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In The Laws of Truth, Smith translates utterances of the form "P unless Q" as (¬Q → P) and takes the further suggestion that (Q → ¬P) to be an implicature of the utterance. The justification for this approach is given (on pages 115-116) as follows:

That "P unless Q" says (i.e. is properly translated as) (¬Q → P) and (in many contexts) implicates (Q → ¬P) - rather than saying (¬Q → P) ∧ (Q → ¬P) - is suggested by the following two facts:

  1. First, one can cancel the suggestion that (Q → ¬P). For example, it would make perfect sense for me to say, "I'll come swimming with you, unless it rains - and even that might not stop me" (or "and even then I might come anyway", or "even if it rains, I still might come," etc.). If (R → ¬S) were part of what is said by "I'll come swimming with you, unless it rains," however, then this addition would make little sense: it would be akin to "P and Q - but Q might not be true."
  2. Second, it would also make perfect sense for me to say "I'll come swimming with you unless it rains - in which case I won't come." This statement is properly translated as (¬R → S) ∧ (R → ¬S). However if "I'll come swimming with you, unless it rains" were already properly translated as (¬R → S) ∧ (R → ¬S), then adding "in which case I won't come" would sound redundant: it would be akin to "P and Q - and Q."

There are a few aspects of this justification that I don't quite understand, but before I get into them I'd just like to clarify the difference between "saying" and "implicating". We can think of an utterance as conveying three types of information:

  • What is said - this is the proposition which underlies the utterance, and is what we aim to capture when we translate the utterance from natural language to propositional logic.
  • What is implied - these are all the propositions which follow logically from what is said.
  • What is implicated - there are all the things which follow from the assumption that the utterance is correct, where "correct" means that the utterance fully conforms to the general and special norms of conversation. General norms are stipulated by the Gricean maxims of the Cooperative Principle while the special norms are those which are attached to the individual words within the utterance. There are two types of implicatures:
    • Conversational implicatures - arise from adherence to the general norms; can be cancelled.
    • Conventional implicatures - arise from adherence to special norms; cannot be cancelled.

My questions:

  • I interpret "P unless Q" as conveying two pieces of information: (¬Q → P) and (Q → ¬P), but I don't understand why Smith regards "P unless Q" as saying the former and implicating the latter. Why not the other way around (i.e. why not take "P unless Q" to say (Q → ¬P) and implicate (¬Q → P))?

  • The first point of the justification (in the quoted excerpt above) says that this implicature can be cancelled (and is therefore a conversational implicature), but I'm not sure about this.

    Earlier in the book, Smith gives an example of a conventional implicature: "The prime minister's speech, in which she praised Senator Bellinghausen, was magnanimous - although I do not mean to suggest there is any rivalry between the prime minister or the senator, not that the prime minister is his superior." Here, the conventional implicature that the speaker believes the PM to be the senator's superior or rival comes from the norm attached to the word "magnanimous" and therefore cannot be cancelled. Likewise, couldn't we argue that the implicature in the case of "P unless Q" is also a conventional implicature, since the only thing in the utterance that conveys it is the word "unless"?

  • How would taking (R → ¬S) as part of what is said by "I'll come swimming with you, unless it rains," be akin to "P and Q - but Q might not be true"?

  • The second point of the justification says that translating "P unless Q" as (¬Q → P) ∧ (Q → ¬P) would render the second part of "P unless Q, in which case not P" redundant. But couldn't we argue that the second part is itself redundant, since it is implicated by "P unless Q", so that the correct translation of "P unless Q, in which case not P" is simply (¬Q → P)?

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  • Didn't bother to review before posting? Oct 19, 2023 at 3:13
  • I did, I thought the Latex would render once it was posted since sometimes on Maths SE it takes time to render in the preview.
    – user51462
    Oct 19, 2023 at 3:18
  • I'll fix it now.
    – user51462
    Oct 19, 2023 at 3:18
  • You know this forum doesn't support, latex, right? What I have done in the past is write the equations on the math forum and copy and paste the rendered version. Oct 19, 2023 at 3:25
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    The usual reading in English of "P unless Q" is "P, if not Q". Consider e.g. "I won't go the library unless I need a book", we have: "I won't go the library, if I do not need a book." Thus, in symbols: (¬Q → P). Oct 19, 2023 at 6:36

2 Answers 2

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I interpret "P unless Q" as conveying two pieces of information: (¬Q → P) and (Q → ¬P), but I don't understand why Smith regards "P unless Q" as saying the former and implicating the latter. Why not the other way around (i.e. why not take "P unless Q" to say (Q → ¬P) and implicate (¬Q → P))?

The apparent issue with translating the phrase "P unless Q" as (Q → ¬P) would be the fact that (Q → ¬P) does not prevent the occurrence of (¬P ∧ ¬Q), resulting in the possibility of a direct violation of the condition "P unless Q", as "P unless Q" means that P must be the case unless Q . Seeing that that is the case, one cannot take (Q → ¬P) to be the original, rather than an extrapolated, statement.

The first point of the justification (in the quoted excerpt above) says that this implicature can be cancelled (and is therefore a conversational implicature), but I'm not sure about this. Earlier in the book, Smith gives an example of a conventional implicature: "The prime minister's speech, in which she praised Senator Bellinghausen, was magnanimous - although I do not mean to suggest there is any rivalry between the prime minister or the senator, not that the prime minister is his superior." Here, the conventional implicature that the speaker believes the PM to be the senator's superior or rival comes from the norm attached to the word "magnanimous" and therefore cannot be cancelled. Likewise, couldn't we argue that the implicature in the case of "P unless Q" is also a conventional implicature, since the only thing in the utterance that conveys it is the word "unless"?

The author put forth very lucid reasoning as to why it cannot be a conventional implicature. Note that in the two examples he brought, in each case he appended onto each sentence a clarifying phrase, with the appended phrase not lending itself to confusion nor leading to a baffling statement, but rather specification of meaning. Let us contrast this with the use of the word "magnanimous", as the author stated earlier in the book:

It is appropriate to call an act “magnanimous” only when the beneficiary of the act is less powerful than, or a rival of, the person who performs the act.

Hence for one to use the term magnanimous in a statement, and then append a clarifying statement negating the two aforementioned implications of the term, the individual being spoken to would identify the usage of magnanimous in the sentence as being mistaken. Conversely, the word "unless" does not have a rigid usage, its exact meaning is gleaned from context.

How would taking (R → ¬S) as part of what is said by "I'll come swimming with you, unless it rains," be akin to "P and Q - but Q might not be true"?

The author was juxtaposing this with the statement "I'll come swimming with you, unless it rains - and even that might not stop me". If we were to understand the (initial part of the) statement, to the exclusion of the clarifying phrase, to contain in its meaning (R → ¬S), then the clarifying phrase (and even that might not stop me) would contradict that.

To clarify this: Take R as P and take ¬S as Q then assume P. Given P, Q must follow, hence P and Q. Yet given the clarifying phrase, Q is not necessarily there, so: "P and Q - but Q might not be", resulting in the initial implication rule not holding.

The second point of the justification says that translating "P unless Q" as (¬Q → P) ∧ (Q → ¬P) would render the second part of "P unless Q, in which case not P" redundant. But couldn't we argue that the second part is itself redundant, since it is implicated by "P unless Q", so that the correct translation of "P unless Q, in which case not P" is simply (¬Q → P)?

No, one could attempt to argue that the initial condition (¬Q → P) of the conjunction is redundant (which it cannot be due to the reason put forth in my response to the first question), but the second condition is crucial to the statement. If we were to confine the statement "P unless Q, in which case not P" to the condition (¬Q → P), we would be disregarding the phrase "in which case not P". As the phrase "P unless Q, in which case not P" states that "P must be the case unless Q is the case, in which case (that being if Q is the case) then P is not the case". Yet with the condition (¬Q → P), there is nothing to preclude P and Q from both being the case.

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  • Thank you @Max Maxman.
    – user51462
    Oct 19, 2023 at 22:33
  • @user51462 My pleasure
    – Max Maxman
    Oct 19, 2023 at 22:47
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P unless Q can be interpreted as:

  1. P v Q = ~P -> Q = ~Q -> P
  2. Q -> ~P = P -> ~Q = ~P v ~Q = ~(P & Q)

P: x > 10
Q: x < 100

Sufficient & Necessary Conditions issue.

😁

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