# Talk:Interpretation (logic)

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## Satisfiability and validity article

I just created Satisfiability and validity for reasons I mentioned on Talk:Logical connective#Boolean bias, and forgetting about the existence of this article. Does it make sense to keep that article as a navigable placeholder for the four concepts it mentions, or should it be nixed? — Charles Stewart (talk) 20:53, 26 June 2009 (UTC)

## Why logical semantics and plain old semantics have nothing to do with each other

This is meant to explain why we should use the word "extension" in place of "meaning" for the purposes of this article. Perhaps there is a strict use of the word "meaning" that allows us to equate the two, but common usage doesn't support this.

Take the formal predicate: "Tall," and the singular terms "a" and "b". a = Abraham Lincoln and b = Benjamin Franklin (since Kripke it's been generally assumed that proper names behave like singular terms). Our interpretation function assigns {a} or {Abraham Lincoln} to the extension of the predicate Tall. If interpretation assigns "meaning" (as the article previously stated) then we're committed to saying that Tall just means Abraham Lincoln here. It doesn't, it just has Abraham Lincoln in its extension. Therefore interpretations don't assign meaning, but extension. There's no point in using the term "meaning," which comes with extra baggage even in Philosophy of Lanugage, when we already have the term extension at hand. To do otherwise would be misleading.--Heyitspeter (talk) 23:21, 23 September 2009 (UTC)

No, I don't think this is correct. The point of an interpretation is to assign meaning. In a Tarskian structure, the meaning of a predicate symbol is its extension, but Tarskian structures are not the only possible interpretations. Even within very conventional 20th-century math logic terms, for example, the predicate "is a set" has no extension, because the collection of all sets cannot be a completed totality. Nevertheless set-theoretic utterances, in which the variables range over all sets, have meaning.
The word interpretation is more general than the strict sense you're giving it, and the article needs to reflect that. --Trovatore (talk) 23:35, 23 September 2009 (UTC)
There are also issues such as modal logic, where the interpretations of square and diamond in terms of Kripke models are certainly "interpretations" but they do not give "extensions" in any clear sense. Indeed, it's only really relation symbols that are really given "extensions" – what is the "extension" of a universal quantifier? Of the conjunction symbol? — Carl (CBM · talk) 01:35, 24 September 2009 (UTC)
Response to both, I guess: Tarskian semantics don't have anything to do with "meaning" as commonly conceived either. I agree with you that noted logicians have used the words "semantics" and "meaning" to describe logical systems, but I think it's evident that this terminology is misleading, and that there are alternative terminologies that don't mislead. An interpretation is just a function that takes a symbol and yields a set of things. That's all. If interpretation assigned meaning, then coextensional predicates would mean the same thing. If, in our domain (let's take the domain to be our universe), all things with hearts had livers, then "having a heart" would mean the same thing as "having a liver." This simply isn't the case. Though the word "meaning" can be taken to mean the same thing as "extension," it just doesn't in common parlance. Using the same words leads to confusion (some of which we see here).
To Carl in particular, your comments make me wonder whether I've learned an entirely different logic than you. I don't mean that as an attack, just a note; I know there are many ways to construct a logical system. But as I learned it, you start with an interpretation function I. Then define a valuation function v that takes a variable x as argument and yields an object o. All(x)P(x) is true in our model iff for all valuations v(x), v(x) is in the extension of P as given by our interpretation function (i.e., is in the set I(P)). As you say, there is no extension for a universal quantifier. There is also no interpretation (logic) given to it.
But note the difference between interpretation (logic) and interpretation (mental event). The former can only define the extension of P. The latter can make the claim that P means plant, or some such. These are different processes.--Heyitspeter (talk) 03:38, 24 September 2009 (UTC)
Tarskian semantics don't have anything to do with meaning??? Sure they do. Popper dropped his objection to the notion of "truth" precisely in response to Tarski.
I think you're trying to force an overly deflationary agenda on this article. I'm not arguing against that agenda per se, just that it doesn't represent the whole spectrum of the notion of interpretation in logic. --Trovatore (talk) 03:56, 24 September 2009 (UTC)
You're equating truth-conditions with meaning. Tarksi never made this conflation. He said that "'snow is white' is True iff snow is white. He didn't equate the meaning of 'snow is white' with its truth condition. As reductio: "'snow is white' is True iff snow is white and 1+1=2. But 'snow is white' doesn't mean that snow is white and that 1+1=2. Tarski's T-schema can successfully give the truth-conditions for a statement, but it can't give the meaning. Davidson made this extrapolation, and this shows him to be wrong. Tarski never did. See Soames' "Truth, meaning, and understanding" if you like.--Heyitspeter (talk) 04:04, 24 September 2009 (UTC)
But "snow is white" does mean that snow is white. This is an interpretation, in the sense of logic, and on-topic in this article. --Trovatore (talk) 04:10, 24 September 2009 (UTC)
It does, but that's not my point. Sorry if I'm being brusque I've got stuff I'm in the process of doing. I just mean that Tarski's T-schema / semantics give the truth-conditions, but not the meaning.
Take these two accurate T-schemas for the sentence 'snow is white':
1) 'snow is white' iff snow is white
2) 'snow is white' iff (snow is white and 1+1=2)
(1) and (2) both give the truth-conditions for 'snow is white'. The left half is true iff the right half is in either case. But the right half is not the meaning of the left half in either case. Therefore, truth-conditions of sentences do not give the meanings of sentences. As to "interpretation", I can interpret logical sentences without interpreting them in the logical sense. The former is a thing I think, the latter is a mathematical function.--Heyitspeter (talk) 04:22, 24 September 2009 (UTC)
I believe you are too narrowly focusing on a single notion of logical interpretation. The reason I don't state this more strongly is that I'm not all that familiar with the alternatives. But take the case I mentioned, interpretation of formulas from set theory. Surely you don't want to exclude these, merely on the grounds that there is no extension for the predicate V? --Trovatore (talk) 04:25, 24 September 2009 (UTC)
I guess I figured that's why we created a page called "Interpretation (logic)" instead of weaving it into the page for "Interpretation" tout court. And insofar as the interpretations of formulas from set theory do not concern extensions, yes, I do want to exclude those from this article. You'd find that sort of thing under "Philosophy of logic," "Set theory," or "Philosophy of Set Theory." The "intepretation" you're latching onto here is synonymous with "analysis" or "evaluation." Those aren't synonyms for interpretation as it's used in formal logic.--Heyitspeter (talk) 04:35, 24 September 2009 (UTC)
Well, it seems to me that this is too narrow a focus for an article called interpretation (logic). The subject you're talking about can be treated in a handful of lines, and indeed is so treated, at structure (mathematical logic). A more general approach is needed here. --Trovatore (talk) 04:49, 24 September 2009 (UTC)
I agree in the sense that I think this article doesn't need to exist. Be that as it may, this article treats of interpretation functions, and it is more than a handful of lines long. And I think it needs to be more than a handful of lines long. I don't know about you, but I find the section on interpretation functions in the article you cite to be nearly incomprehensible.--Heyitspeter (talk) 04:54, 24 September 2009 (UTC)
IDEA!! What if we renamed the article "Interpretation function"? That would clear up everything. I don't know why I didn't think of that before.--Heyitspeter (talk) 04:56, 24 September 2009 (UTC)
Because that was never what this article was supposed to be limited to. Anyway an article called interpretation function would be redundant with structure (mathematical logic) -- just make a section redirect, pointing to structure (mathematical logic)#Interpretation function. --Trovatore (talk) 04:59, 24 September 2009 (UTC)
(went ahead and did it while it was on my mind) --Trovatore (talk) 05:02, 24 September 2009 (UTC)
Yeah, I guess I just feel that this general "explanation" of an interpretation function is *so* abstract it becomes worthless. Why wasn't this article supposed to be limited to "Interpretation function"?--Heyitspeter (talk) 09:06, 24 September 2009 (UTC)
Doing that would limit the article to predicate logic, but its goal is to cover interpretations in other logics as well. — Carl (CBM · talk) 10:39, 24 September 2009 (UTC)

### Interpretations of modal logic

Let me go back to interpretations that do not have extensions. In modal logic, there are two modal operators, $\square$ and $\diamond$. The usual interpretation of these is:

$\square \phi$ means φ is necessarily true
$\diamond \phi$ means φ is possibly true

This is formalized via Kripke models. However, there is another intepretation of the modal operators. One can interpret them as:

$\square \phi$ means φ is provable in PA
$\diamond \phi$ means φ is not disprovable in PA

This interpretation is part of provability logic.

In each case, the assignment of meaning to the modal operators (either via Kripke models, or by identification of the modal operators with provability) is an interpretation. But in neither case is there a clear "extension" of the modal operators. So I am going to change the article to using the word semantics, and put "meaning" in quotes for those who want a quick idea of what semantics means. — Carl (CBM · talk) 10:14, 24 September 2009 (UTC)

Under Kripke semantics there is no interpretation of $\square$ and $\diamond$. A fully interpreted Kripke model makes no claim as to whether these symbols concern possible worlds, ought-statements, propositional attitudes, beliefs, etc. See below.--Heyitspeter (talk) 19:17, 24 September 2009 (UTC)
In a Kripke model, $\square \phi$ holds at a particular node w if every descendant node of w satisfies φ. What would you call this, if not an interpretation? The fact that we don't know what w "really is" doesn't enter into the picture; what matters is that we have a way of determining the truth value of $\square \phi$ at each node of the model. Any rule that allows us to achieve that is an "interpretation" in the sense of this article. — Carl (CBM · talk) 19:29, 24 September 2009 (UTC)
I agree that not knowing what w "really is" has no logical implications. Again, you're mixing terms. An interpretation is an interpretation function. An interpretation of a model is the interpretation function that gives the extension of the relevant non-logical symbols. The reason that our definition of truth in a model (which you're laying out here) is not an interpretation in the logical sense is that it is not an interpretation function.
Right. In the philosophical circles where the distinction between extension and intension arose, modal statements serve as canonical examples of nonextensional semantics. — Emil J. 15:16, 24 September 2009 (UTC)
No that isn't true. Kripke semantics for modal logic is extensional, and e.g., Quine never thought intensional semantics were viable (his "modal logic" employed first-order non-modal predicate logic). Here's how Kripke constructs an, e.g., propositional modal logic: Our interpretation function I for model M takes an ordered pair of a sentence P and a possible world w and yields its extension: True or False. $\diamond P$ is true at world w iff there is some world v such that w R v and the interpretation function assigns Truth to the extension of $P$ at v. Thus if I(P,v) is {True} and v is accessible from w, $\diamond P$ is true at w.
Again, the confusion here is between two uses of interpretation. The one I describe is logical, while the one you describe is exegetical. --Heyitspeter (talk) 19:05, 24 September 2009 (UTC)
Logics, under interpretations, don't have meaning in the usual sense. I can fully interpret a modal logic (with an interpretation function) and still make no claim as to whether the logic concerns possible worlds, belief-states, or ought-statements. Maybe this is where we're getting hung up. I'm not sure. Can you show me an interpretation that is intensional? I've never heard of or seen this. I honestly can't imagine what that would entail, though this may be my fault. In any case I like the quotes around meaning that seems alright with me. --Heyitspeter (talk) 19:13, 24 September 2009 (UTC)
I think the issue I have is that I would not call a truth value an "extension". The usage in mathematical logic, at least, reserves the word "extension" for the set of elements that satisfy a particular unary relation, or more generally for the collection of elements of some set. So to say that a truth value is an "extension" is odd to me. What is being defined by your "interpretation function" is the semantical meaning of the formula $\square \phi$. Similarly, I have no idea what the "extension" of the logical constant $\land$ would be, but the T-schema provides an interpretation of the meaning of this symbol in terms of the natural-language term "and". — Carl (CBM · talk) 19:26, 24 September 2009 (UTC)
The "extension" of a logical constant P is just the output (a set) which is yielded when our interpretation function takes P as argument. I don't see that as problematic. (Sidenote: Again, the T-schema doesn't give the meaning of a sentence nor does it give an interpretation in any respect, logically or linguistically. It gives truth-conditions. You're picking up on a practical alternative usage for the T-schema structure, posited by Davidson, and discredited. Even he never thought his use of the T-schema to determine meaning in natural language had anything to do with formal languages.) I'm beginning to think this doesn't matter anymore. Scare quotes satisfy me, I just think it's going to look shitty if we put scare quotes on every usage of the term "meaning" in this article. We can avoid this by using the term extension. --Heyitspeter (talk) 21:05, 24 September 2009 (UTC)

Maybe it helps to think of it this way. Take Sider's and/or Fitting and Mendelsohn's Logic / modal logic textbooks as cues. The interpretation I of a model gives the extension of each sentence letter. In both of their books, the extension is drawn from the set {0,1}. So I(P)=1, while I(Q)=0. This is all it takes to give an interpretation of a model (a logical interpretation). You assign each sentence letter in the domain one member of the set {0,1}. Later, we, in metalanguage, speaking as philosophers or mathematicians decide that 1 will mean "True" and 0 will mean "False." This is what it means to give an interpretation in the natural language sense (an exegesis). Nothing in the logical machinery says anything about this, though. The logical system is "formal," it's just a bunch of symbols. --Heyitspeter (talk) 21:12, 24 September 2009 (UTC)

When I am talking about "meaning" here, I am not interested in natural-language statements at all. By "meaning" of a sentence I simply mean the truth value that is assigned by the interpretation, which you seem to call "extension". On the other hand, the "meaning" of a term is the object denoted by the term; would you also call than an "extension"?
The T-schema does, in this sense, determine the "meaning" of symbols such as $\land$, which in an uninterpreted language have no meaning at all, but in the usual interpretations corresponds to conjunction. Indeed, I could change to a different T-schema in which $\land$ is interpreted as disjunction.
Now, I have never seen a mathematical logic book that uses the word "extension" to denote the truth value of a sentence, nor to denote the object represented by a term. If this is actually standard in some other field, I'd be interested to read about it. — Carl (CBM · talk) 21:28, 24 September 2009 (UTC)
To be honest, I'm not sure if it's standard "everywhere." I haven't read every textbook. All of my past teachers, and all of the textbooks I've read, have referred to the extension of sentence letters as truth-values. I don't know how to justify this as "standard" without name-dropping, though.--Heyitspeter (talk) 08:30, 25 September 2009 (UTC)

## Motion to rename article "Interpretation function"

• Approve. For reasons stated in the foregoing section of this talkpage.--Heyitspeter (talk) 09:09, 24 September 2009 (UTC)

See above.--Heyitspeter (talk) 19:08, 24 September 2009 (UTC)
OK, I think I see what you mean now. You use the term "interpretation function" to mean any function that assigns truth values to elements of some formal language. I would call that assignment just an "interpretation" in general. Either way, this is not the same as the special case of "interpretation function" in structure (mathematical logic). In that context, the thing you are calling an interpretation function is also called a "valuation" and is denoted using the symbol $\models$, while the "interpretation function" only assigns truth values to atomic formulas.
If I have your meaning correct, then it seems like there is no actual change in scope under the rename; we could simply rename everything that is currently "interpretation" to "interpretation function". But that seems to make things more complicated than necessary, using extra words without extra meaning. Am I missing something? — Carl (CBM · talk) 20:08, 24 September 2009 (UTC)
Yeah. In propositional logic it assigns True and False (strictly it assigns 0 and 1). In predicate logic it assigns non-logical constants. I don't think much of the content of this article would have to change, just the wording. I'm sorry I feel like I'm being so stubborn or nitpicky or something. Seeing this article is like seeing a physics article where all the electrons are referred to as "pixies." It doesn't make a difference in content, strictly speaking, but it infuses it with this kind of magical quality that just isn't there.--Heyitspeter (talk) 21:16, 24 September 2009 (UTC)
Well, this response sort of confirms my suspicion that you're trying to make the page conform to a deflationary point of view. --Trovatore (talk) 21:31, 24 September 2009 (UTC)
I'm trying to make this page conform to the point of view of contemporary philosophers and logicians. --Heyitspeter (talk) 21:34, 24 September 2009 (UTC)
Edit since I'm sidetracking from addressing your point: You're right that it's deflationary in some sense, but this is good, and it's endorsed by contemporary experts. For an analogy, Aristotle talked about objects "wanting to move towards the ground." This explains gravity but it's misleading in that it connotes will-power. There is a sense in which it's deflationary to take the agency out of this statement and replace it with "objects fall towards the ground." But that shouldn't be taken as an argument against our doing so.--Heyitspeter (talk) 21:42, 24 September 2009 (UTC)
Well, I'm not arguing against your deflationary POV per se here, just that I think you may be too narrowly focused on one side of the debate. You admit that some authors use the term "meaning" here; your position seems to be that they don't really mean it. I'm skeptical on that point. --Trovatore (talk) 23:30, 24 September 2009 (UTC)
Re Heyitspeter, you may well be an expert on the subject; I don't know. You may not know that Trovatore and I each know something about it. In any case, I really would be interested to read the actual words of contemporary logicians who use the word "extension" to refer to the truth value of a sentence. I find that usage somewhat remarkable, but it may be an issue of terminology varying by field. — Carl (CBM · talk) 23:49, 24 September 2009 (UTC)
Though Wikipedia's relative anonymity would allow me to claim professional status, I won't. I'm a student at Reed College. So from first-hand experience, Mark Hinchliff and Paul Hovda use this terminology. The logic textbooks I've used at this particular school include Ted Sider's "Logic for Philosophy" which you can find here, and Fitting and Mendelsohn's "First-Order Modal Logic," most of which you can read on google books. Also, W.V.O Quine makes the distinction. An extensional logic (according to him) is one where each sentence letter gets assigned one truth-value under a given interpretation. An intensional one is where a sentence letter gets assigned more than one truth-value. He discusses this in his "Notes on Existence and Necessity," if my memory serves me, towards the middle. Intensional logics would include modal logics, according to him, which is why he thought they weren't feasible. Then Kripke came along and showed that there could be extensional modal logics. In summary I think "Philosophy" is the field where this is common parlance. It's not as though people don't make mistakes in this field, though.--Heyitspeter (talk) 08:46, 25 September 2009 (UTC)
Sider's book uses 'extension" in the way I mentioned above: to refer to the set of objects that are satisfied by a predicate. For example, on page 230, he says "the truth value of Fa at world r will be 0 (false), since the denotation of a isn't in the extension of F at world r". The terminology you are proposing for the article would have read something like "the extension of Fa at world r will be 0 (false), since the extension n of a isn't in the extension of F at world r". Sider does not appear to use the word "extension" to refer to a truth value either there or in the section on propositional logic.
Sider also says, on p. 21, "In the next section we will introduce a semantics for propositional logic. A semantics for a language is a way of assigning meanings to words and sentences of that language". So I don't think your concerns about the word "meaning" are completely shared by Sider, either. — Carl (CBM · talk) 10:15, 25 September 2009 (UTC)
Setting the word "semantics" aside, which isn't in question, look at what he says about interpretation: "Definition of interpretation: A PL-interpretation is a function that assigns to each sentence letter either 1 or 0... Instead of saying “let P be false, and Q be true”, we can say: let I be an interpretation such that I(P ) = 0 and I(Q) = 1... The valuation function... assigns truth values to complex sentences as a function of the truth values of their sentence letters—i.e., as a function of a given interpretation... Intuitively: we begin by choosing an interpretation function, which fixes the truth values for sentence letters. Then the valuation function assigns corresponding truth values to complex sentences depending on what connectives they’re built up from: a negation is true iff the negated formula is false, and a conditional is true when its antecedent is false or its consequent is true." An interpretation (for Sider, a logic professor at the number one university in the United States for Philosophy according to the Philosophical Gourmet Report) is a technical term. It is only a function that assigns either 0 or 1 to sentence letters. No more, no less.--Heyitspeter (talk) 21:10, 25 September 2009 (UTC)
He is not defining "interpretation" generally: First, he is working there in the explicit context of propositional logic in that section. Second, he goes out of his way to call it a "PL-interpretation" to make clear he is defining a technical term. But this article is not about the technical term for interpretations of propositional logic; it is about the general assignment of meaning to formulas of arbitrary formal languages. For example, see the "example" section near the top of the article, in which certain strings are interpreted as representing binary numbers. How would you fit this into the "PL-interpretation" framework? — Carl (CBM · talk) 21:17, 25 September 2009 (UTC)
It's technical, yes. That's been my whole point throughout this 'debate'. I don't know what you want from me. I've explained everything. For another, more explicitly general definition of an interpretation, in layman's terms, see page 8 of Fitting and Mendelsohn's First-Order Modal Logic book. You can find it on google books. They're at CUNY, number 9 in the nation or something: "Think of an interpretation i as a particular assignment of truth-values to the non-logical particles in the proposition, with the logical particles understood as the usual boolean functions. On the simplest reading of $/box$ and $/diamond$, a proposition is necessarily true if it comes out true for every interpretation..." (8). Notice that he uses the term reading to talk about our own hermeneutic of the symbols (this avoids confusion) but uses the term interpretation to talk about the specific assignment of truth-values to the non-logical particles, whereas connectives are not interpreted at all and are rather boolean functions. --Heyitspeter (talk) 06:10, 26 September 2009 (UTC)
I'm asking how the example near the top of the article, which is obviously an assignment of semantics to a formal language, fits into the framework you are proposing. It does not involve an assignment of truth values at all. (Also, the sentence "connectives are not interpreted at all and are rather boolean functions" has something amiss, because a connective is a symbol, not a function.) — Carl (CBM · talk) 11:25, 26 September 2009 (UTC)
No, a negation connective isn't a symbol, a tilde is a symbol. In any case, it seems like bad policy to argue with Fitting and Mendelsohn about this. As for the example, I don't understand it, to be honest. Can you find me anything like that in the literature? It doesn't make sense to me.--Heyitspeter (talk) 22:01, 26 September 2009 (UTC)
• Oppose - it isn't itself primarily a "function", same is true for "relation." It is primarily an idea which is expressed as a function or relation, etcetera.Pontiff Greg Bard (talk) 21:05, 24 September 2009 (UTC) [incidentally, I find the latest edits by Heyitspete and CBM to be just fine (although I don't really think the Abe Lincoln stuff works too well). I just want to thank you for reformulating rather than deleting.] Pontiff Greg Bard (talk) 21:08, 24 September 2009 (UTC) [Ok now heyitspete is taking out too much. Please elucidate on the relationship between meta and object language at some point in the article as appropriate]-GB

## Intended interpretations

I'm not sure how much more can be said about Intended interpretation to justify a separate article. I propose a merge to the existing section here. Tijfo098 (talk) 11:11, 9 April 2011 (UTC)

## A sign is more general than a symbol

The article states: “An interpretation is an assignment of meaning to the symbols of a formal language.” Would it not be more precise to say that an interpretation is an assignment of meaning to the objects of a formal language?

I mention this because for Charles Sanders Peirce “A symbol is a sign fit to be used as such because it determines the interpretant sign” (C.S. Peirce, New Elements: 1904, Essential Peirce vol. 2. Not to be confused with The New Elements of Mathematics by Peirce later edited by various people).

So for Peirce, a sign is more general than a symbol, and he distinguishes three types of sign: icon, index, and symbol.--Semeion (talk) 17:11, 12 November 2013 (UTC)

I think "objects of a formal language" is a bit confusing, because one might easily read it as "objects of discourse of a formal language", which is exactly wrong (those are the meaning being assigned, not the things to which meaning is assigned).
Peirce is a very interesting thinker, but I don't think his sometimes-odd choice of terminology needs to determine our wording here. I think "symbol", as used here, is pretty standard for the intended meaning. --Trovatore (talk) 17:37, 12 November 2013 (UTC)
Trovatore,
I am sympathetic to the cause to guard against a notion of interpretation as “discourse.” But this is why a symbol, in Peirce’s view, is something that is already interpreted as such.
In and up to the limits defined by Godel’s theorems it is the object (understood as a symbol of bivalence) that determines the meaning of its combinations. Beyond this limit it is the “Phaneron” that determines the content of the symbol, and this semantic content is derived from experience, not bivalence. Hence the use of formal language, mathematics and logic, to interpret and structure the understanding of the experience in question.--Semeion (talk) 13:54, 27 November 2013 (UTC)