Can we define a well formed formula? [closed]

Would it be possible to define (even circularly) what a well formed formula is through logic, rather than presupposing φ?

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closed as off-topic by James Kingsbery, Swami Vishwananda, Camil Staps, Joseph Weissman♦Mar 4 at 20:59

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I don't know what "presupposing φ" means, but the question is about mathematical logic and belongs on math SE. – Conifold Feb 23 at 20:59
Well formed formula is defined in the first page of every math log textbook. – Mauro ALLEGRANZA Feb 23 at 21:05

Sure, you do it this way:

1. If phi is an atomic sentence, phi is a wff.
2. If phi is a wff, then so is "not phi".
3. If phi and psi are wff, then so are "phi and psi", "phi or psi" "if phi, then psi".
4. nothing else is a wff.

You need some extra clauses if your logic includes quantifiers or modal operators, but the basic idea is the same. The definition of a WFF isn't circular--it relies upon the primitive notion of an atomic formula, but that isn't circular.

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What about adding the negation "non phi" as wff? – Jo Wehler Feb 23 at 20:15
+1, nice catch. – shane Feb 23 at 20:16
Thanks. If I wanted to express a set of all wffs of first order logic, how would I do it? – esnafga Feb 23 at 22:06
You're going to define the notion of a wff for your first-order language in the meta-language. You can express that metalanguage in first order terms if you want. Let the universe of discourse be the set of strings of symbols, then we could define the set W of all wff to be every string in the domain such that it conforms to those rules above. But note this is a NEW first order language, not the original one we started with. That's not itself a problem, it is just something to keep in mind. – shane Feb 23 at 23:06

Well-formedness is a syntactical not a logical concept.

Well-formedness of a formula is alike to syntactical correctness of a sentence or proposition. But whether the formula or the proposition is true, that's a second issue.

Well-formed formulas may be true or false. While ill-formed formulas are neither true nor false. They are just meaningless.

Hence well-formedness of a formula is a prerequisite in order to apply logical reasoning.

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Sure, but after presupposing it, can you express it through the language? – esnafga Feb 23 at 20:01
@esnafga Well-formedness is encapsulated by a grammar. The grammar of many formal languages, e.g., like programming languages, is not specified by the language in question. Instead, general data structures from informatics like trees are used. – Jo Wehler Feb 23 at 20:06
If I wanted to express a set of all wffs of first order logic, how would I do it? – esnafga Feb 23 at 22:19
@esnafga The set of all wffs is infinite. For which purpose do you need the set of all wffs? The answer of shane shows the principle how to build this set bottom up, – Jo Wehler Feb 23 at 23:19

This is really a mathematical logic question, rather than logic thought philosophically.

A formula or sentence of a certain formal language is well-formed when it happens to follow the rules of grammar for that formal language.

Panini in India, was the first to write a comprehensive formal grammar of a natural language - in his case - Sanskrit.

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Yes, so is it possible to refer to entirety of such sentences through the rules of grammar that it claims? How would you define φ without saying "for every wff φ"? – esnafga Feb 23 at 19:51
@esnafga: it's done recursively: sentences are built of from fragments of sentences. – Mozibur Ullah Feb 23 at 19:54
Actually, the example I gave there is of co-recursion; recursion is its dual; this was probably the point of your question which did slightly confuse me: how does one define something when it's defined in terms of itself - this is a good question; but see what I wrote above in the comments. – Mozibur Ullah Feb 23 at 19:54