is always true, such as Pv¬P (v = or).)

rockysheedy

Member

Registered: 2006-06-11

Posts: 12

Re: What is rule P and T in Predicate Calculus?

Nevermind you have give a direction in which it will be easier for me to proceed now on.

Do you have any idea about what are these {1}, {3} and  {1,3} used in the above problem?

does T means syllogism?

Last edited by rockysheedy (2007-06-29 03:33:43)