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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.

user525966
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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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Mauro ALLEGRANZA
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  • 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? – user525966 Sep 05 '18 at 19:42
  • What is "true under every interpretation" exactly? Is there an example of this? – user525966 Sep 05 '18 at 19:44
  • Is it not possible for something to be sound, but invalid? – user525966 Sep 05 '18 at 19:45
  • @user525966 - in classical propositional logic, "true under every interpretation" are tautologies; see truth table. – Mauro ALLEGRANZA Sep 05 '18 at 19:45
  • Does "every interpretation" mean "all possible values across all possible inputs"? e.g. $a \land b$ has interpretations $(0 \land 0)$, $(0 \land 1)$, $(1 \land 0)$, $(1 \land 1)$? – user525966 Sep 05 '18 at 19:47
  • @user525966 Yes, that's pretty much what it means in this semantics. If $a$ and $b$ are propositional variables, then for sentences just involving $a$ and $b,$ there will be four relevant classes of interpretation corresponding to $(a,b) = (0,0), (0,1), (1,0), (1,1)$ ('classes' because the other propositional variables aside from $a$ and $b$ can take various truth values, but who cares what they are). We see $a\land b$ takes the value $0\land 0 = 0$, $ 0\land 1 = 0$, $1\land0 =0$, and $1\land 1 = 1$ respectively. Thus we conclude $a\land b$ is not a valid formula, nor is its negation valid. – spaceisdarkgreen Sep 06 '18 at 02:24
  • @spaceisdarkgreen But if $a \land b$ is invalid, why do we use it in practice? It seems like quite a fundamental operator? – user525966 Sep 06 '18 at 02:29
  • @user525966 You can use something without asserting it is always true... I use the equation $3x + 4y =0$ even though it isn't true for all $x$ and $y.$ – spaceisdarkgreen Sep 06 '18 at 02:31
  • But then this would imply the only valid things are tautologies and everything else is invalid, no? That seems strange to me that only tautologies can be considered logically valid – user525966 Sep 06 '18 at 02:40
  • @user525966 Valid means it is true by virtue of its form, i.e. its truth is not contingent on the circumstances. If I told asked you if "(proposition 97) and (proposition 12)" is true, you'd look at me strange and say, "well, that depend what proposition 97 and proposition 12 are, in particular, whether they're true or false." On the other hand if I asked you if " either (proposition 80) or the negation of (proposition 80)" is true, you would say 'yes' without needing to know what proposition 80 is. – spaceisdarkgreen Sep 06 '18 at 02:42
  • @user525966 That is exactly what it implies. Why is that strange? (Perhaps the connotation of invalid? We generally say 'not valid' rather than 'invalid' as the negation of 'valid' in this context, presumably for this reason.) – spaceisdarkgreen Sep 06 '18 at 02:43
  • @user525966 Sometimes other qualifiers are used to highlight the distinction between "valid" and the colloquially synonymous but mathematically very different "true". For instance you often hear "universally valid". (and it doesn't help that 'valid' has other non-mathematical meanings as well, such as pertaining to arguments rather than statements). – spaceisdarkgreen Sep 06 '18 at 02:48
  • Does this mean true (i.e. just $\top$) is valid/a tautology? And false (i.e. $\bot$) is invalid / not a tautology? – user525966 Sep 06 '18 at 03:02
  • @user525966 sure. But we can say something stronger about false. It is the negation of a valid formula... i.e. something that is false in all interpretations, also known as a contradiction or an unsatisfiable formula. – spaceisdarkgreen Sep 06 '18 at 03:29
  • @spaceisdarkgreen So the $p \to q$ statement is also invalid because it is false when $T \to F$? Even though it isn't possible for it to be false when the premise is true? – user525966 Sep 06 '18 at 03:34
  • So a sound statement is one that is a tautology and happens to have true premises? – user525966 Sep 06 '18 at 03:38
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    @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.) – spaceisdarkgreen Sep 06 '18 at 03:38
  • Yes; the propositional constant $\top$ is a tautology, because we interpret it as denoting TRUE, and thus, as a formula, it is always true. The propo const $\bot$ is a contradiction, i.e. always false, for the same reason. – Mauro ALLEGRANZA Sep 06 '18 at 06:17
  • $p \to p$ as well as $\lnot p \lor p$ are tautologies. $\lnot p \land p$ is a contradicition. $p \to q$ is neither taut nor contra (someone call it : contingent). You have to try to use truth table: it is the best way to learn the machinery of the (elementary) semantics of prop logic. – Mauro ALLEGRANZA Sep 06 '18 at 06:32