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I am trying to keep the concepts of metalogic and logic as separate as I can to avoid confusion, but I still get a little lost when we talk about validity and soundness. I look at past answers and they seem a little hazy.

Are these metalogical or logical concepts? Is there such a thing as syntactical validity/soundness? Semantic validity/soundness? How are these terms used and defined? Why are they important exactly? Are we able to say anything useful even if something is invalid or unsound?

Do these definitions change depending on the logic we're talking about? (propositional calculus, predicate calculus, Hilbert-style, ND-style, etc). Does the concept of validity/soundness only apply to a 2-valued logic system with true/false?

I suppose in particular I am focusing on classical propositional calculus but it would be nice to know if things change outside of that context as well.

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Definitions

An argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. It is not required that a valid argument have premises that are actually true, but to have premises that, if they were true, would guarantee the truth of the argument's conclusion.

A formula is valid if and only if it is true under every interpretation, and an argument form (or schema) is valid if and only if every argument of that logical form is valid.

In propositional logic, a valid formula is also caleld a tautology.

An argument is sound if it is valid and all the premises true.

A logical calculus (language + formation rules (defining well-formed formulas) + axioms + inference rules) with its semantics is sound (or has the soundness property)

if and only if every formula that can be proved in the system is logically valid with respect to the semantics of the system.

A logical calculus is complete

with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems.

The term "complete" is also used without qualification, referring to the property of semantical validity. Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true.


Logical systems different from the classical ones, like e.g Intuitionistic logic and Modal logic, are sound and complete wih respect to the relevant semantics : see Kripke semantics.


The study of classical propositional logic is useful because we can see how the basic "machinery" works in a very simple case.

We have the language made of propositional variables : $p_1, p_2, \ldots$, the usual truh-functional logical connectives and the rules for producing well-formed formulas.

We have to define the truth-valuation i.e. a function

$v : \text {Prop} → \{ \text T, \text F \}$,

where Prop is the set of propositional variables of the language.

Then we extend the valuations to all formulas of the language using the usual truth-tables for the propositional connectives.

Example : if formula $\alpha$ is $(p_1 \land p_2)$, we have that $v(\alpha)= \text T$ iff $v(p_1)= \text T$ and $v(p_2)= \text T$, and so on.

Thus, to check that our preferred calculus is sound is an easy task. We have to :

(i) verify that all axioms are tautologies (using truth table);

(ii) verify that the inference rules preserve truth. In the case of modus ponens, this is again a simple application of the truth table for $\to$.

The completeness of the calculus is a mathematical problem that is much harder to solve.

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  • $\begingroup$ Are we defining premise and conclusion as $p \to q$ or is it instead $p \vdash q$, i.e. are we talking about validity within-the-system or outside-the-system? $\endgroup$ – user525966 Sep 5 '18 at 19:42
  • $\begingroup$ What is "true under every interpretation" exactly? Is there an example of this? $\endgroup$ – user525966 Sep 5 '18 at 19:44
  • $\begingroup$ Is it not possible for something to be sound, but invalid? $\endgroup$ – user525966 Sep 5 '18 at 19:45
  • $\begingroup$ @user525966 - in classical propositional logic, "true under every interpretation" are tautologies; see truth table. $\endgroup$ – Mauro ALLEGRANZA Sep 5 '18 at 19:45
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    $\begingroup$ @user525966 This is the last I'll answer here (other things to do... also so as to not flood Mauro's inbox) Remember what I said about saying 'not valid' rather than 'invalid'. Yes, it is not a valid formula provided $p$ and $q$ are sentence variables. The fact that it is true under some interpretation is expressed by saying it is satisfiable. (And since you got it in on time: in mathematical logic, 'sound' is usually a term that describes a deductive system, not a statement.) $\endgroup$ – spaceisdarkgreen Sep 6 '18 at 3:38

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