Mon 16 Jun, 2008 05:24 pm
In this thread I will be introducing you to more complex truth tables. This is a VERY useful tool if you run up against poorly constructed arguments because if you familiarize yourself with the method well enough, you will be able to see right through a faulty argument before considering any of the facts involved or you could break down an argument to find out how it can be contradicted and then try to corner your opponent with that contradiction.

WHAT YOU SHOULD KNOW AT THIS POINT----------------------------------------------------------

At this point you should understand the following from the previous symposiums in order to fully grasp the methodology of complex truth tables;

Symposium 2 - The basics of truth table formation and composition.
Symposium 3 - the fundamental operations of connectives (&,v,-->,<-->, ~)
Symposium 4 - Translating what you want to know the truth value of into Logical language.
Symposium 5 - Knowing how to attack truth values, etc.

RECAP OF TRUTH TABLES---------------------------------------------

So now on to complex truth tables. You can probably gather by now that a truth table is constructed to show you how to determine all the possible truth values a statement can have.

It is at this point that is best to refer to Symposia 2 to show how a basic truth table is composed because it may be hard to follow if you start from this point. Here's a brief rundown of the basics and method. If you are confused, ask by replying to this thread or refer to symposium 2.

COMPLEX TRUTH TABLES AND ARGUMENT VALIDATION------------------------------------------

If Alan is running then Bob is running. Bob is not running. Therefore, Alan is not running.

Now from this point, we need to utilize a method that will come in handy with proofs in the next symposium. Look carefully at the argument. The first thing you should note is that there are three separate sentences. These sentences are the components of your arguments. In the introduction symposium (1), I explained that an argument is a set of premises followed by a conclusion. Now, take the component sentences of this argument and give each sentence its own line.

1. If Alan is running then Bob is running.
2. Bob is not running
3. Therefore, Alan is not running.

Line 1 and line 2 are the premises and line 3 is the conclusion, denoted by the "therefore." Now that we have the lines identified, we can translate these English lines into logic.

1. A --> B
2. ~ B
3. ~ A

Now that we have translated the sentences to their respective line, we can construct a complex truth table. This is what the structure of a complex truth table looks like. (and I also threw in a simple truth table so that you can compare them side by side.)

The complex truth table is pretty much the same as the regular truth table. The only difference is that more solution columns have been put in to facilitate the lines of the problem. So if the argument you were evaluating had 5 lines, you would have 4 premises columns and a conclusion column.

Now that we understand how to construct a complex truth table, let's solve the argument at hand.

STEP 0 - Translate the argument into logic and lines.
STEP 1 - Draw your complex truth table. It is very much similar to a simple truth table like we have seen. The only additions are the extra columns to facilitate each premise and the conclusion in the argument. In this case, there are two premises and 1 conclusion, so there are three columns to work with.
STEP 2 - Insert the premises and the conclusion into the top right column sections. Remember each premise gets their very own column.
STEP 3 - Insert all the variables in the argument into the top left section in order to work out a truth probability matrix.
STEP 4 - Input the truth probability matrix for the two variables in the argument.
STEP 5 - Write in all the truth probabilities for each variable in the bottom right section.
STEP 6 - Solve each column independently.
STEP 7 - Now here is how you solve the complex truth table. You have two main things to keep in mind.

1.If one of the rows in your truth table contains all True's in your premises columns but a False in your conclusion column, then the argument is invalid

2.If there are now rows which contain true premises and a false conclusion, the the argument is valid.

Since there are no such rows where all the premises are true and the conclusion is false, the argument is indeed valid.

So that is how you do a complex truth table. But there is actually a quicker way to perform a truth table.

PARTIAL TRUTH TABLE METHOD---------------------------------------

That's pretty much the main parts of complex truth tables and partial truth tables. There are a few other tricks, like tautology and contingency finders, but that may be a little too much. But let me know if you would like to know how to do them and I'll be sure to post them.

AS ALWAYS, ASK ANY QUESTIONS AND I'LL BE HAPPY TO ANSWER THEM AS WELL AS PROVIDE SAMPLE PROBLEMS!!!

VideCorSpoon

Tue 17 Jun, 2008 07:22 am
@VideCorSpoon,
Here are some sample problems to do. I posted the link to the answer at the bottom of the post. For the complex truth tables, I just listed the main connective truth values so you can see the answers more clearly.
(remember that the / denotes the conclusion, which has its own line.)

Complex truth tables.

1.
A v B
B v C / A v C

2.
A --> B / B --> A

3.
A --> (B -->C) / C --> ( B --> A )

Partial truth tables

1.
A -->B
W-->S
B v S / A v W

2.
~ (A & B) / ~A

Complex Truth Tables - http://i32.tinypic.com/27yq4v4.jpg
Partial Truth Tables - http://i25.tinypic.com/1059rmo.jpg

Arjen

Tue 17 Jun, 2008 07:51 am
@VideCorSpoon,
Say VideCorSpoon, is it not true that B v C / A v C can also be spelled as B v C --> A v C?

VideCorSpoon

Tue 17 Jun, 2008 09:14 am
@Arjen,
BvC / AvC could be translated as; B or C, thus A or C, which can be interpreted as "If" B or C, "then" A or C, which is the conditional (BvC) --> (AvC).

de budding

Tue 17 Jun, 2008 03:25 pm
@VideCorSpoon,
May I request a table with the connective values (v, &, --> etc.) in it with their word equivalents (either, or, and etc.) and the rules that apply to make the connective true/false. This would be a very useful tool while I practice.

Also could you reiterate the rules...

1.If one of the rows in your truth table contains all True's in your premises columns but a False in your conclusion column, then the argument is invalid

2.If there are now rows which contain true premises and a false conclusion, the argument is valid.

Aren't these both the same, if the premises of a row are true and the conclusion false the argument is invalid and valid?

Also do you mean ALL values in a row (main variables and connectives) or just one or the other?

Thanks for another great instalment by the way Vide,
Dan.

Arjen

Wed 18 Jun, 2008 09:50 am
@de budding,
Hold on a second de budding, things aren't as simple as you may think at this point. Words have multiple meanings sometimes. It all depends what one means with it.

examples:
I saw A but not B.
I see A but not when I see B.

The word 'but' means 'or' in the first sentence and 'if' in the second. Perhaps the example isn't that great, but I had a hard time of thinking of one. The point is that it depends on the situation. Think of people with lack of knowledge of a language for instance. They can make sentences which mean the opposite of what it would have ment if someone with good knowledge of the laguage had spoken it. I hope you see this point.

The truth tables have a different value then you think I am afraid. All conditions may be true, but the conclusion may be false. The logical reasonings have nothing to do with reality, remember. I could say that A, B and C are prensent and because of that D is also. That would look like a solid reasoning, but D may be present due to other circumstances than the presence of A, B and C. This would leave the reasoning incorrect, even thought the conditions are all met. For the same reason the conclusion can be invalid with all arguments valid and therefore the reasoning would be correct. I think trouble like this can only exist when the arguments are not correctly transformed into logical sentences and therefore do not correlate with 'reality'.

This clearly shows my earlier point that a reasoning may be correct, but that it has nothing to do with what actually takes place.

de budding

Wed 18 Jun, 2008 10:11 am
@Arjen,
I get the first part about word use, but as long as I keep this in mind such a table, as requested, would not be misused. So thanks! I'll keep my wares about me.

And yes I see your point about reasoning being correct within the context of logic but completely absurd in actuality but I am still interested to get a grasp of the popular system within the context of logic, just so I can use it to annoy people if nothing else

Dan.

Arjen

Wed 18 Jun, 2008 10:21 am
@de budding,
de_budding wrote:
I get the first part about word use, but as long as I keep this in mind such a table, as requested, would not be misused. So thanks! I'll keep my wares about me.

Such a table cannot be constructed because it would be a table used for the creators personal opinions of the usage of such words.

Quote:

And yes I see your point about reasoning being correct within the context of logic but completely absurd in actuality but I am still interested to get a grasp of the popular system within the context of logic, just so I can use it to annoy people if nothing else

Dan.

I'll not give you a hard time on actuality and reality....

Anyway, the popular use of logic contains the understanding that correct reasoning has no bearing on reality; only on an actuality.

VideCorSpoon

Wed 18 Jun, 2008 12:24 pm
@Arjen,
de_budding,

Thanks for the thanks! Sorry for the delay, I needed to make sure I put in all the things I could for the tables. You shed light on a very good idea, which is to make a kind of single page reference sheet so all the rules and such will be displayed on a single page. I'll get to work on that.

This is the connectives and associated words...

The primary words are law... they cannot be disputed. The additional words are the ones I am aware of that are acceptable. This is not to say that other words cannot denote connectives. Arjen brings up some good points about words like "but" can in some instances mean "or. They can also in some cases denote a conditional. But it is not a precise word and it is not used much in formal logic syntax because it is informal. I'm not quite sure if I put all the additional words in though, so if you or anyone else have any more additions, I'll mend the chart.

Also, this is a table with the connective truth value rules...

You are right when you said that 1 and 2 from the validity rules are the same. I just find it easier to remember validity and invalidity in those two distinct ways.

Also, you had asked if "all values in a row (main variables and connectives) or just one or the other?" in regards to the truth table rows. The only values that matter are the main connectives of each column. The other truth values are just needed to derive the value of the main connective, so you can just cross them out when you get the main connective truth value. But don't erase them in case you make a mistake somewhere along the line. The explanation chart below elaborates on the complex truth table procedure at step 7 (and also proves that I do not practice what I preach because I erased the irrelevant truth values, LOL!)

de budding

Wed 18 Jun, 2008 02:43 pm
@Arjen,
Arjen wrote:
Such a table cannot be constructed because it would be a table used for the creators personal opinions of the usage of such words.

I'll not give you a hard time on actuality and reality....

Anyway, the popular use of logic contains the understanding that correct reasoning has no bearing on reality; only on an actuality.

God I really need to figuer out what the difference is between the two (reality/actuality) >.<

So feel free to give me a hard time

de budding

Wed 18 Jun, 2008 02:48 pm
@VideCorSpoon,
I like you art work, it's been coming along nicely and thank you very much for the time and effort you have put into these posts. Most helpful and now I get it
Dan.

Arjen

Wed 18 Jun, 2008 03:35 pm
@de budding,
de_budding wrote:
God I really need to figuer out what the difference is between the two (reality/actuality) >.<

So feel free to give me a hard time

Perhaps a new topic would be appropriate?

de budding

Wed 18 Jun, 2008 03:50 pm
@Arjen,
Ok, what sub forum would be best suited?
And how are the questions...

What is actuality?
What is reality?
How do actuality and reality differ?

Dan.

Arjen

Wed 18 Jun, 2008 08:16 pm
@de budding,

Holiday20310401

Wed 22 Oct, 2008 09:24 pm
@Arjen,
Arjen wrote:
Such a table cannot be constructed because it would be a table used for the creators personal opinions of the usage of such words.

I'll not give you a hard time on actuality and reality....

Anyway, the popular use of logic contains the understanding that correct reasoning has no bearing on reality; only on an actuality.

I'm sorry but I have to disagree with you here, but that is probably because we do not have the same definitions of actuality and reality in our heads, because I struggle to entertain myself with your version when mine just makes so much more sense. Perhaps you could start a thread to clarify, Arjen.

Also Vide, I don't quite understand step 8 at the very beginning. When we dealt with connectives it was all with two variables, not 1. So vT = T and vF = F, because single variables will always imply themselves?

And are partial tables mandatory?

VideCorSpoon

Thu 23 Oct, 2008 07:35 am
@Holiday20310401,
Step 8 is the culmination of the entire truth table. Steps 1 - 6 were obviously just a set up for the entire problem. In other words, we had to get the truth values that we worked out in steps 4 into the main problem area so that we can work out the connectives. Once you reach 7, step 7 and 8 have to be done simultaneously. Let me explain. Looking at the conjunction and disjunction, we can see we have to determine which is in fact the main connective

ValidandNotinUse

Tue 14 Apr, 2009 07:34 am
@VideCorSpoon,
I have two questions:

1) Do the premises and the conclusion go in the same order along the top of your truth table as it is initially?

2) How does one find the value for a connective that is outside of a parantheses? For example, ~F v (G & H)

Thus, how would one complete the following truth table?

~F v (G & H)
P --> F
--------------
~H --> ~P

thanks!

VideCorSpoon

Tue 14 Apr, 2009 08:54 am
@ValidandNotinUse,
For your first question, the premises and the conclusion do go in the order along the top of the truth table. But that does not mean that a conclusion is necessary for you to perform a truth table analysis. Say you did A | ~(A v~A), with the A being the only variable in your analysis. The truth value of A, though True and False are the only values provided in the truth probability matrix, can be manipulated in any way depending on your formula. You could for all intents and purposes have a truth table A | A, which would be a short truth table, but there you go. This even applies to complex truth tables as well if you need to use them.

For your second question, to find the value for a connective that is outside of the parenthesis, you have to first establish what is the primary connective. Here is your example from #2 in truth table form.

However, there is a way to do long proof truth tables without all of the work, which is by doing a complex truth table. I have not done a tutorial on it, but if you want me to show you I would be happy to show you. But if you are unsure about anything I said or anything else, don't hesitate to ask me to clarify because I usually don't do a good job in that respect. I'm happy to help!

ValidandNotinUse

Mon 20 Apr, 2009 12:08 am
@VideCorSpoon,
Hi Spoon!

Okay, so I have some more questions about the order of operations regarding Truth Functional Logic.

I'm pretty sure I did it incorrectly, specifically the ~(P & Q) part, but I don't how/why it is wrong.

Problem:
~(P & Q)
R --> P
--------
~R

Example #2 Problem
(P-->Q) v (R --> Q)
P & (~P --> ~R)
----------------
Q

***yes, I realize I forgot to do line 8!!!***
example 1 questions:
As you can see, I'm not sure how to get the '&' values from the P & (~P --> ~R).

I know you've addressed this before, but how do you address ~(P & Q)?

Do you ignore the tilde first and attack the (P & Q)?

Does the tilde make the P negative and the Q negative? Or just the P variable?

Can you explain in more detail how to determine the main connective?

Thank you so much for your help!!!!!!

VideCorSpoon

Mon 20 Apr, 2009 07:56 am
@ValidandNotinUse,
No worries, you get the hang of it the more times you go through it. For the first problem (i.e. ~(P&Q) :: R -->P |- ~R) It's not exactly correct, but you are definitely on the right track. You have definitely got the format of the truth table down, the only thing now is the way in which you attack the table. Here is the way I did the table.

So, the first thing you do is set up your truth table which you did perfectly and set up your truth possibility matrix which is correct. The best thing to do in any truth table, and even though it takes a little bit more time to do, is to transfer what is under the truth possibility matrix to the variables you have in the truth table. So what we have put under P (namely TFTFTFTF) would go under every P in the truth table. The same goes for the Q's and the R's. Once you have those columns in place, then comes the part of identifying the main connective. Take a look at what I did if we were just looking at ~(P& Q) alone

first. After parentheses, you attack problems that are then encapsulated in brackets and so on. So say we had; [B v (P&Q)]. This is the standard compound statement that you will encounter throughout propositional logic. So, if you had this problem, you know that Bv(P&Q) are meant for each other because of the brackets ([]), and chances are the problem demands you handles these variables together. Right off the bat, you can identify the main connective. If you were attacking the problem in a truth table, the P&Q would of course come first, with the ampersand (&) being the connective of that isolated problem. Once you have the truth value, you can then break open the parentheses as it were to put together the value of the ampersand and B v __.

So now look at our ~(P&Q) truth table. We know now that a) we attack what is in the parentheses first and b) we need to break down that parentheses to get a final truth value to apply to things outside it. In box 3, the green squares represent the values we need to determine are correct or not. The middle column is the solution of our connective (i.e. &). Now remember the rules of a conjunction. A conjunction is true if both conjuncts are True. With this rule in mind, we can then fill in the values of the conjunction. Now that we have the truth values of what are inside the parentheses, the issue is now what to do next. We want to do what is outside the parenthesis. You essentially want to work from the inside out. You now take the value of the conjunction and essentially flip it. The main connective is therefore the negation because in order to get to the negation we had to work out the contents of the parentheses first. The last connective done is the primary connective.

Now onto the rest of the bigger truth table.

With R-->P, we don't have the same issue that we had with the negation in ~(P&Q). Like we had done in the previous problem, you had shifted the values to R and P into the truth table. Now you have to find the truth value of the conditional. A conditional is false when the antecedent is false and the conclusion is true.