Propositional Logic Symposia - [6] – Complex and Partial Truth Tables

[VideCorSpoon]

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 1 – Not a dang thing…
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.

NOTE: This particular example I give has 4 extra rows as a 2 variables only require 4 rows of truth probabilities… this was not intentional, it was a mistake… my bad. But suffice to say that even if you add extra rows, you just repeat the process with redundant information. It does not hurt the entire truth table if you mess of like this.



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

This is almost identical to the type of truth tables we have been constructing so far, but the difficulty is a little more complex and the composition is slightly different… but you will see that it is certainly no complex at all.

To start off with, we will examine a basic argument and break it down into a complex truth table to discover if the argument is valid or not.

The Sentence: 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---------------------------------------

In most if not all of your truth table needs, you will have many different components. The partial truth table method can help you do a truth table without doing a full page of matrices.

Say we have an argument that looks like this in translated form…

1.A -->B
2.B-->C
3.C-->D
4.D-->E / E -->A

This is how to go about doing a partial truth table…



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]

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

Answer Key links

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

[Arjen]

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]

Exactly! If you had BvC / AvC all by itself, you could in certain circumstances interpret it as the conditional BvC --> AvC. You can do that because that slash symbol is basically the same as saying therefore or thus because it denotes a conclusion much like a conditionals “then.”

So… 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]

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]

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]

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]

Originally Posted by de_budding

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.

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]

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]

Originally Posted by Arjen

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