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Thu 5 Jun, 2008 07:02 pm
Translating is probably one of the most interesting parts of the introductory phase of logic. This will thread will show you how to convert everyday sentences into logical symbolization so that you can calculate it later in proofs.
It is a step by step process and gradually incorporates many sentences and connectives, so if you get lost in one phase of the explanation, go back to the beginning and recap to see if you missed out on any information.
How to symbolize a single sentence-----------------------------------------------------------------------------------
Take this simple sentence; "John is a funny guy"
1.It is up to you to pick a letter to symbolize this sentence, but it is easiest to go for the most obvious subject, which is John.
So we can symbolize "John is a funny guy" as the letter "J" which symbolizes the entire sentence.
How to symbolize a compound sentence----------------------------------------------------------------------------
Now take the compound sentence; "John is a funny guy and Mary is a funny girl."
1.First, isolate the two sentences you see in the compound sentence; "John is a funny guy and Mary is a funny girl."
2.Now take what we know about connectives from symposium 3 and translate the connective "and" into the connective "&."
3.Now we can translate the compound sentence into; J & M.
How to symbolize a sentence with more than one connective.-----------------------------------------------
Example 1.
Take this sentence; "John is funny and Mary is funny or Alan will be there."
1.First, isolate the simple sentences; "John is funny and Mary is funny or Alan will be there."
2.Now translate the connectives; and (&), or (v).
3.Now from left to right translate; J & M v A
But this is where it can become difficult. You have to group the symbols in order to simplify. The grouping work the same way they do in math, and this is why I suspect people are afraid of formal logic, because it looks like math, But don't think of it in that sense. Group the way you would in math; (z),[y ( Z) ], {x [ y ( z)] } judging on the configuration of the sentence.
4.In this case the sentence can be translated as either (J & M) v A or J & (M v A)
You can do it either way.
Example 2.
Take the sentence; "Either John walks and Mary walks, or Ann walks and Barry walks."
1.First isolate the compound sentences; "Either John walks and Mary walks, or Ann walks and Barry walks."
2.Break down the isolated compound sentences into simple sentence and their connective; J & M, A & B.
3.Then identify the main connective. It may seem obvious what the main connective is. The beginning of the sentence begins with "Either" and the middle of the sentence has the connective "or." A disjunction is formally identified as "either?or"
So put it all together and group the compound sentences together joined by the main connective.
4.So the sentence translates as; ( J & M ) v ( A & B)
How to symbolize a single sentence with a Negation--------------------------------------------------------------------
But what if we say; "It is not the case that John is a funny guy."
1.If you remember the characteristics of a negation from Symposium 3, this sentence incorporates a negation. Now to symbolize this sentence, we first look at the whole sentence and isolate the main point, which is "John is a funny guy." Now this sentence is translated as "J" BUT!!!!! The sentence is negated when we incorporate the rest of the sentence, "It is not the case that?" When you see the phrase "It is not the case that?" you can substitute it with a negation symbol, (~).
So we can symbolize "It is not the case that John is a funny guy" as ~A.
How to symbolize a compound sentence with a negation-------------------------------------------------------------
Negations are annoying and are very difficult, so don't be annoyed if you cannot get it right off the bat. Usually, arguments that use proper English never pose a statement this way, but people who do not use proper English make state something thusly.
Take this sentence; "It is not the case that John is funny and Mary is funny."
1.First way, isolate the simple sentences; "It is not the case that John is funny and Mary is funny."
2.Translate with the main connective; J & M
Now you can interpret the sentence again this way? please hang in there with me.
3."It is not the case that John is funny, but Mary is funny"
Translating the sentence again this way makes it easier for you to translate.
4.Isolate the simple sentences; "It is not the case that John is funny, but Mary is funny"
5.Isolate the connective; but.
Here's the thing, in so many words, "but" is equivocal with "and." Just remember that.
6.Translate as; J & M
7.Incorporate the negation, "It is not the case that?" as ~.
8.Translate the whole sentence as; ~J & M.
UP UNTIL NOW!!!!!--------------------------------------------------------------------------------------------------------------
At this point, you are aware of simple, compound, multiple connective, and negation sentences.
Now for translating when it comes to conjunctions, disjunctions, conditional, and bi-conditionals
Translating connectives---------------------------------------------------------------------------------------------------------
Conjunctions
Take this sentence for example; "John is funny and Mary is funny."
1.Identify the simple sentences; "John is funny and Mary is funny."
2.Identify the connective; "John is funny and Mary is funny."
3.Translate; A & B
Disjunctions
Take this sentence for example; "Either John is funny or Mary is funny."
1.Identify the simple sentences; "Either John is funny or Mary is funny."
2.Identify the connective; "EitherJohn is funny or Mary is funny."
3.Translate; A v B
Conditional
Take this sentence for example; "If John is funny then Mary is funny."
1.Identify the simple sentences; "If John is funny then Mary is funny."
2.Identify the connective; "If John is funny then Mary is funny."
3.Translate; A --> B
Bi-Conditional
Take this sentence for example; "John is funny if and only if Mary is funny."
1.Identify the simple sentences;" John is funny if and only if Mary is funny."
2.Identify the connective; "John is funny if and only ifMary is funny."
3.Translate; A <-->B
REACP!!!!-----------------------------------------------------------------------------------------------------------------------------
Suffice to say, just keep the foremost translation method in mind;
1.Read the sentence (simple, compound, etc.)
2.If a simple sentence, simply translate as a single letter.
3.If compound sentence, identify the individual sentences in the compound sentence.
4.Identify the connective, or connectives in the compound sentence.
5.Translate the entire sentence.
6.Group according to the syntactical structure of the argument.
PRACTICE!!!------------------------------------------------------------------------------
1.The Songbirds will sing and the Owls will hoot, or the Eagles will scream.
2.Either Ted will jog and Bill will jog, or Alan will jog.
3.Either John or Mary will jog, or either Bill or Dennis will jog.
4.Alexia will run and Barry will run, and Charles will run.
5.John will play a tune and either Alan will play or Mary will play.
6.John and Mary both won't run.
Answer key (highlight the empty area for the answers)
1.(S & O) v E
2.(T & B) v A
3.(J v M) v (B v D)
4.(A & B) & C
5.J & (A v M)
6.~ (J & M)
I'm sure this can be said better, so I'll be updating this often. There is ALOT more to say on translating, but explaining every little instance and idiosyncrasy may be hard to follow.
PLEASE ASK ANY QUESTIONS YOU HAVE BECAUSE I KNOW I WAS NOT VERY CLEAR ON MUCH OF THIS.
@VideCorSpoon,
VideCorSpoon,
May I ask for answers in a different notation? I think this will prove to be very confusing. Perhaps the anwers will be in plain sight when using word and sticking it up there, but at least your explanations will be consistent.
@Arjen,
Of course! This is a more visual version of the answer key.
@Arjen,
Thanks, I guess??? Is there some other way you wanted the answers?
Also, I'm sensing some hostility?
@VideCorSpoon,
VideCorSpoon wrote:Thanks, I guess??? Is there some other way you wanted the answers?
I was just thinking of replacing the whited answers with formal logic notations is all. You went the other way. I guess it doesn't relly matter.
Quote:
Also, I'm sensing some hostility?
Not at all. It just made me laugh. It seemed more logical to me to replace the whited answers is all.
@Arjen,
Arjen... those are formal logic notations. There are many ways to symbolize a connective, the ones I'm using is the Pospesal method because it is easier to type, other wise I would be using horseshoe, turnstyle, etc. There are many other ways.
As to your other comment... some would call that being a di... ahem, being a disingenuous comment.
@VideCorSpoon,
Ok I got the hang of it now
, I made up a couple of random scentences of my own and put them in truth tables but I am not sure what is the 'answer'... When all the connectives are true? when the main connective is true? etc.
e.g. if the cat is in at night then I will not be able to sleep if he meows.
If the cat is in at night (c) then [-->] I will not be able to sleep (~s) if [v] he meows (m)
C --> (~S v M) or (C --> ~S) v M (i'm not sure which:o).
right?
if so can you make a truth table for me and highlght what I should look at for the 'solution'... pleeease
,
Dan.
@de budding,
You are extremely close!
You cited the variables correctly and cited the first conditional correctly, but look at the last part of your sentence "?if he meows." You have an open conditional without a conclusion so the sentence in a way is incomplete.
Your sentence, "If the cat is in at night then I will not be able to sleep if he meows" would look something like this; C -->~S ? M ? ?
The M is not logically connected, even though it sounds like it is. It is a floating premise without a conclusion.
Your logical translations are perfect but in this particular case do not match the sentence composition.
When you translated the sentence as; C --> (~S v M) The sentence translates back to," if the cat is in at night, then I will not be able to sleep or he meows.
When you translated he sentence as; (C --> ~S) v M The sentence translates back into, "If the cat is in at night, then I will not be able to sleep. Or he meows."
You could say this; "If the cat is in at night and meows, then I will not be able to sleep."
This translates into (C & M) --> ~ S
As to the truth tables, Ill post the complex truth table in a little bit to (C & M) --> ~ S, but you are one step ahead in the game. I'm impressed that you are absorbing the method down so quickly. This is a few months into predicate logic classes. But I think you are really going to enjoy the next thread on complex truth tables.
@VideCorSpoon,
VideCorSpoon wrote:Arjen... those are formal logic notations. There are many ways to symbolize a connective, the ones I'm using is the Pospesal method because it is easier to type, other wise I would be using horseshoe, turnstyle, etc. There are many other ways.
As to your other comment... some would call that being a di... ahem, being a disingenuous comment.
I think I now know what hostility you sensed.
Asking for consistency is disengenial, but not understanding the basics andthe limits of the system is perfectly normal? I beg to differ.
@Arjen,
Arjen,
I am very glad you now know what hostility I sensed.
But besides this, why is this the only conversation that has nothing constructive to lend to the topic?
@VideCorSpoon,
VideCorSpoon wrote:Arjen,
I am very glad you now know what hostility I sensed.
But besides this, why is this the only conversation that has nothing constructive to lend to the topic?
Ah, more hostility. Perhaps it is not constructive because you do not allow it to be?
@VideCorSpoon,
Ok you know the statement "The songbirds will sing and the Owls will hoot, or the Eagles will scream".
You translate it to this. (S & O) v E
However, what's to say it can't be S & (O v E)? Can we establish a rule for determining the placement of brackets in comparison to the english?
Even though I somehow intuitively grasped the right answer I wonder why I did so, naturally.
What if the comma was moved to precede the 'and'. Is that the determinate of placing the brackets, or does the subjects affected by and take priority to the subject affected by or, and the brackets are placed for highest priority.
@Holiday20310401,
That's a good point.
If you translate "the songbirds will sing and the owls will hoot, or the eagles will scream" it will indeed translate into (S & O) v E. It could translate as S & (O v E), but it will not retranslate back into English? which is a key point that it is a faulty translation. If you retranslated S & (O v E), you would get "the songbirds will sing and either the owls will hoot or the eagles will scream." So you can see that the two sentences are not the same. Correct placement of the parenthesis is vital. But also, when you are translating, look at the commas or semi-colons are in the sentences. This is usually an indicator that there is some sort of division in the logical sentence. I think it came naturally to you because you saw that comma as a division and understood proper logical syntax. That fantastic! When you start seeing thing like that, you are subconsciously putting yourself in a logical frame of mind which does nothing but help you create more logical and cogent sentences.
If the comma was moved before the and, I would suppose that there would then be some ground for the sentence to translate into S & (O v E) supposing that the comma between hoot and or is removed. Otherwise, it might not be a well formed sentence. Certainly, it would be a well formed formula, but syntactically not good.
Is the comma the determinate of the brackets? Yes and no. Good observation. It depends on how the sentence is laid out. Commas are usually a good determinate that there is a break or a conditional nearby. Thinking about it, the more you become acquainted with syntactical structure and logical structure, the easier it will be discern what goes where.
Darn good point though.
@VideCorSpoon,
if the comma is the determinate in the statement, how would something like this be symbolised?
it's cold, but it's not windy or foggy
i was thinking (C ~(W v F)). am i close at all? is it appropriate to have a basic proposition followed by a negated compound?
C: it's cold W: it's windy F: it's foggy
@facepuncher,
Actually yes for the most part. You could say "it's cold, but it's not windy or foggy" or " It is neither windy or foggy, but it is cold." Same thing, just different structure.
The translation (C ~(W v F)) is alright except for a few things. First, You need to have proper parenthetical devices (symposia 5, post 1,3) first. So change the outer parenthesis to brackets.
[C ~(W v F)]
Then remember the rule for the well formed formula (symposia 5, post 1). A well formed formula cannot have a floating variable, namely C.
With that in mind, you need to add a sentence operator (i.e. &, v, -->, <-->). The word "but" functions in many respects like a conditional. So basically when you see the word "but," think "if,then." So all you need to do is add the conjunction symbol between the C and the ~(W v F). So it becomes;
[C --> ~(W v F)]
But you can definitely have a single variable followed by a negated compound sentence. Its all a matter of translation.
@VideCorSpoon,
[INDENT]
much appreciated sire, its difficult to find help with this kind of stuff anywhere. if i could just ask pose more question thats been bugging me.
obviously up to you whether you want to try at it:
If you'll stay healthy if and only if you exercise and don't eat junk food, then even if you exercise, if you eat junk food you won't stay healthy.
(H ≡ (E ∧ ?J) was about as far as i got, addressing the first part.
H = stay healthy E = exercise J = eat junk food
≡ is material biconditional or iff (for some reason i seem to use different operators than everyone else)
muchly appreciated
[/INDENT]
@VideCorSpoon,
VideCorSpoon wrote:
With that in mind, you need to add a sentence operator (i.e. &, v, -->, <-->). The word "but" functions in many respects like a conditional. So basically when you see the word "but," think "if,then." So all you need to do is add the conjunction symbol between the C and the ~(W v F). So it becomes;
Nice thread, but just had to comment on this. "But" usually would be translated into an AND operator, not a conditional. Now maybe if it were worded something like "A but for B", it could be translated differently, though it seems that standard practice is usually to take AND from "but".
Now that you've got this posted we just need a follow-up on using truth tables to check validity of arguments. That would come in useful around here. :cool:
facepuncher wrote:If you'll stay healthy if and only if you exercise and don't eat junk food, then even if you exercise, if you eat junk food you won't stay healthy.
(H ≡ (E ∧ ?J) was about as far as i got, addressing the first part.
That's just a very convoluted, redundant sentence. You might as well delete everything after the first comma, (and the "if" at the beginning, is that a typo?) and you get the same statement, where your translation is correct (if you remove the one parenthetical in front of the H, and if that ^ symbol is an AND).
@Pangloss,
pangloss sire can you halp with my above question?
I KNOW YOU HAVE THE BRAIN POWER OF A GREAT LOGICIAN
yeah i can see now that it is quite redundant, but can the second half still be symbolized simply for the sake of it, not for any argumentative purposes?
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