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The Logical Structure of Propositions

1 Logically Structured Propositions

1. Methods for determining if an inference is valid only apply to inferences whose propositions have a clear logical structure, so that it is clear that they are either negations, conjunctions, disjunctions or conditionals. Negations, conjunctions, disjunctions, and conditionals are complex propositions. For example:

(a) The cat is on the mat.

(b) Jones owns a Ford or Brown is in Barcelona.

(a) is a simple proposition. (b) is complex.

2. Consider four types of logically structured complex propositions and a fifth case that is complex but not logically structured:

(a) It is not the case that Smith is at school.

(b) Jones owns a Ford or Brown is in Barcelona.

(c) Willie Mosconi was a pretty good pool player and he won the World Pool Championship fifteen times.

(d) If the summer sports camp is cancelled today, then it is hot.

(e) Jack believes that the cat is on the mat.

(a) is a negation, a 'not' proposition. The simple proposition being negated is "Smith is at school.".

When we translate from English into Logically Structured English, we assign each simple proposition a lower-case letter. Any letter can be used for any proposition, though a different one for each proposition. For purposes of familiarity, the letter used is usually the letter the verb begins with, except in case where the verb is a form of "is" or "have", in which case the adjective's or object's first letter is used.

Thus (a), above, might be symbolized using the letter "a", after the word "at". If we let the lower-case letter "a" stand for "Smith is at school.", we can translate (a) as "not a". "Not" is short for "It is not the case that ...". We set out our translation key and translation into logically structured English as follows:

a = Smith is at school.

not a

In general, negations have the form "not S", where the capital letter "S" stands for any proposition, whether simple or complex. (This last part — whether simple or complex — is important enough that we will return to it explicitly in the first part of the next section.)

(b) is a disjunction, an 'or' proposition. The first disjunct is "Jones owns a Ford.", and the second is "Brown is in Barcelona.". If we use the translation key …

o = Jones owns a Ford. i = Barnes is in Barcelona.

… we can express the proposition in the logically structured English as

o or i

In logically structured English, disjunctions have the form "S or T", that is, a proposition (the left-hand disjunct) followed by the word "or", followed by another proposition (the right-hand disjunct).

(c) is a conjunction, an 'and' proposition. The first conjunct is "Willie Mosconi was a pretty good pool player.", and the second conjunct is "Willie Mosconi won the World Pool Championship fifteen times.". Our translation key and translation might be:

p = Willie Mosconi was a pretty good pool player.w = Willie Mosconi won the World Pool Championship fifteen times.

p and w

In logically structured English, conjunctions have the form "S and T", that is, a proposition (the left-hand conjunct) followed by the word "and", followed by another proposition (the right-hand conjunct).

(d) is a conditional, an 'if … then ...' proposition. The antecedent (the "If …" part of the conditional) is "The summer sports camp is cancelled today.", and the consequent (the "then …" part of the conditional) is "It is hot.". Our translation key and translation for example (d) could be:

c = The summer sports camp is cancelled today.h = It is hot.

if c, then h

In logically structured English, conditionals have the form "if S, then T", that is, the word "if" followed by a proposition (the antecedent) followed by the word "then", followed by another proposition (the consequent).

It is important to realize that in the context of an inference conditionals aren't causal propositions. In other words, in the context of an inference the conditional proposition "If the summer sports camp is cancelled, it is hot." does not mean that the cancellation causes the heat. Rather, it means that if we believe that the sports camp is cancelled, then we can believe that it is hot.

You are not likely to think this particular example is a causal statement because you know that cancellations don't cause hot weather, and so the speaker is unlikely to mean that the cancellation causes the heat. But it is easy to make this mistake with lots of conditionals because causal statements are often expressed as conditionals. For example, the fact that the excess heat has caused the sports camp to be cancelled might be expressed by saying "If it is excessively hot, the camp is cancelled.". In the context of an inference, however, this conditional would only mean that if we believe that it is excessively hot, then we can believe that today's camp is cancelled.

A conditional being used to justify a conclusion is called a material conditional, as distinct from a causal conditional. In P&C, we are concerned only with material conditionals; in I&S we will investigate causes.

(e) is a bit tricky. It is a complex proposition: it has the proposition "The cat is on the mat." as a part. But it is neither a negation, nor a disjunction, nor a conjunction, nor a conditional. Because of this, we will treat it as a simple proposition in what follows, and we use a single letter (e.g. "b" for "believes") to stand for the entire thing. (All compound propositions are complex propositions, but not all complex propositions are compound. Negations and sentence like (e) are complex but not compound.)

2 Very Complex Propositions, Ambiguous Propositions

1. In general terms, when making a translation key for the propositions in a passage, assign a letter to every simple proposition, that is to every proposition in the inference that remains once you have set aside the words indicating a negation, disjunction, conjunction, or conditional.

Consider, for example, the following propositions:(a) The cat is on the mat.

(b) Jack is nice and Henry is mean.

In making a translation key for (a), you should use a letter to stand for the entire thing: after all, it is a simple proposition and is neither a negation,

nor a disjunction, nor a conjunction, nor a conditional. So, our translation key might simply be:

o = The cat is on the mat.

and the proposition in logically structured English is:

o

In contrast, in making a translation key for (b), you would not use a letter to stand for the entire thing because it is a conjunction. Instead, you should use a letter to stand for "Jack is nice.", and a different letter to stand for "Henry is mean.". Our translation key and translation into logically structured English might thus be:

n = Jack is nice.m = Henry is mean.

n and m

2. Now consider the following proposition: Jill is coming camping and if Smith is coming, Jones is coming.

This proposition is fundamentally a conjunction ("S and T"). But the right-hand conjunct ("T") is itself a complex proposition, in this case a condition with "Smith is coming camping." as the antecedent and "Jones is coming camping." as the consequent. If our key is as follows:

g = Jill is coming camping.s = Smith is coming camping.j = Jones is coming camping.

in logically structured English the whole proposition would be

g and if s then j

The lesson here is that complex propositions can become even more complex because the parts of a complex proposition can themselves be complex.

3. Now consider the following proposition: Jill is coming camping and either Smith or Jones is coming.

This proposition is fundamentally a conjunction ("S and T"). And the right-hand conjunct ("T") is itself a complex proposition, a disjunction of "Smith is coming camping." and "Jones is coming camping.". If we use the

same key as before, in logically structured English the whole proposition would be

g and s or j

But note that, as written, it is not clear what the fundamental logical word of the proposition is, "and" or "or". The proposition "g and s or j" is ambiguous. It could be understood as an 'and' proposition (a conjunction) with "g" as the left-hand conjunct and "s or j" as the right-hand conjunct, or, it could be understood as an 'or' proposition (a disjunction) with "g and s" as the left-hand disjunct and "j" as the right-hand disjunct. If we wanted to make clear that the fundamental logical word of "g and s or j" is the "and", we would insert parentheses as follows:

g and (s or j)

Since the "and" does not occur within parentheses, it is the fundamental logical word. If we were trying to translate a proposition with "or" as the fundamental structuring word, we would insert parentheses as follows:

(g and s) or j

Here are some more examples. Consider the following propositions in logically structured English, where "c" is "Jones is in Cleveland." and "o" is "Jones is in Ohio." and "t" is "Jones is traveling on business.":

(a) If c then o and t

(b) c and o or t

(c) Not c and t

Each of these propositions is ambiguous. In the first case, the speaker might be saying "If Jones is in Cleveland then he's both in Ohio and traveling on business." or "If Jones is in Cleveland then he's in Ohio. And also, he's traveling on business.". The ambiguity can be removed in this and the other propositions by using parentheses:

(a) (if c then o) and t vs. if c then (o and t)

(b) (c and o) or t vs. c and (o or t)

(c) (not c) and t vs. not (c and t)

4. Parentheses can be used in order to make any complex proposition easier to understand, even where there is, strictly speaking, no ambiguity. Consider the following proposition:

if a and b then c

Unlike "g and s or j", this proposition is not ambiguous, since the words "if" and "then" perform the function of parentheses, marking "a and b" as the antecedent and "c" as the consequent. However, since "and" and "or" are a frequent source of ambiguity, we can remove any doubt by using parentheses, as follows:

if (a and b) then c

In this case, it is clear that the proposition is fundamentally a conditional.

Parentheses can be used to make the structure of a complex proposition easier to see, but they must be used whenever the structure is ambiguous. One interesting case of ambiguity is when a proposition uses multiple "and"s or multiple "or"s. For example:

(d) a and b and c(e) a or b or c

In these cases, it is not clear with "and" or which "or" is the fundamental structuring word. We thus have to insert parentheses. But where? Which "and" is the fundamental word in (d)? Which "or" is the fundamental word in (e)? In fact, it doesn't matter and you can simply pick whichever "and" or "or" you want:

(d) (a and b) and c OR a and (b and c)

(e) (a or b) or c OR a or (b or c)

5. We can limit the need for parentheses somewhat by adopting the following rule: the logical word "not" modifies what follows it immediately. Consider the following proposition:

not a and b

Using our rule, in this proposition, the "not" is understood to modify only "a" and not "a and b". To negate "a and b" use parentheses, as follows:

not (a and b)

3 Some Extras On Negations, Disjunctions, & Conjunctions

1. Some propositions which have the logical structure of a negation and which should be translated as "not S" (or in full "It is not the case that S.") are disguised. Here are three such propositions:

(a) The cat is not on the mat.

(b) No doctors are rich.

(c) Socrates is unmarried.

All three of these are negations. (a) comes to "It is not the case that the cat is on the mat.". (b) comes to "It is not the case that some doctors are rich.". (c) comes to "It is not the case that Socrates is married.".

2. Some propositions which have the logical structure of a disjunction and should be translated as "S or T" are disguised:

(a) Jack is either a philosopher or a bus driver.

(b) The team will either win this week or lose.

(c) At least one of the two girls, Jill and Kofi, will get to the top of the mountain.

(a) comes to "Jack is a philosopher or Jim is a bus driver.", and (b) comes to "The team will win this week or the team will lose this week.". (c) does not use the word "or" at all; even so it can be translated as "Jill will get to the top of the mountain or Kofi will get to the top of the mountain.".

Note that in re-writing these propositions there is a complete proposition to the left of the word "or" and there is a complete proposition to the right of the word "or", even though in English speakers often abbreviate.

Note that "S or T" does not rule out the possibility of both S and T. English-language propositions, however, might describe a situation in which it would be impossible to have both. (a) does not rule out the possibility that Jack is both a philosopher and a bus driver, so translating it as "S or T" is appropriate. Proposition (b), on the other hand, should be translated as "(S or T) and not (S and T)". In English, careful speakers can rule out the 'both S and T' possibility either by doing so explicitly ("Jack is either a philosopher or a bus-driver, but not both.") or by using "(either) … or else …" as in "Jack is a philosopher or else he's a bus-driver." but they often fail to do so. In cases such as (b), the contents of the proposition are mutually exclusive, and these can be translated as "one or other and not both". For example, because a proposition such as "Jack is either on base or in the field." is understood to mean that he cannot be in both places at once), it should be translated (using the obvious letters) as "(b or f) and not (b and f)".

3. Some propositions which are, or involve, the logical structure of a conjunction and which should be translated as "S and T", are disguised:

(a) S but T.

(b) Although S, T.

(c) S; moreover, T.

In addition, the structure ...

(d) Neither S nor T.

… is equivalent to "Not S and not T". The expression "S if and only if T" is a conjunction but it will be treated in the next section because it also involves conditionals, and these are trickier than the conjunction to handle.

Consider, for example, the following propositions:(a) Jill loves country music but she hates rap.(b) Although the meeting isn't until tomorrow, the members came

yesterday.(c) Aquinas's First Way is invalid; moreover, it has three false

premises.(d) Jack is neither nice nor good at algebra.

(a) comes to "Jill loves country music and Jill hates rap.". In addition to conjoining the propositions, the word "but" highlights the contrast between, on one hand, Jill's attitude towards country music and, on the other hand, her attitude towards rap. (b) comes to "The meeting isn't until tomorrow and the members came yesterday.". Other words such as "nonetheless" are also logically equivalent to "and". (c) comes to "Aquinas's First Way is invalid and Aquinas's First Way has three false premises.". Other words such as "in addition" are also logically equivalent to "and". In addition to conjoining the propositions, words such as "but" in (a) and "although" in (c) highlight the fact that it is a bit surprising that the second proposition is true given that the first is true. We cannot capture this surprise in logically structured English and it is lost in translation.

(d) comes to "Jack is not nice and Jack is not good at algebra." and is of the general form "Not S and not T". (This is further equivalent to the negation proposition "It is not the case that (Jack is nice or Jack is good at algebra)." or "Not (S or T)".)

"Neither S nor T" is not equivalent to "not both S and T", such as "Jack not both nice and good at algebra.". "Not both S and T" should be translated into logically structured English as "Not (S and T)" or "not S or not T". Consider, for example, the proposition "Gore and Bush were not both elected President.". This propositions does not say that neither Gore nor Bush was elected President. Rather, it is saying that either Gore was not elected President or Bush was not elected President.

Thus, with "n" standing for "Jack is nice." and "g" for "Jack is good at algebra.", "Jack is neither nice nor good at algebra." is translated as "not n and not g" or "not (n or g)". "Jack is not both nice and good at algebra" on the other hand, is translated "not (n and g)" or "not n or not g".

Exercises

Part 1. For each proposition, underline "Negation" if (fundamentally) it is a negation, underline "Disjunction" if it is a disjunction, underline "Conjunction" if it is a conjunction, underline "Conditional" if it is a conditional.

1. The rabbit went either to the left or to the right.Negation Disjunction Conjunction Conditional

2. Marshall Swain isn't a coherentist.Negation Disjunction Conjunction Conditional

3. Laurence BonJour used to be a coherentist but now he's an old-school foundationalist.

Negation Disjunction Conjunction Conditional

4. If it's not too cold, Jack will go camping.Negation Disjunction Conjunction Conditional

5. Although he promised otherwise, Henry ate the last cookie.Negation Disjunction Conjunction Conditional

6. Some people drive too fast and others use their phones while driving.Negation Disjunction Conjunction Conditional

7. Jack is tall, dark and handsome.Negation Disjunction Conjunction Conditional

8. Jack hates country music, whereas Gill loves it.Negation Disjunction Conjunction Conditional

9. If this rain doesn't stop, we'll have to stay in, but if it does, we go can fly a kite.Negation Disjunction Conjunction Conditional

10. Either Jack will show up late and offer some lame excuse or Jill will get them both here on time.Negation Disjunction Conjunction Conditional

11. Henry isn't worried about the impending snow-storm.Negation Disjunction Conjunction Conditional

Part 2. For each proposition, use the translation key below and underline the correct translation.

j: John is at the party.s: Sue is at the party.g: Gale is at the party.

12. Neither Sue nor Gale is at the party.

a. if not s, then not gb. not s and not gc. not s or not gd. (if not s, then not g) and (if not g, then not s)e. s and g

13. Sue and Gale aren't at the party but nonetheless John is there.

f. if (not s and not g), then jg. if j, then (not s and not g)h. (not s and not g) or ji. (if (not s and not g), then j) and (if j, then (not s and not g))j. (not s and not g) and j

14. Gale and John are at the party, or Gale and Sue are at the party.

a. (g or j) and (g or s)b. (j and g) or (j and s)c. if (g and j), then (g and s)d. (g and j) or (g and s)e. (if g, then j) or (if g, then s)

Part 3. For each proposition, use the translation key below to put the assertion into Logically Structured English. Use parentheses as necessary.

Translation Keyb: Jack is a bachelor. (Read as: Let “b” stand for the proposition “Jack is a bachelor.”.)h: Jack has a good sense of humor.j: Jill will go out with Jack.l: Jack likes the outdoors.m: Jack is married.s: Jack says that he is a bachelor.t: Jack is tall.

15. Either Jack is a bachelor or he is married.

16. Jack does not like the outdoors.

17. Although Jack isn't married, Jill will not go out with him.

18. Jack is not both tall and likes the outdoors.

19. It's not the case that Jack either likes the outdoors or has a good sense of humor.

20. If Jack is tall or likes the outdoors, then Jill will go out with him.

21. If Jack is tall, likes the outdoors, and has a good sense of humor, then, if he’s a bachelor, Jill will go out with him.

Answers To Select Exercises

6. Some people drive too fast and others use their phones while driving.Negation Disjunction Conjunction Conditional 8. Jack hates country music, whereas Gill loves it.Negation Disjunction Conjunction Conditional 10. Either Jack will show up late and offer some lame excuse or Gill will

get them both here on time.Negation Disjunction Conjunction Conditional

Part 3. For each proposition, use the translation key below to put the assertion into Logically Structured English. Use parentheses as necessary.b: Jack is a bachelor.h: Jack has a good sense of humor.j: Jill will go out with Jack.l: Jack likes the outdoors.m: Jack is married.s: Jack says that he is a bachelor.t: Jack is tall.

16. Jack does not like the outdoors.not l

20. If Jack is tall or likes the outdoors, then Jill will go out with him.

if (t or l) then jThe parentheses are not strictly necessary.

4 Some Extras On Conditionals

1. A variety of English expressions can be translated as "if … then …":

(a) T if S.

(b) S only if T.

(c) Provided that S, T. (or "T provided that S.")

(d) In order that T, it is sufficient that S.

Crucial Advice: With conjunctions and disjunctions, it actually doesn't matter what order the parts appear in: "S and T" means the same as "T and S"; "S or T" means the same as "T or S".

But with conditionals, the order of the sub-propositions is vitally important.

"if S, then T" does not mean the same as "if T, then S": "S only if T" does not mean the same as "T only if S"; and so on.

So: Pay attention to the order that the sub-propositions appear in. Each of (a) through (d) is equivalent to "if S, then T" and none of them means "if T, then S".

A few other common English expressions involve "if … then …" along with other logical words:

(e) Unless S, T. (or "T unless S.")

(f) In order that S, it is necessary that T.

(g) S if and only if T.

(e) is translated as "if not S, then T". (f) is translated as "if not T then not S." (which is further equivalent to "if S then T."). (g) is translated as "(if S then T) and (if T then S)".

Consider, for example, the following propositions:(a) Jack got an A if he got every possible point on the homework

assignments and exams.

(b) Henry graduates only if he passes his logic course.

(c) You can watch TV provided that you have done your homework.

(d) In order that Bill receives his allowance, it is sufficient that he cleans his room.

(e) In order that Henry graduates from college, it is necessary that he passes logic.

(f) You can leave your room if and only if you have apologized to your brother.

(g) Unless you pay up front, you're paying more than you need to.

In proposition (a), the "if" part of the proposition (the antecedent) appears second. This is perfectly natural English, but in logically structured English, the antecedent has to come first. We can rewrite (a) as "If Jack got every possible point on the homework assignments and exams, then Jack got an A." and see that it matches the structure of a conditional: "if S, then T".

Many of the other words and phrases that are translated using some version of "if … then ..." can be found in either the first or second half of the English sentence. For example, "provided that" can appear at the start of the English sentence or in the middle; "only if" can appear at the start of the sentence or in the middle; and so on.

Concerning propositions such as (b), it might seem that "S only if T." should become "if T, then S", rather than "if S, then T". But think about whether (b) is equivalent to (i) or to (ii):

(b) Henry graduates only if he passes his logic course.

(i) If Henry passes his logic course, then Henry graduates.

(ii) If Henry graduates, then Henry passes his logic course.

"only if" means that the condition that follows is one condition. Henry's registrar might say, for example, "You graduate only if you pass logic.". By this the registrar would mean that passing logic is one condition needed for graduation. The registrar does not mean, however, "If Henry passes logic, he graduates." since there are (or could be) other requirements for graduation. So, option (i) is not equivalent to (b). It is equivalent to (ii): if Henry graduates, then he passes logic. (This sentence sounds awkward because "If ..., then … ." in English is often used to express a causal connection, and causal connections involve a temporal direction (from earlier to later). But, as mentioned in section 2, above, in the context of an inference we are only concerned with the conditional truth or believability of two propositions, regardless of any causal or temporal relationship between what they describe.)

Quick Trick: To quickly turn "S only if T." into an "if …, then …" proposition of the standard form, replace the words "only if" with "then" and place an "if" at the beginning of the proposition. (b) comes to "If Henry graduated last spring, then Henry passed his logic course last fall.". In other words, go through these steps:

S only if T. ⇒ S only if then T. ⇒ if S, then T

Proposition (c) involves "provided that". In everyday English the phrase "provided that" is typically used to stipulate a condition(s) under which a certain action or event will take place. If this is satisfied, the event will take place. Thus "You can watch TV provided that you brush your teeth." can be understood as "If you brush your teeth, then you can watch TV." and in logically structured English as "if b then w", using "b" for "You brush your teeth." and 'w' for "You can watch TV.".

To translate (c), we must first re-arrange the proposition so that "provided that" appears at the front: "Provided you have done your homework, you can watch TV.". This then comes to "If you have done your homework, then you can watch TV.".

Proposition (d) comes to "If Bill cleans his room, then he receives his allowance.". Using "c" for "Bill cleans his room." and "r" for "Bill receives his allowance.", we translate into logically structured English as "if c then r".

There are other variations expressing the same idea in English, such as ...

Cleaning his room is sufficient for Bill to receive his allowance.Bill cleaning his room is a sufficient condition for receiving his

allowance.To receive his allowance, Bill need only clean his room.

(In formal English, the subjunctive can be used …

In order that Bill receive his allowance, it is sufficient that he clean his room.

In order that Bill should receive his allowance, it is sufficient that ...In order that Bill might receive his allowance, it is sufficient that …

… but nobody seems to talk like this any more.)Proposition (e) describes one thing as being necessary for another, in

this case, passing logic is necessary in order for Henry to graduate. One state of affairs being necessary for another means that if the one is not present or does not occur (passing logic), then the other is not present or does not occur (graduating). "In order that Henry graduates from college, it is necessary that he passes logic." can thus be understood as "If Henry does not pass logic, then he does not graduate.”. In logically structured English, using obvious letters, we get "if not p then not g" and in general "if not S, then not T".

English has other ways of expressing the idea of one state of affairs being necessary for another besides “In order that …, it is necessary that …” such as ...

In order to graduate from college, Henry must pass logic.In order for Henry to graduate, he must pass logic.Henry’s passing logic is necessary if he is to graduate from college.Henry’s passing logic is a necessary condition for his graduation from

college.Passing logic is necessary for Henry to graduate.

(Here are the subjunctive forms …

In order that Henry graduate from college, it is necessary that he pass logic.

In order that Henry should graduate from college, it is necessary that ...In order that Henry might graduate from college, it is necessary that … )

When translating propositions such as (e) and (f), be sure to re-arrange the proposition so that "it is sufficient that" and "it is necessary that" is in the middle; in English it is natural to reverse this order and say “It is sufficient/necessary that … in order that …” and so on for all of the alternatives.

Concerning propositions such as (d) and (e), notice that in the alternative English expressions the parts are not, grammatically, propositions. In the proposition "Henry’s passing logic is necessary if he is to graduate from college.", for example, they are gerunds that point to a certain state of affairs, such as 'Henry's passing logic'. In giving a translation key for a logically structured English translation, however, these phrases should be turned into propositions.

The thought that one thing is sufficient for another and/or is necessary for another, can be applied to lots of different contexts. Necessary and sufficient conditions are used constantly in the law, the tax-code, computer-programming, and modeling the natural world. They might be tricky to grasp at first, but you will soon see them everywhere you look.

Proposition (f) contains the phrase "S if and only if T." which is a quick way of saying "S if T, and, S only if T.". (It can also be written, even more briefly, as "S iff T.". Yes, there are two "f"s in that "iff".) As we saw above concerning (a), the first conjunct in logically structured English is "if T then S". As we saw above concerning (b), the second conjunct is "if S then T". The whole proposition, therefore, is translated as "(if T then S) and (if S then T)". In short, "S if and only if T." is a conjunction of two conditionals, with their antecedent and consequent reversed.

Thus (f) comes to "You can leave your room if you have apologized to your brother and you can leave your room only if you apologize to your brother." which in turn comes to "If you have apologized to your brother then you can leave your room and if you can leave then you have apologized to your brother.". Using obvious letters, the logically structured English translation is "(if a then l) and (if l then a)".

Finally, proposition (g) comes to "If it is not the case that you pay up front, then you are paying more than you need to.". Note that English allows the "unless" to occur in the middle of the proposition; in such cases, the constituent propositions should be re-ordered before translating.

"Unless" is tricky. Here is the procedure for moving from English propositions using "unless" to logically structured English. First, ensure that the proposition is of the form "Unless S, T". If the proposition has the form "T unless S.", change it first to "Unless S, T.". Second, in a proposition of the form "Unless S, T.", replace "unless" with "if not …", and add a "then" before T, to get "If not S, then T.". In other words:

T unless S. ⇒ Unless S, T ⇒ Unless if not S, T. ⇒ if not S, then T

For example, translating the sentence "Jill will not go out with Jack unless Jack is a bachelor." into logically structured English goes like this. Move first to "Unless Jack is a bachelor, Jill will not go out with him.". Then replace "unless" with "if not", yielding "If Jack is not a bachelor, then Jill will not go out with him.". As an extra complication in this example, both the antecedent (the 'if' proposition) and the consequent (the 'then' proposition) of the sentence are negations, so we also have to move the negations to the front of each part of the proposition, as follows: "If it is not the case that Jack is a bachelor, then it is not the case that Jill will go out with him.". Using "b" for "Jack is a bachelor." and "g" for "Jill will go out with Jack." we write: "if not b then not g".

Exercises

Part 1. For each proposition, underline "Negation" if (fundamentally) it is a negation, underline "Disjunction" if it is a disjunction, underline "Conjunction" if it is a conjunction, underline "Conditional" if it is a conditional.

1. Provided that oxygen is present and not neon, the splint will flare up.Negation Disjunction Conjunction Conditional

2. Only if Jill is both alive and in Ohio is she walking in a park in Columbus.


3. Although he bombed the mid-term, theoretically Jack can still pass the class.


4. Either he got there on time but without the wine, or he got there with the wine but not on time.


5. Unless you tell the truth, I can't help you.Negation Disjunction Conjunction Conditional

6. Plato wrote Socrates's Defense and also Euthyphro, but not The Odyssey.


7. Having wings is a necessary condition for being a bee.Negation Disjunction Conjunction Conditional

8. Bolt will win provided that he doesn't pull his hamstring.


9. We'll have a party whenever Johnny comes marching home.Negation Disjunction Conjunction Conditional

10. We can play today only if at least 10 people show up.Negation Disjunction Conjunction Conditional

Part 2. For each proposition, use the translation key below and underline the correct translation.


11. Gale is at the party if John is.

a. g and jb. g or jc. if g then jd. if j then ge. s or j

12. If Gale is at the party, John is, and if John is there, so is Gale.

a. g and jb. g or jc. if g then jd. if j then ge. (if g then j) and (if j then g)

13. A sufficient condition for Gale's being at the party is that John isn't there.

a. g and not jb. if not j, then gc. if j, then gd. if g, then not je. if g, then not s

14. Sue is at the party provided that John isn't.

a. if s, then not jb. s and not j

c. if not j, then sd. (if s, then j) and (if j, then s)e. j or not s

15. John is at the party unless Gale is there.

a. j and gb. if g, then jc. if j, then gd. if not g, then je. (if j, then not g) and (if not g, then j)

Part 3. For each proposition, use the translation key below to put the assertion into Logically Structured English. Use parentheses as necessary.b: Jack is a bachelor.h: Jack has a good sense of humor.j: Jill will go out with Jack.l: Jack likes the outdoors.m: Jack is married.s: Jack says that he is a bachelor.t: Jack is tall.

16. Jill will go out with Jack if he likes the outdoors.17. It's not the case that if Jack says he's a bachelor then he is a

bachelor.18. Jack is neither tall nor likes the outdoors.19. Unless Jack says he's a bachelor, Jill will not go out with him.20. Jack's being unmarried is a necessary condition for Jill's going

out with him.21. Only if Jack is a bachelor will Jill go out with him.

Answers To Select Exercises

2. Only if Jill is both alive and in Ohio is she walking in a park in Columbus.

Negation Disjunction Conjunction Conditional 4. Either he got there on time but without the wine, or he got there with

the wine but not on time.Negation Disjunction Conjunction Conditional (The simple answer is Disjunction, based on the "or". But Conjunction is also a possibility, since he cannot do both (exclusive-or), which is perhaps suggested by the "either … or …".)6. Plato wrote Socrates's Defense and also Euthyphro, but not The

Odyssey.Negation Disjunction Conjunction Conditional

8. Bolt will win provided that he doesn't pull his hamstring.Negation Disjunction Conjunction Conditional

Part 2.


14. Sue is at the party provided that John isn't.

c. if not j, then s

Part 3

b: Jack is a bachelor.j: Jill will go out with Jack.h: Jack has a good sense of humor.l: Jack likes the outdoors.m: Jack is married.s: Jack says that he is a bachelor.t: Jack is tall.

17. It's not the case that if Jack says he's a bachelor then he is a bachelor.

not (if s then b) 21. Only if Jack is a bachelor will Jill go out with him.if j then b

5 Summary

1. To translate a proposition in English into a logically structured English proposition, follow these steps:

(i) Make a translation key. That is, identify the simple propositions in the proposition and assign each simple proposition a lower-case letter.

(ii) Rewrite the proposition using the letters chosen in step one and only the four logical structure words "not ...", "... and ...", "... or ...", "if … then …".

The logical structure words ("not ...", "... and ...", "... or ….", "if …, then …") are often disguised as other English words. Here is a brief translation dictionary from English to Logically Structured English:

"Although S, T." = S and T"At least one of two …" = S or T"S but T." = S and T"either S or T" = S or T (But see "or else", below.)"S if and only if T" = (if S then T) and (if T then S)"It is not the case that S." = not S"It is false that S." = not S"S; moreover, T." = S and T"S is necessary for T" or "In order that T it is necessary that S." = if not S, then not T = if T, then S"neither S nor T" = not (S or T) = not S and not T"No …" = "It is false that some …" = "not some …" (e. g. "No Scotsman puts sugar on his porridge." = "It is not the case that some Scotsmen put sugar on their porridge.")"S only if T" = S then T"(Either) S or else T." = (S or T) and not (S and T)"S is sufficient for T or "In order that T it is sufficient that S." = if S then T"un-" as a prefix (e. g. "Jack is unmarried.") = not S"Unless S, T." or "T unless S." = if not S, then T

5 summary - web view, it is not clear what the fundamental logical word of the proposition is,...

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