Talk:Validity

From Wikipedia, the free encyclopedia

Contents 1 Disputed 2 Fixed 3 Corrections 4 Models of Propositional Calculus 5 A better definition 6 Appeal to authority

[edit] Disputed

The content of this article is contradicted by an authority. See Talk:Logical_argument#RV_20040302.27s_edit for details. ---- Charles Stewart 13:02, 28 Oct 2004 (UTC)

I read the linked article regarding the dispute. Different sources may provide different definitions for validity and soudness (and adequacy and correctness). It is generally accepted that valid statements are true under every interpretation. Wittgenstein 1921 calls such statements 'tautologous' or 'tautologies'. The same applies to sets of statements (i.e. arguments). If all the members of S are true under every interpretation, S is a valid argument. Or you could simply say that an argument is valid iff its corresponding statement is true under every interpretation. I.e. {P->Q, P, Q} is valid iff P->Q->P->Q (or [(P->Q)&P]->Q) is true under every interpretation (is valid). Statements aren't said to be 'sound'.

I've seen in numerous texts soundness being applied to informal arguments - arguments that are interpreted. They are not called 'valid' because they are not true under every interpretation. They are sound, however, because i)the premises are true; ii)the conclusion is true; iii)the premises and conclusion are relevantly related; in other words, they do not commit any informal fallacy. Soudness may also be applied to theories. Nortexoid 04:01, 11 Nov 2004 (UTC)

"The same applies to sets of statements (i.e. arguments). If all the members of S are true under every interpretation, S is a valid argument."

This is wrong. A valid argument is not a set of tautologies (think how useless this would be); it is a set of statements where the conclusion necessarilly follows from the premises.

It's a Larry's Text page. It might be better to start again from scratch. Charles Matthews 10:28, 22 Nov 2004 (UTC)

[edit] Fixed

I struck some text but kept what read properly to me. You are invited to improve it. Ancheta Wis 04:09, 28 Feb 2005 (UTC)

Do not remove the definition of 'valid formula'. It is as primary a use of 'valid' as that applied to arguments.Nortexoid 02:17, 2 Mar 2005 (UTC)

[edit] Corrections

I made a few corrections. First, it was said that a valid formula is true under every valuation. In model theory, a valuation is a part of an interpretation (or model or structure), but it is not the whole thing. A valid formula is true under every interpretation. (In sentential logic, a valuation is an assignment of truth values to sentence letters. Here valuation and interpretation are the same. But the concept of validity is not restricted to sentential logic.) Second, tautologous was cited as synonymous with valid. But tautology is a term of sentential logic. Again, this is too restrictive.

I also added a bit on the relation between valid argument and valid formula. I suppose that expands a bit on the Fixed comment above by Charles Matthews. --JMRyan 00:20, 12 August 2005 (UTC)

An atomic valuation and an interpretation amount to the same thing in predicate logic. An interpretation will induce a unique atomic valuation. Conversely, an atomic valuation v (on a domain D) will have an associated interpretation which defines each relation (or property) P as the set of n-tuples <a1,...,an> from D such that P(a1,...,an) is true under v. It doesn't really matter how it's phrased unless someone is unaware of their equivalence. Nortexoid 05:44, 12 August 2005 (UTC)

[edit] Models of Propositional Calculus

Presumably any model of propostional calculus would consist of a truth assignment to each atomic proposition (and then to every wff using the standard semantics of ^,v,¬ etc.) such that the axioms hold. Is this right?

[edit] A better definition

Here's the current definition of a valid argument: "An argument is said to be valid if, in every model in which all premises are true, the conclusion is true."

Under this definition it's ambiguous whether an argument is valid when it contains a self-contradictory premise (since there would be no model in which all the premises are true). At least the way I learned it, we want such arguments to be considered valid. So a better definition would be:

"An argument is valid if and only if it is impossible for all the premises to be true and the conclusion false."

It's not ambiguous. If an argument has an inconsistent set of premises then it is vacuously true that "an argument is valid if, in every model in which all premises are true, the conclusion is true". Note that the antecedent of that conditional fails in the case of inconsistent premises, and so the conditional will be "vacuously" true.

Your definition invokes these strange locutions involving impossibility that we'd be better off without. Nortexoid 19:10, 21 December 2005 (UTC)

Okay, whatever. I'm fine with the other definition... but if there are inconsistent premises, are you sure "the conclusion is true in every model in which all premises are true" would be false? To me it makes just as much sense to call that true, if there are no models in which all premises are true.

That is precisely what was said. If there is no model for the premises taken together, then it is vacuously true that "the conclusion is true in every model in which all the premises are true". Is is not false. Nortexoid 02:17, 23 December 2005 (UTC)

[edit] Appeal to authority

I'd like a second opinion in the discussion at Talk:Appeal to authority...I could be wrong, or arguing poorly... NickelShoe 15:46, 2 February 2006 (UTC)