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Consider:

(p ∧ q) ↔ r

p ∧ (q ↔ r)

These two formulae have different meanings, and yet, if one attempts to translate them into ordinary English in the usual way, they both seem to come to

p and q if and only if r

How can preserve the difference in their meanings when rendering them in English?

jsw29
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Stewart Jean
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    You should ask that on a math-related SE site, not here. – tchrist Jan 17 '23 at 21:20
  • They told me on ask on here haha. – Stewart Jean Jan 18 '23 at 00:53
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    Why do you need to differentiate them in English? Who is your audience? What form of writing do you want (e.g. notes, powerpoint, academic, general audience, legal document, or other formal prose; do you want to use symbols p and q or particular propositions in plain English)? Is your goal brevity or clarity? Can you use examples in your writing (e.g. "For example if A is true but not B then the full proposition is not true")? This seems more like writing advice than anything specific to the English language though. – Stuart F Jan 18 '23 at 13:50
  • This looks like a question purely about propositional logic (a mathematical question) which is answerable by many methods but truth tables are the most automatic. For this to be on-topic on English, we could either answer about how these symbols work in natural language semantics, or, and I think this may be the main thing), how would you -pronounce- these sentences so that they are distinguishable. So what -English- question are you looking to answer? – Mitch Jan 18 '23 at 14:24
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    I have taken the liberty of reversing a recent edit which inadvertently made the point of the question less than clear. The question is really about how to represent the role that parentheses play in logical (and, more generally, mathematical) notation using the syntax and punctuation of ordinary English (in which parenthesis cannot be used in the same way); the meaning of the logical operators for *and* and *if and only if* is not at issue here. (@Mitch, I think that this edit, which restores the OP's intentions, clarifies the matters that you raised.) – jsw29 Jan 18 '23 at 17:14
  • Does this answer your question? Precedence of "and" and "or" A and B or C ... [A and B] or C vs [A] and [B or C]. In writing, A and B –/,/ ... or C vs A –/,/ ... and B or C. Parallel structures for iff (though I can't see this arising in everyday English). – Edwin Ashworth Jan 18 '23 at 20:13
  • @EdwinAshworth, the two questions are related, but not duplicates: this one is about the relationship of *and* and *if and only if, while the other is specifically about the relationship of and* and *or*. – jsw29 Jan 18 '23 at 21:58
  • @jsw29 I'm saying the analysis is the same. And 'p and q if and only if r' hardly belongs on a site looking at everyday English. – Edwin Ashworth Jan 19 '23 at 16:15
  • @EdwinAshworth, perhaps the analysis is the same, but that deserves to be explicitly stated in an answer to this question; it cannot be simply assumed. Moreover, the core of the question is about the aspects of English syntax and punctuation that are not affected by its involving p, q, r; they can be replaced by sentences/clauses of ordinary English, and the question would still be the same. – jsw29 Jan 19 '23 at 16:34

2 Answers2

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Logical notation has been invented precisely because it is difficult to express the structure of complex propositions precisely and unambiguously in ordinary language. One thus generally cannot expect that translating such formulae into ordinary English will result in something that sounds natural and can be understood effortlessly (as effortlessly as the formulae are understood by those who are familiar with the notation used in them).

The first formula can be rendered as

r is true if and only if both p and q are true.

and the second one as

p. q if and only if r.

Note that I have taken the liberty of flipping the sides of the biconditional in the first formula and splitting the second formula into two separate sentences. The task would be more difficult if such manoeuvres were outlawed.

What made the task relatively manageable in these two cases is that the formulae were relatively simple. Natural-language translations of more complex formulae may require lengthy paragraphs and/or the use of numbered clauses of the kind that one sees in legal documents.

jsw29
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    Yeah, this is why people invented truth tables and other graphical representations as well. – Stuart F Jan 17 '23 at 22:41
  • Ah yes, thank you! I was also suggested the use of commas, would that work as well? – Stewart Jean Jan 18 '23 at 00:54
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    Commas may work in relatively simple cases, such as these, but (1) they don't have unambiguous counterparts in spoken language, (2) they have many other purposes, which makes it difficult to be sure that a particular comma serves as the equivalent of a parenthesis in logical notation, and (3) logical notation allows for multiple layers of embedded parentheses and commas cannot clearly represent their hierarchies. – jsw29 Jan 18 '23 at 16:22
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There are two separate kinds of answers, one is about written English and one is about spoken English.

Contra the enlightenment philosophes, the written version is both boring and easy - either keep the notation (this is the best) or if you find that somehow (I can't think of why) you -must- literally spell everything out:

(p ∧ q) <---> r is written as "p and q, if and only if r"

and

p ∧ (q <---> r) is written as "p, and q if and only if r"

This gives a hint (or presages, or anticipates, or is backwardly inspired by) the spoken version.

In general, mathematical formulas, to speak them out as unambiguously as positive, implement parantheses, which mark subformulas, by appropriate pauses. For example,

a + b*c

where b is multiplied by c then that quantity is added to a, is pronounced

a (pause) plus b times c

and pauses are often written as commas.

If you wanted to pronounce (a + b)*c You'd pronounce it as:

a plus b (pause) times c

or often, to emphasize the constituents and remove more ambiguity:

The quantity a plus b times c

or

a plus b quantity times c.

The logical statements you have work the same:

p and q (pause) if and only if r

and

p (pause) and q if and only if r

(It is an exercise for the reader to use 'quantity' here)

Of course, mathematical notation, being so formal, is often more easily transmitted more accurately via writing. But if you must pronounce it outloud (as in a class) 'pauses' work well for short phrases.

Mitch
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    Professor Lawler has argued at a number of places on this site that it is a fiction that commas correspond to pauses in speaking, but that may not affect the core of this answer, as he does acknowledge that they correspond to something that can be heard; see How to use conjunction or punctuation for two or three adjectives next to each other?, Comma Usage in "but if we do", and Comma before "not". – jsw29 Jan 18 '23 at 22:30
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    The last sentence of this answer is very important: pauses/commas may work well for shorter phrases, but they cannot represent the relationships among multiple embedded parentheses in logical notation, nor is it always going to be clear when a pause/comma is used for this purpose and when for some entirely different purpose (think in particular of the cases in which we are not dealing with p, q, r, but actual sentences/clauses of natural language, each containing commas within itself). – jsw29 Jan 18 '23 at 22:44
  • @jsw29 To Prof. Lawler's point, commas in writing and pauses in speech are not identical, but in this very particular context a comma indicates some kind of pause. – Mitch Jan 19 '23 at 15:32