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I don't understand this definition of validity from Wiki:

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.

When we are talking about premises and conclusions are we specifically defining this as $P \to Q = \text{True}$ where $P$ is the premise and $Q$ is the conclusion? This validity technically requires us to have given semantic meaning to the operator $\to$, correct? Or is premise and conclusion something we can talk about outside of a logical system, e.g. $P \vdash Q = \text{True}$, if this is even a concept, or does it not even make sense to talk about semantics on a metalogical level?

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.

What does this even mean? "True under every interpretation"? What is an argument form/schema? How we defining formula here?

user525966
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  • I don't believe the ideas of 'valid argument' and 'argument scheme' are mathematical, although there are mathematical ideas they closely resemble. The part about valid formulas is certainly mathematical. To understand what 'true under every interpretation' means, you need to understand what an interpretation is and what it means to be true under an interpretation (which depends on the specific language and the specific semantics for it that you are using)... click through to https://en.wikipedia.org/wiki/Interpretation_(logic) – spaceisdarkgreen Sep 06 '18 at 00:16
  • For instance, what we might colloquially call a 'valid mathematical argument' from premises $\Gamma$ of the conclusion $\phi,$ can be formalized into a proof in some formal system. The argument 'taking a form' such that the premises can't be true and the conclusion false, corresponds to us using a sound system, i.e. one with true axioms and truth-preserving rules of inference. Of course, we must know what 'true' means for this to make sense, whereas we don't need any internal notion of truth for statements in a formal system to talk about what it proves and what it doesn't. – spaceisdarkgreen Sep 06 '18 at 00:39
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    A formula of prop logic is valid (i.e. a tautology) when it is true for every assignment of truth values to its prop variables. Consider e.g. $\lnot p \lor p$ and check its validity with truth table. – Mauro ALLEGRANZA Sep 06 '18 at 06:34
  • An argument (of prop logic) like e.g. $p, p \to q \vDash q$ is valid because every truth assignment to the prop variables that evaluates to TRUE all the premises evaluates to TRUE also the conclusion. Check it with truth table. – Mauro ALLEGRANZA Sep 06 '18 at 06:36
  • See van Dalen's textbook, page 15, for the semantics of propositional logic. See page 18 for the formal def of tautology (i.e. "true under all valuations" (or interpretations)) and the def of "a formula $ϕ$ being a semantical consequence of a set of formulas $Γ$" (that formalize the concept of valid argument (or "following logically from")). – Mauro ALLEGRANZA Sep 06 '18 at 09:06
  • Regarding "form/schema", see the post : Is an argument valid simply if its form is valid ? – Mauro ALLEGRANZA Sep 06 '18 at 13:21

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An argument, as intended in the page you mentioned, consists of a collection of premises, used to establish the truth of one (or more) conclusion.

If you were to model this in, say, propositional logic, you would call the premises $p_1, \dotsc, p_n$ and the conclusion $c$. Then, the argument would be encoded by the formula $$ p_1 \land \dotsb \land p_n \implies c $$ To attach a semantic meaning to this formula, i.e. if we want to establish if it is true or false, we need two ingredients:

  1. The truth values of $p_1,\dotsc,p_n$ and $c$ - you need to fix such values to obtain the truth value of the whole formula; the way you assign this truth values gives you an interpretation.
  2. A "meaning" for the logical connectives. This means, for example, that the truth value of the conjunction $\land$ can be computed by means of a function (and same goes for the implication).

If we call our interpretation $I$, we say that a formula is satisfied by $I$ (or true under that interpretation) if by assigning the truth values of all the variables as specified in $I$ and then computing the truth values of the logical connectives, the output is true.

As a mathematical convention - this is how implication is defined - a formula of the form $A \implies B$ is false when $A$ is true and $B$ is false; in all the other cases, it is true. This means that, if the premise $A$ is false, the overall formula is true, no matter the value of $B$. But if $A$ is assumed to be true, then $B$ must be true for the argument to be true.

This means that for an argument to be valid you must be free to give any possible value to each of your variables and still obtain a true formula. This can be generalized to arbitrary formulas (not only the one in argument form), and that is what the concept of tautology is about.

As an example, the formula $p \lor \neg p$ is a tautology: here, you only have two possible interpretations, one that makes $p$ true, the other makes $p$ false. You can choose any, and the formula turns out to be true.

Another example of a valid argument is $p \implies p$: assume that something is true; then, that thing is true. Here, you can again choose between two interpretations and no matter what your choice is, the formula is true.

According to the language you are using, there are different ways of defining formula and truth values. You can distinguish between propositional formulas (the ones described above), first-order formulas (as an example, $\exists{x}. p(x) \implies q(x)$), modal formulas and many others. You can choose how many truth values are there: true and false, or true, false and unknown, or infinitely many. Depending on the choices that you make here, the notion of truth and validity change. Above, I introduced the ones related to classical propositional logic.

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Let me try to give an answer at the basic level. I'll try to explain the problem in terms of distinction between material implication ( the ordinary --> ) and logical implication ( the metalevel relation symbolized by " ==> ")

Reference : on the distinction between " material implication" and " logical implication" see Lipschutz, Schaum's Outline Of Set Theory ( Ch14 Algebra of propositions) - Available at Archive.org

Your problem is that (1) the validity of a reasoning can be expressed using the " --> " operator in terms of validity of a conditional statement corresponding to the reasoning examined and -2) you want to conclude from this that the operator " -->" must belong to the metalevel, which would require giving a semantics to this operator at the metalevel...

Your objection contains 2 correct observations : first, the concept of validity actually belongs to the metalevel; you cannot say , inside the logical language itself (the object language) that a reasoning is valid, or that a formula is valid. That is why a special symbol is used , namely the symbol |= .

So we say , at the metalevel :

|= (A-->B)& ~A --> ~B

( Read " the formula (A-->B)& ~A --> ~B is a valid formula)

and

{ (A-->B), ~B} |= ~A

( Read " the reasoning with premises (A-->B)and ~B and with conclusion ~A is valid , in other words ~ B is a logical consequence of the premises (A-->B)and ~B ).

The second thing that is correct is that there are two equivalent ways to express the fact that a reasoning is valid.

(1) First you can do that in terms of sets of interpretations ( An interpretation is the fact of attributing a truth value to each atomic proposition, each possible interpretation - more precisely class of interpretations - corresponding, to a row in the truth table). So , here, we will say that a reasoning is valid iff

the set of interpretations in which all the premises are true is included in the set of all the interpretations in which the conclusion is true.

(2) The second, and equivalent, way to express the validity of a reasoning is to rephrase it in terms of validity of the corresponding conditional. Here you will say that a reasoning is valid iff :

in case the (conjunction of) the premises are true in an interpretation, the conclusion is also true in that interpretation, which means that the reasoning is valid just in case

in all possible interpretation ( IF the premises are true THEN the conclusion is true)

or, in other words , a reasoning is valid just in case

the FORMULA "( Premise 1 & Premise2&.....& Premise n --> Conclusion)" IS VALID ( = TRUE IN ALL POSSIBLE INTERPRETATIONS)

And here you can see what was wrong in your observation : we did not use the " --> " operator as a metalevel operator. What we are doing is that, at the metalevel we talk about a formula belonging to the object language level ( a formula that has --> as ordinary object language level operator), and we say, of this formula that it is a tautology. In what we have said, only the concepts of interpretation and of validity belong to the metalevel. The formula itself does not.

I however think that this mistake can be explained. Sometimes the symbol " ==> " is used to express at the metalevel that a conditional statement is not , in a sense, an ordinary one, on account of the fact that this statement is a tautology.

This symbol X ==> Y means " X implies logically Y" or, equivalently " Y is a logical consequence of X". This is not a new operator, to which you should attribue a semantics at the metalevel. It is symply an **abbreviation for the metalevel statement

" the object language level formula X --> Y is valid, is a tautology".**

So, instead of writing

{ (A-->B), ~B} |= ~A

or of writing

|= (A-->B)& ~A --> ~B

one can, more briefly, write

[(A-->B) & ~B] ==> ~ A ( Remark : I wrote " ==> " not " --> " )