Basic Mathematics - 2018-12-09

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tarit goswami

06:23

@user21820 Hi, :) Can we continue our discussion?

@user21820 What does it mean then by validity checking?

user21820

@taritgoswami I don't use such terminology ("premise" and "conclusion" and "valid") because they are very misleading. Instead, what you need to know is the meaning of propositions (defined via truth-tables) and the rules for a deductive system.

I presume you already know truth-tables.

tarit goswami

@user21820 Yes, I know

user21820

Based on them, we can ask whether a propositional sentence is true under every interpretation (of propositional variables).

If so, we call it a tautology.

tarit goswami

user21820

We may also work in the context of some assumptions, called axioms. In that case, we would be interested in what sentences are true under every interpretation that satisfies all the axioms.

tarit goswami

06:39

Understood

user21820

@taritgoswami In many simple cases, such as the one you asked about, we do not need any axioms. Instead, that should be expressed as an implication:

> ( A and B implies C ) and ( C implies D ) implies ( not D implies not A or not B ).

Where A = ( I try ) and B = ( I have talent ) and C = ( I will become a mathematician ) and D = ( I will be happy ).

Take a while to understand the above, and remember the precedence rules I told you about. You can see that without precedence rules it will be much harder to grasp the logical structure of the sentence (due to too many brackets)!

tarit goswami

Ok, got it

user21820

Ok so the ultimate question, in this case, is whether or not this is a tautology.

(That is all that actually matters; there is no need or reason to talk about "premises" and "conclusions", which have in my experience confounded numerous students.)

Firstly, you can via truth-tables establish that indeed this is a tautology.

Secondly, you should learn to use a deductive system to actually prove this tautology.

Let me know when you're done with the first.

tarit goswami

@user21820 Ok

So an implication, to be valid it need to be always true or tautology, right?

user21820

@taritgoswami I didn't and won't use the term "valid".

tarit goswami

06:51

I mean, the question useing the term "validity"

user21820

I know, but you have to understand what you actually want to know, which in this case is whether the implication is a tautology or not. Think about the meaning of the implication. The question you cited is unimportant.

tarit goswami

@user21820 ok

@user21820 Yes, all are true.

From TT

user21820

Good. So the thing is that truth-tables work for propositional logic, but will be completely useless for first-order logic. So we need a deductive system, and we might as well start with one for propositional logic.

You can refer here:

A: Predicate logic: How do you self-check the logical structure of your own arguments?

Truth tables are not enough to capture first-order logic (with quantifiers), so we use inference rules instead. Each inference rule is chosen to be sound, meaning that if you start with true statements and use the rule you will deduce only true statements. We say that these rules are truth-preser...

Up to the section "Boolean operations".

tarit goswami

Ok, I will read it slowly and let you know :)

user21820

07:19

@taritgoswami: You can take a look at simple examples here. (Both are linked from my profile.) And here are some exercises:

(1) A and ( B or C ) implies ( A and B ) or ( A and C ).
(2) A or ( B and C ) implies ( A or B ) and ( A or C ).
(3) ( A or B ) and ( A or C ) implies A or ( B and C ).
(4) ( A or B ) and ( B or C ) and ( C or A ) implies ( A and B ) or ( B and C ) or ( C and A ).
(5) ( A implies B or C ) implies ( A implies B ) or ( A implies C ).

Post your attempts here and I will check them. Let me know if you have any questions.

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