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Really confused on the terminology here.

According to a few resources as-linked:

Valid is defined as a logical form where it is impossible to have true premises leading to a false conclusion. It doesn't mean the premises are necessarily true. Just that if they were true, the conclusion would need to be true as well.

Sound is defined as a valid logical form where the premises are indeed true.

But then we also have "semantic consequence" denoted $\Gamma \vDash \varphi$ which normally means "if everything on the lefthand side is true, then the stuff on the righthand side is true." Sometimes this is also referred to as "validity" or "tautology."

But this also seems like it is clashing with the definition of "sound" as well.

So I'm really quite confused how we're supposed to be defining and separating these terms out. I'm seeing different answers on Wiki, different answers in books, different answers in other Math StackExchange posts, etc.

I'm going nuts over here trying to separate all these concepts (metalogic vs. logic, soundness vs. validity, semantic vs. syntactic, etc) when every other resource is constantly merging stuff together in vague ways.

Once and for all what is the definition of validity? Soundness? Completeness? Syntactic consequence? Semantic consequence? Logical consequence? Tautology? Theorem? Axiom? Inference? Formula? Sentence? Expression? Proposition? Statement? Relation? Connective? Operator? Are any of these actually synonyms of each other? How are they different?

user525966
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  • That's something of a laundry list at the end, which I'm afraid makes the question too broad to fit our format here well. – hmakholm left over Monica Sep 10 '18 at 23:30
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    The definitions you're asserting here certainly don't match how I usually see the words used. – hmakholm left over Monica Sep 10 '18 at 23:32
  • That doesn't make those definitions match how I usually see the words used. If you're looking for a terminology that's consistent with those definitions, you might need to find a particular book that's using those words in your favorite way, and then ignore everyone who uses them differently. – hmakholm left over Monica Sep 10 '18 at 23:34
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    I can't tell if you're being snarky or not. I clearly can't just "ignore everyone". I want to understand the correct definitions of these words. I can't tell which resources have it right and which do not since there are so many differences, even among answers on this very website. Or am I left to surmise that there is no consistency and it's all very much informal and however people want to define them? – user525966 Sep 10 '18 at 23:35
  • You start by asserting definitions of the words that according to my experience are quite at odds with how they're usually used. I have trouble seeing which kind of answer you're expecting other than "Yes, user525966, we're going to throw our entire existing terminology away and instead start using the definitions you have graciously supplied in the disguise of asking a question." – hmakholm left over Monica Sep 10 '18 at 23:38
  • Are you saying the Wiki has it incorrect? – user525966 Sep 10 '18 at 23:39
  • I'm saying that if you want an explanation of anything you should not start by fixing what the only explanation you're prepared to accept MUST be. That's not a question, that's just fishing for people to agree with YOUR favored definition. – hmakholm left over Monica Sep 10 '18 at 23:40
  • It's not fishing. It's a direct question meant to separate the usual from the not usual. Does the wiki have it wrong or not? You say it's normally used in a different way -- how does the Wiki have it wrong, and what's the usual way you refer to instead? – user525966 Sep 10 '18 at 23:41
  • Your question here starts by ASSERTING some particular non-standard definitions of the words "valid" and "sound". Then you want definitions for a laundry list of other words -- presumably definitions that would work together with your asserted definitions of "valid" and "sound". I maintain that this is not a question. It is an attempt to tell the world how you have decided we must all use the words. – hmakholm left over Monica Sep 10 '18 at 23:43
  • Okay you're going off the rails here. I'm clearly not doing that and you're very clearly avoiding my very simple/direct question (twice now), which would easily prove/disprove your initial assertion or not. – user525966 Sep 10 '18 at 23:44
  • The first three paragraphs of the "question" are CLEARLY trying to assert some definitions of the word "valid" and "sound". That's what happens in English when you state: "Valid is defined as ..." – hmakholm left over Monica Sep 10 '18 at 23:46
  • My post is clearly asking for the definitions. I'm not "asserting" anything other than posting the definitions as-listed on Wiki and other resources (all of which are linked). "I'm" not asserting these things -- Wiki is, these other posts are, etc. It's the entire purpose of the post -- untangling all these definitions to get something consistent and commonly-used/understood. And yet when I ask you to tell me how the Wiki is wrong (which you imply it is), you refuse to elaborate. – user525966 Sep 10 '18 at 23:52
  • @user525966 People use the same words to mean different things. This is quite common in mathematics in general and is hardly unique to mathematics. This is just something you have to deal with. Yes, it does make it harder to understand what's going on. You simply have to look at how a word is being defined in each context. This is especially true in mathematical contexts, as there is often a need for a variety of words for similar concepts that are different for technical reasons that may or may not be important in other contexts. – Derek Elkins left SE Sep 10 '18 at 23:53
  • @HenningMakholm I added the line "According to a few resources as-linked:" to make it extra clear that I am not asserting these definitions, but if you click the links as-mentioned in the paragraphs, that's where the definitions are coming from. Usually that's the whole point of presenting links but I'm being extra explicit here. You say these definitions are non-standard. So I am asking you, how are they non-standard? How is the Wiki wrong? – user525966 Sep 10 '18 at 23:54
  • The definitions are non-standard because they do not match how the words are used in practice. "Valid" is used about a formula whose truth value is "true" in every interpretation of the non-logical symbols in it. "Sound" is used about a proof systems that proves only valid formulas (when used without non-logical axioms), and only proves formula that are semantic consequences of the non-logical axioms when such axioms ARE used. – hmakholm left over Monica Sep 10 '18 at 23:59
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    @user525966 The definitions you give for "valid" and "sound" strike me as more common in philosophical literature than in mathematics or (formal) logic. Obviously there is some connection intended between these more "philosophy" definitions and the mathematical ones, but they aren't the same. Personally, I find a lot of "logic for philosophy" to be quite bad and to create more confusion than it dispels. – Derek Elkins left SE Sep 10 '18 at 23:59
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    I have literally never heard the $\vDash$ relation referred to as a "tautology". A tautology is the same thing as a valid formula of propositional logic. – hmakholm left over Monica Sep 11 '18 at 00:01
  • One example is https://en.wikipedia.org/wiki/Double_turnstile#Meaning or Logic and Structure (Dirk van Dalen), page 18 (the book Mauro Allegranza recommended), etc – user525966 Sep 11 '18 at 00:12
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    In $\vDash\varphi$ is is not $\vDash$ that is a tautology. Rather, $\vDash\varphi$ asserts that $\varphi$ is a semantic consequence of nothing -- which happens to be the case when $\varphi$ is a tautology. – hmakholm left over Monica Sep 11 '18 at 00:39
  • Many of the above concepts are already discussed in your previous post : Defining premise and conclusion. – Mauro ALLEGRANZA Sep 11 '18 at 06:26
  • @HenningMakholm if something is a semantic consequence of nothing then that implies it's always true is it not? – user525966 Sep 11 '18 at 07:19
  • Or is it that $\vDash$ is the same as "valid" and happens to be the same as "tautology" is propositional logic? – user525966 Sep 11 '18 at 07:53
  • @user525966: Yes, something that is a semantic consequence of nothing is always true. However $\vDash$ is neither "valid" nor a tautology. $\vDash$ is the symbol for (a) the semantic consequence relation and (b) the evaluation relation in a particular interpretation. The latter of these is used to define validity (and tautologies) but it makes no sense to say that the $\vDash$ relation itself is "valid" or a "tautology". These are words that are used about fomulas, not metalogical relations! – hmakholm left over Monica Sep 11 '18 at 10:15
  • @HenningMakholm I'll rephrase. If we have the string $\vDash \varphi$ does this mean "$\varphi$ is valid" and "$\varphi$ is a tautology" and "$\varphi$ is always true" and "$\varphi$ is a semantic consequence of the empty set" and "$\varphi$ is sound"? – user525966 Sep 11 '18 at 10:51
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    @user525966: It means that $\varphi$ is "valid" and "always true" and "a semantic consequence of the empty set". If $\varphi$ is a propositional formula this is also expressed by the word "tautology". The word "sound" is not used about formulas. – hmakholm left over Monica Sep 11 '18 at 11:00
  • @HenningMakholm How can $\varphi$ be always true and yet not a tautology outside of propositional logic? – user525966 Sep 11 '18 at 11:11
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    @user525966: For example, in first-order logic the formula $\forall x(x=x)$ is always true, but is not a tautology. To the extent one uses "tautology" about non-propositional formulas at all, it means a substitution of a propositional tautology. The only propositional formula that can be substituted to give $\forall x(x=x)$ is a single propositional variable, and a single propositional variable is not a tautology. – hmakholm left over Monica Sep 11 '18 at 11:13
  • Interesting, makes sense, thanks – user525966 Sep 11 '18 at 11:19
  • While I like answering your questions on here, learning what the terminology means will not teach you logic (or any kind of math) very well alone. You have to go through books (and video lectures if you can find them) and do exercises. If you haven't seen this guide, it's a good place to start. http://www.logicmatters.net/resources/pdfs/TeachYourselfLogic2017.pdf When you commit to a book, you can simply trust its terminology and definitions, and then look to wikipedia and other books to contextualize. – spaceisdarkgreen Sep 12 '18 at 04:03
  • @spaceisdarkgreen Of course, but not knowing the terminology makes it impossible to go further into doing anything more with it – user525966 Sep 12 '18 at 07:35
  • @spaceisdarkgreen I have indeed seen that document before, though I found it super verbose and I had frustrations with most of his recs. – user525966 Sep 12 '18 at 08:11

2 Answers2

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It is important to stick to mathematical logic here as opposed to the definitions from philosophy. The first sentence of the wikipedia

In mathematical logic, a logical system 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.

while not fully precise, is the sense in which 'sound' is generally used in mathematical logic. This corresponds to a system having valid axioms and validity-preserving rules of inference, so you can see how it corresponds loosely to the philosophical notion of true premises and correct (i.e. truth-preserving) argumentation.

As for validity, the wikipedia sentence

A formula of a formal language is a valid formula if and only if it is true under every possible interpretation of the language. In propositional logic, they are tautologies.

uses 'valid' in the way it is used in mathematical logic. Here the connection to the philosophical notion of a valid argument that you cite is a little less direct. Let $A$ and $B$ be your premises and $C$ your conclusion. And say $$ (A\land B) \to C $$ is valid in the mathematical sense. That means that $A\land B \to C$ is true in any interpretation, which means that it is impossible for $A$ and $B$ to be true and for $C$ to be false. So the two notions of validity connected, but even more loosely than was the case for soundness.

Note that the validity of $(A\land B)\to C$ is the same thing as $C$ being a semantic consequence of $A$ and $B,$ i.e. $$ A,B\models C$$ which has more of the flavor of "a valid argument from $A$ and $B$ to $C$," though strictly speaking, there's no 'argument' here. (However, the syntactic variation $A,B\vdash C,$ which is equivalent in the presence of a completeness theorem, means there's a proof of $C$ from assumptions $A$ and $B$... that's more of an 'argument'.)

I've tried to be conciliatory to the philosophical definitions here, but in math it's important to get comfortable with the fact that precise definitions vary from treatment to treatment. (Heck, the other day I learned that not all books have the same definition of 'compact set' in a general topological space, which was a definition that I thought was more or less sacrosanct.) It should also go without saying that the definitions don't need to conform to the colloquial meaning of the words, though it's nice when they're at least not wildly misleading.

In a field with as many moving parts in mathematical logic we will often need to adjust and adapt our terminology as we develop new ideas and try to apply mathematical logic to more exotic situations. Even in the beginning of my answer where I gave some 'general purpose' definitions and remarks, I had some particular contexts in mind, and we will need to make these things fully precise in any detailed treatment. For instance, when we move to the specific context of classical propositional logic, "valid" becomes synonymous with "tautology" and we make precise the ideas of "interpretation" and "truth" in a given interpretation. When we flesh things out in a given context (especially when working in more exotic situations than, say, classical propositional or predicate logic) everything I said is subject to revision.

spaceisdarkgreen
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    As a rough guideline, if you see words like "argument", "logical form", "syllogism" or a heavy emphasis on "premises" and "tautology", you are probably in "philosophy logic" land. In mathematical logic land, you'll more see "(well-formed )formulas", "proof", "derivation", "rules( of inference)", "model", "semantics", and, obviously, a heavy emphasis on symbolic expressions. This isn't a perfect discriminator though, especially for older works. – Derek Elkins left SE Sep 11 '18 at 00:31
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    @Derek Agree with all of these except “tautology” is used quite a bit in mathematical logic. – spaceisdarkgreen Sep 11 '18 at 17:46
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The notation $\Gamma \models \varphi$ means $\varphi$ is true in every structure in which all statements in $\Gamma$ are true.

The notation $\Gamma \vdash \varphi$ means $\varphi$ can be proved by using the statements in $\varphi$.

The latter depends on some notion of proof. One wants such a notion to satisfy three desiderata:

  • Soundness, i.e. if $\Gamma\vdash\varphi$ then $\Gamma\models\varphi.$
  • Completeness, i.e. if $\Gamma\models\varphi$ then $\Gamma\vdash\varphi.$
  • Effectiveness, i.e. there is a proof-checking algorithm, which will correctly identify its input as either a valid proof or not.

Note that $\Gamma$ is allowed to be a set of statements whereas $\varphi$ is just one statement. So why not conjoin all the statements in $\Gamma$ into one? The problem here is that if one is allowed to join infinitely many statements into one, then putting such an infinite conjunction in the role of $\varphi$ has the result that one cannot satisfy all three desiderata simultaneously. Any proof of that takes a lot of work.

Michael Hardy
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  • Does $\vDash \varphi$ (empty left-hand side) mean $\varphi$ is a tautology? – user525966 Sep 11 '18 at 07:45
  • @user525966 : It means $\varphi$ is true in every "structure" (I haven't been explicit here about what a "structure" is). Sometimes people reserve the word "tautology" for things in propositional logic rather than predicate logic (predicate logic uses universal and existential quantifiers; propositional logic only uses Boolean connectives such as "and", "or", and "not"). Sometimes people express it by saying this means $\varphi$ is "universally valid". – Michael Hardy Sep 11 '18 at 17:56
  • How would you describe "structure"? Is this the same as "interpretation" or "model"? What does it mean? – user525966 Sep 11 '18 at 18:25
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    @user525966 In the standard semantics of classical first order logic, sentences are interpreted in structures, i.e. structures are the interpretations. For instance, the first order sentence $\forall x \exists y(x R y)$ can be interpreted in the structure $(\mathbb R, <)$ (where we're interpreting the relation symbol $R$ as $<$) to mean the reals have no maximal element. So it's true in this structure. If a set of first order sentences are all true in a structure, we say the structure is a model of the set of sentences. https://en.wikipedia.org/wiki/Structure_(mathematical_logic) – spaceisdarkgreen Sep 12 '18 at 03:28