Advanced Symbolization

If we want to be able to use our new system of derivations effectively, we need to be able to apply them to English language arguments. But if we want to do that, we need to know how to go back and forth, from English to symbols, and from symbols to English.

Symbols to English

As before, let’s begin by thinking about how we can translate sentences in our formal language into English, and some simple ways in which we can “clean up” our translations in order to make them more readable.

Symbols to English
  1. If the sentence to be translated is in unofficial notation, then (if it makes things simpler for you) restore any parentheses that would be there if the sentence were in official notation.
  2. Locate the main connective of the formal sentence.
  3. If the main connective is , so that the sentence is (ϕ → ψ), then write “If ϕ, then ψ”.
  4. If the main connective is , so that the sentence is (ϕ ∧ ψ), then write “it’s both the case that ϕ and that ψ”.
  5. If the main connective is , so that the sentence is (ϕ ∨ ψ), then write “either ϕ or ψ
  6. If the main connective is so that the sentence is (ϕ ↔ ψ), then write “if ϕ then, and only then, ψ
  7. If the main connective is ¬, so that the sentence is ¬ϕ, then write “It is not the case that ϕ.”
  8. If either ϕ or ψ contains any formal language connectives, then apply this procedure to ϕ and ψ as well.
  9. If there’s no main connective, and you have only a sentence letter, replace the letter with its English meaning according to the scheme of translation.
  10. Apply this procedure to each of the sub-formulas of the target sentence.

Here are some examples, using the following scheme of abbreviation.

P  =  Language is powerful
Q  =  We try to understand each other
R  =  All is lost

Ex. 1

  1. P ∨ R
  2. Either P or R
  3. Either Language is powerful or all is lost

Ex. 2

  1. P ∧ Q → ¬R
  2. If P ∧ Q, then ¬R
  3. If it’s both the case that P and that Q, then it’s not the case that R
  4. If it’s both the case that language is powerful and that we work hard to understand each other, then it’s not the case that all is lost.

Ex. 3

  1. P ∧ Q ↔ ¬R
  2. If P ∧ Q then, and only then, ¬R
  3. If it’s both the case that P and that Q then, and only then, it’s not the case that R
  4. If it’s both the case that language is powerful and that we work hard to understand each other then, and only then, it’s not the case that all is lost.

Ex. 4

  1. ¬P ∨ R
  2. Either ¬P or R
  3. Either it’s not the case that P or R
  4. Either it’s not the case that language is powerful or all is lost.

As with our first procedure for translating from symbols to English, this gives us a unique translation for every sentence, and the translation will have the right logical properties. But, once again, the sentences are not automatically easy to understand. Why do we do it this way, then? Well, the key idea is this: expressions like “It’s both the case that … and …” make it easy to figure out what a connective has inside its left side. For example, with “both … and …”, whatever’s in the left side is whatever’s between “both” and “and”. This makes it easy to recover the parsing structure of our original logical sentence from the English translation, and ensures that the translation is logically unambiguous. For example, (P ∧ (Q ∨ R)) and ((P ∧ Q)∨R) are translated, respectively, as “It’s both the case that P and that either Q or R” and “Either it’s both the case that P and that Q or R”. On the other hand, if we just replace “” with “and”, and ‘’ with “or”, then both (P ∧ (Q ∨ R)) and ((P ∧ Q)∨R) become “P and Q or R”, which is logically ambiguous.

How can we make the sentence more readable? This is more of an art than a science, but the following four tricks may help.

Commas

Commas can be used to mark logical breaks in a sentence. For example, “Either P or it’s both the case that Q and R” has two main parts: “P” and “it’s both the case that Q and that R”. So, one can write it as “Either P, or it’s both the case that Q and R”. Note that one would not write this as “Either P or it’s both the case that Q, and R”.

Stylistic Variants

If it creates no ambiguity about the parsing structure of the sentence (for example, if you have made clear what the main parts are by using commas), then we can replace some of our standard logic phrases with stylistic variants, in the same way that we can replace the standard phrase “if … then” with something like “given that …, …”. The following possible stylistic variants all are correct:

  1. “It’s both the case that P and that Q” can be replaced by
    1. ϕ and ψ”,
    2. ϕ but ψ”,
    3. ϕ although ψ”,
    4. ϕ even though ψ”,
    5. ϕ despite the fact that ψ”, and similar phrases.
  2. “Either ϕ or ψ” can be replaced by
    1. ϕ or ψ
    2. ϕ unless ψ”, and similar phrases.
  3. “If ϕ then, and only then, ψ” can be replaced by
    1. ϕ just in case ψ”,
    2. ϕ exactly on the condition that ψ”,
    3. ϕ if and only if ψ
    4. “Just in case ϕ, ψ”,
    5. “Exactly on the condition that ϕ, ψ”,
    6. “If and only if ϕ, ψ”, and similar phrases.

Subjects and predicates containing connectives

When we have a cluster of simple sentences which are combined using only English phrases that translate ‘’, ‘’ or ‘¬’ we can usually “put this in the noun phrase”, by creating a new sentence whose main noun phrase is “both the subjects of the previous sentence”, “either of the subjects of the previous sentence” or “neither of the subjects of the previous sentence.”

For example, the sentence

  1. “It’s both the case that Polk was a president and that Quincy was a president”

can be paraphrased as “Both Polk and Quincy were presidents” and the sentence

  1. “Either Polk was a president or Quincy was a president”

can be paraphrased as “Either Polk or Quincy was a president”. Although in English, we use a singular, rather than plural form of the verb here—I wonder why that is. In the same way,

  1. “It’s not the case that it’s both the case that Polk was a president and Quincy was a president”

can be paraphrased as “Not both Polk and Quincy were presidents”.

There are, however, some cases where this doesn’t work. For example, consider “Either Polk was not a president, or Quincy was not a president”. We can’t paraphrase this as “Either not Polk or not Quincy was a president”—I suppose this is because “not Polk” and “not Quincy” aren’t noun-phrases that can be joined into a larger noun phrase.

We an also sometimes take a logical cluster like this and “put it in the verb phrase”. For example

  1. “Emma sings and Emma dances”

can be paraphrased as “Emma sings and dances”.

  1. “Either Emma sings or Emma dances”

can be paraphrased as “Emma either sings or dances”.

  1. “Either it’s not the case that Emma sings or it’s not the case that Emma dances”

can be paraphrased as “Emma either doesn’t sing or doesn’t dance”. Here, we can paraphrase a sentence with embedded negations—I suppose because “doesn’t sing” is still a verb-phrase that can be joined into a larger verb phrase.

Non-Restrictive Relative Clauses

One last type of “cleaning up” which we sometimes see is the use of non-restrictive relative clauses. These allow you to take a verb phrase, like “dances” and attach it to a noun-phrase (like “cat”), where sometimes this noun-phrase is inside of a more complex verb-phrase. For example:

  1. “Twyla is a cat and Twyla hunts well.”

can be paraphrased as “Twla is a cat who hunts well”

  1. “Tackles hurt the players, and tackles are forbidden”

can be paraphrased as “Tackles, which hurt the players, are forbiddden”. Though not as “Tackles which hurt the players are forbidden”. What’s the difference here?

  1. “Genies grant wishes and genies are often malicious”

can be paraphrased as “Genies, who grant wishes, are often malicious”.

English to Symbols

Translations from English to symbols are correct if they result in a symbolic sentence that, when translated back into English, means the same as (is a stylistic variant of) the original sentence. When we’re dealing with sentences we find “in the wild”, we sometimes have to rely a little bit on our intuition for natural language. But the same approximate algorithm that we used before is still sometimes helpful.

English To Symbols
  1. Replace all the stylistic variations on “if ϕ, then ψ”, “it is not the case that ϕ”, “It’s both the case that ϕ and that ψ”, “Either ϕ or ψ”, and “if ϕ then, and only then, ψ” with their corresponding non-variant phrases.
  2. Replace the English main connective with the appropriate symbol. If you are replacing anything but “it is not the case that”, remember to wrap the two clauses you’re replacing in parentheses, if they’re not already wrapped.
  3. If there are any sentences remaining that do not occur in the scheme of abbreviation, repeat this procedure on those sentences.
  4. If there are just sentences that occur in the scheme of abbreviation, replace them with the corresponding sentence letters.

As with our previous translation exercises, the hard part is step 1. Once you have done that, you can always locate the main connective by looking at the leftmost word. Whatever phrase that word is a part of (whether it is “if ϕ, then ψ”, “it is not the case that ϕ”, “It’s both the case that ϕ and that ψ”, “Either ϕ or ψ”, or “if ϕ then, and only then, ψ”), that phrase is the English main connective which we need to replace in step 2.

Problem Set 10

Please translate the following English sentences into our formal language, using this scheme of abbreviation:

P  =  Peter is proud.
Q  =  Queens is a short ride away.
R  =  Real estate is harder than it seems.
S  =  Susan is getting a deal.
T  =  Tom is Proud.
U  =  Susan is buying an umbrella.

exercise 10.1
[70,81,66,93,73,34,88,61,63,76]
exercise 10.2
[70,81,66,94,73,94,39,78,48,63,73]
exercise 10.3
[70,81,66,95,73,32,43,78,62,76]
exercise 10.4
[70,81,66,88,73,91,39,76,82,63,64,92,43,65,62]
exercise 10.5
[70,81,66,89,73,35,75,76,82,76,61,92,43,65,63,76]
exercise 10.6
[70,81,66,90,73,35,88,61,56,65,87,83,90,51,76]
exercise 10.7
[70,81,66,91,73,39,90,95,63,48,70,38,87]
exercise 10.8
[70,81,66,84,73,91,35,76,82,60,64,92,43,73,56,65,87,38,94,65]
exercise 10.9
[70,81,66,85,73,91,39,61,67,56,64,47,88,73,57,65,87,32,94,65]
exercise 10.10
[70,81,66,93,89,83,95,53,65,82,57,47,88,51,69,65,87,13,95,52,67,48,58,90,87]

Problem Set 11

Please translate the following English sentences into our formal language, using this scheme of abbreviation:

P  =  Professor Farnsworth has built a robot.
Q  =  Kif’s quips are rather funny.
R  =  Robots can be trusted.
S  =  Kif’s quips sink ships
T  =  There will be a tomorrow.
U  =  Robots can be underestimated.

exercise 11.1
[70,80,66,93,73,94,37,78,48,60,73]
exercise 11.2
[70,80,66,94,73,91,37,61,67,63,64,92,43,48,76]
exercise 11.3
[70,80,66,95,73,13,95,51,48,67,60,90,88,61,56,76]
exercise 11.4
[70,80,66,88,73,91,38,78,48,63,64,92,43,73,60,67,53,13,37,76,82,18,61,90,87]
exercise 11.5
[70,80,66,89,73,91,35,76,82,61,64,92,43,73,62,48,70,13,35,72,76]

Please also prove the validity of the following arguments. DeMorgan’s laws, proof by cases, and the negated conditional rule are very likely to be useful here.

exercise 11.6
11.6 ~(~P->(Q/\R)), ~S\/P :|-: ~S
exercise 11.7
11.7 P\/Q->S, ~(R\/~P) :|-: ~(R<->S)
exercise 11.8
11.8 (P->Q)->T, S\/T, P->(S->Q) :|-: T
exercise 11.9
11.9 (P->Q)->P, Q->P :|-: P
exercise 11.10
11.10 ~(P->Q), ~(R->Q) :|-: ~(P->~R)

Problem Set 12

Please prove the validity of the following two arguments. In both cases, DeMorgan’s laws well be very helpful, as part of a strategy for showing a disjunction.

exercise 12.1
12.1 P<->S, ~Q->~P, ~R->P :|-: R\/(S/\Q)
exercise 12.2
12.2 P\/S, S->~(Q->~P) :|-: P\/R

Please also translate the following sentences, using this scheme of abbreviation

P  =  Genies are playful.
Q  =  Genies are quick
R  =  Genies seek revenge
S  =  Genies are scary

exercise 12.3
[70,83,66,95,73,91,39,78,48,61,64,92,43,51,76]
exercise 12.4
[70,83,66,88,73,35,90,95,68,62,68,77,36,72,76]
exercise 12.5
[70,83,66,89,73,35,88,61,68,62,68,77,36,72,76]