A truth-tables based standard of valid rules of inference?

[blowfly]

Hi all,

I've recently learnt about propositional logic in a fairly unique way - by developing truth table generation software for arbitrarily long & complex propositions. It was a great experience and I learnt heaps about the nuts and bolts of propositional logic. (More than I would have gotten from a text book I dare say!)

In doing so I learnt about the various rules of inference (which I understand to mean the transformation of one theoreom into another) - Conjunction, De Morgans, etc.

And then I forgot them all, because actually, I don't see why they're required.

Surely, in this day and age with computer software, we shouldn't need rules of inference? All we need to do to confirm proposition Y can be inferred from proposition X is to crunch the truth table tables of the proposition "X -> Y", confirm each interpretation results in "True", and hey presto, it's a valid inference.

What do you think? Can the validity of an inference from X to Y be derived simply by checking that the proposition "X -> Y" produces True for all possible interpretations of the variables in each proposition?

Thanks in advance,
-blowfly

P.S. This difference of approach became obvious to me when I wrote an argument in propositional logic (including "3: ~(M & P), 4: P -> ~M (from 3)"), someone said "Uh, what rule of inference gets you from 3 to 4?" and I replied, I have no idea, I just crunched the truth tables

[kennethamy]

Originally Posted by blowfly

Surely, in this day and age with computer software, we shouldn't need rules of inference? All we need to do to confirm proposition Y can be inferred from proposition X is to crunch the truth table tables of the proposition "X -> Y", confirm each interpretation results in "True", and hey presto, it's a valid inference.

What do you think? Can the validity of an inference from X to Y be derived simply by checking that the proposition "X -> Y" produces True for all possible interpretations of the variables in each proposition?

Yes, of course. Every inference has a corresponding conditional whose antecedent consists of the premises of the inference, and the consequent consists of the conclusion of the inference, and the corresponding conditional of every valid inference is a logical truth or a tautology, and the corresponding conditional of every invalid inference is a contingent proposition, which is true under some interpretations of its variables, and is false under others.

[blowfly]

Originally Posted by kennethamy

Yes, of course. Every inference has a corresponding conditional whose antecedent consists of the premises of the inference, and the consequent consists of the conclusion of the inference, and the corresponding conditional of every valid inference is a logical truth or a tautology, and the corresponding conditional of every invalid inference is a contingent proposition, which is true under some interpretations of its variables, and is false under others.

So what's the need for inference rules in the first place? The tautologies we happen to permit as inferences rules seem fairly arbitrary (ANY tautologous proposition "A -> B" should count as a valid rule, not just the "standard" ones), and the rules seem unnecessary to begin with if we can just check truth tables.

Are they just a historical artifact - ie. crunching truth tables was too time consuming before computers came along?

[goapy]

Originally Posted by blowfly

I've recently learnt about propositional logic in a fairly unique way - by developing truth table generation software for arbitrarily long & complex propositions. It was a great experience and I learnt heaps about the nuts and bolts of propositional logic.

Originally Posted by blowfly

So what's the need for inference rules in the first place? The tautologies we happen to permit as inferences rules seem fairly arbitrary (ANY tautologous proposition "A -> B" should count as a valid rule, not just the "standard" ones), and the rules seem unnecessary to begin with if we can just check truth tables.

Are they just a historical artifact - ie. crunching truth tables was too time consuming before computers came along?

I wouldn't say the rules are arbitrary at all. They are instances of some of the most common and often repeated inferences. Understanding these inferences is a worthwhile activity in itself. Producing derivations in a natural deduction system is a challenging exercise in problem solving. Working with derivations often helps in computer science and reasoning in general.

You say you learnt a lot by coding your software. How would those who only use your software accomplish the same understanding? Would the understanding be lost?

[kennethamy]

Originally Posted by blowfly

So what's the need for inference rules in the first place? The tautologies we happen to permit as inferences rules seem fairly arbitrary (ANY tautologous proposition "A -> B" should count as a valid rule, not just the "standard" ones), and the rules seem unnecessary to begin with if we can just check truth tables.

Are they just a historical artifact - ie. crunching truth tables was too time consuming before computers came along?

It is interesting and enlightening to do demonstrations in logic. Formal logic is not just for getting answers. Ahd there is second order (predicate) logic that cannot be done in terms of truth-tables. And there is modal logic too. And there are other kinds of issues in logic like completeness and consistency. Anyway, why don't you look up Godel's theorem which concern proof theory. Logic is a large area, and proofs are only a part of logic.