Propositional Logic Symposia - [5] - Truth Functional Lingo, Syntax, and Calculations

[VideCorSpoon]

Thinking back on it, I could have incorporated this in the last symposium, but some things need to be cleared up before getting into complex truth tables and proofs.

Logical Language
These are some of the basic things to keep in mind…

Sentence Variables.
A sentence variables is basically any letter A to Z. So the variables in the sentence (A&B) --> C are A,B, and C. It is important to remember that any of these variables and any grouping of these variables is a sentence in truth functional logic. So A, B, A&B, AvB, A-->B, A<-->B are all sentences as well.

Sentence Operators
Sentence operators are the connectives; &, v, -->, <-->. (i.e., conjunction, disjunction, conditional, and bi-conditional).

Parenthetical Devices
The grouping devices used to isolate compound sentences from other sentences, such as; ( ), [ ], { }.

So now that you understand the essential elements of logical language, the formation of the sentence s you wish to translate must be well formed in order to be conveyed coherently.

Well Formed Formulas
This is essential in translating and proofs. You have to make sure you symbolize and simplify the translations you do before you get into the proof because if the formula you want to evaluate is not formed correctly, you will not be able to accurately translate is, find its truth value, etc.

There just a few things to remember.

TOO MANY BRACKETS!!!
Say you have this sentence; { [ (A&B) ] }
It seems obvious that those two outer parentheses (i.e. { } and [ ] ) can be dropped, because they are merely extra weight and mean nothing in the overall sentence. But you can also drop the ( ) as well because the only sentence is A & B. However, if the sentence was { [ ~(A & B) ] }, then you can drop the two outer brackets, but must leave the inner parentheses because of the tilde.

WTF!!! Deformed Well Formed Formulas!
The grammar within logic has to be very precise, and when translating sentences to logic may show one how good or bad their English composition is.

It is important to keep in mind that;

1.Variables cannot be side by side. Example: BN & H
2.Negations cannot be placed between two variables. Example: H ~N
3.Two connectives cannot be next to each other. Example: E &v H
4.Parentheses must be used. Example: A & N v C
5.Variables cannot float outside a parentheses. Example: N (D & B)

RECAP!!!!

Letters are variables, connectives are operators, Parentheses are parentheses! Also, to make a coherent formula and subsequent proof, you have to have a well formed formula, which is grammatically correct!

These things are the fundamental elements in the logical meta-language. It is important to cover these things now because you will need to know them in the complex truth tables. But before going into complex truth tables, let’s look at basic truth table calculations which I think is pretty fun.

CALCULATING TRUTH VALUES
Best way to show this is to dive right into a sample problem from the very beginning. Suppose you had the sentence “Alan is playing and Bob is not.”




Step 0. Fist and foremost before you begin, you have to identify the main operator that will determine the truth value of your sentence. This is obviously the conjunction symbol. It is not the negation because it is not a connective.
Step 1. Translate the problem correctly. Remember that if you are off in your translation, you will not be able to get a legitimate truth value.
Step 2. For now, we are supposing the truth values of our sentence. There is another way to do this, but it is much later in logic. Suffice to say that you need to provide the truth values yourself. But if you were looking for all the possible ways to see if the sentence could be true, do a basic truth possibilities matrix on the side and then input those possibilities into your calculation depending on how many variables you have. But for now, we understand that A is true and B is false.
Step 3. Replace the variables with the truth values you set in place.
Step 4. Now that you have replaced the truth values, you have to deal with the components of B to get the final truth value for the left conjunct. The B has a negation attached to it. Simply, what is the negation of a truth… or simply, what is the opposite of true? It is false. So you have to consider the left conjunct a false.
Step 5. This is where you memory comes in. YOU MUST REMEMBER THE RULES OF THE CONNECTIVES!!! So, the rule of the main operator ( & ) is “a conjunct is true if both conjuncts are true.” Since the negation in step 4 turned the value of B to T, both conjunct are true, thus the sentence is true.

NOW FOR SOMETHING HARDER!

Now suppose this already translated sentence; ~ (~A --> ~B) v ~C.



Step 1. The sentence is already translated.
Step 2. Suppose the truth values or do a truth probability matrix calculate the possible truth values.
Step 3. Replace the variables with the truth values.
Step 4. In order to make things easier, do the truth values in an ordered way to keep track of your progress and serve as a back-up point in case you mess up on something.
Step 5. Next ordered step in the process
Step 6. Now that you have two truth values within a parentheses, you need to refer to the conditional rule, which states a conditional is false only when the antecedent is true and the consequent false. Since both variables of the conditional are not false, the conditional is true.
Step 7. Next order step in the process
Step 8. Refer to the disjunction rule to solve, which states that a disjunction is false only when both disjuncts are false.

That’s basically what you have to know before we get into complex truth table analyses.

AGAIN, IF YOU HAVE ANY QUESTIONS, LET ME KNOW AND I’LL BE HAPPY TO ANSWER THEM.

ALSO, IF ANYONE WANT SAMPLE PROBLEMS, I CAN DRAW SOME UP FOR YOU OR CORRECT YOUR OWN.

[Arjen]

It is my understanding that no brackets are needed whatsoever, but I am not 100% sure anymore. I think there is always an order in which to work out the connectives. Do you by any chance have an example of where brackets are needed?

[VideCorSpoon]

That’s a really good point that I didn’t clarify.

Say you have the compound sentence; A & B.

You don’t need any parentheses because if you include them in the simple statement A & B, you are implying that there are other operators and variables to consider. So in the example { [ ( A & B ) ] }, all of the parentheses and brackets are redundant because there is nothing to go in-between those divisions.

Now say you have the compound statement; (A & B) v C

You need those parentheses there to individuate the compound sentence “A & B” from “v C” to satisfy logical grammar. Now if you had all those brackets; { [ (A & B) v C ] }, you don’t need either bracket because they are redundant.

Now say you have the compounded statement; [ (A & B) v C ] --> D

The brackets are necessary because that left antecedent has to be completely solved before the conditional can be solved. The brackets in this instance are necessary. But if we added another bracket; { [ (A & B) v C) --> D }, we wouldn’t need it because it is redundant.

Now say you had the compounded statement { [ (A & B) v C ] --> D } <--> E

Then you would need all the parentheses and brackets.

Connectives

To work out connectives, you have to first identify the main connective of a problem. In the the case of { [ (A & B) v C ] --> D } <--> E for example, it is v (disjunction) because of the brackets which tell us that it has to hashed out before moving to the right bicondtional "E."

In A & B, the main connective is &
In (A & B) v C, the main connective is v
In [ (A & B) v C ] --> D, the main connective is -->
In { [ (A & B) v C ] --> D } <--> E , the main connective is <-->

[VideCorSpoon]

Here are some sample problems… (Highlight the blank areas for the answers)

Main Connectives

1.A & B
2.(A v B) v (L & W)
3.B --> (L & N)
4.[ (B & ~A) --> (M -->L) ] --> Q

Answers
1.A & B
2.(A v B) v(L & W)
3.B --> (L & N)
4.[ (B & ~A) --> (M -->L) ] -->Q

Truth Value Calculations
(suppose A,B,C,D are true and E,F,G,H are false

1.A --> (B & C)
2.(A & B) & (C & D)
3.{ [ (A & B) --> (C v D) ] v (E & F) } v (G & H)

1. A --> (B & C)
a.T --> (T & T)
b.T --> T
c.T

2.(A & B) & (C & D)
a.(T &T) & (T & T)
b.T & T
c.T

3.{ [ (A & B) --> (C v D) ] v (E & F) } v (G & H)
a.{ [ (T & T) --> (T v T) ] v (F & F) } v (F & F)
b.{ [ T --> (T v T) ] v (F & F) } v (F & F)
c.[ ( T--> T ) v (F & F) ] v (F & F)
d.[ T v (F & F) ] v (F & F)
e.( T v F ) v (F & F)
f.T v (F & F)
g.T v F
h.T

[Arjen]

You are right. The brackets are just not always needed. The sequence of important connectives depends on the syntax. Funny how such basic things can slip the mind when not confronted with it.

[Holiday20310401]

Is there a sort of BEDMASS (if you are familiar with that trick) for placing brackets. I think the only hard part for me is understanding the prioritizing of connectives relative to other connectives, and I see a pattern from the sentences, but its fuzzy.

I am also starting to get confused with the idea of propositional logic and what it is intended for. It seems very limited. I mean, just because something is not false, doesn't make it true, right? But that's what you say in the symposium lesson here.

[VideCorSpoon]

Yup! Generally it goes (), [], {}. It all depends on how many compound sentences you have. This was form another post in the symposia. Though the topic is more about the need for the brackets, look at the different separators as more complex formulas are made.

Say you have the compound sentence; A & B.

You don’t need any parentheses because if you include them in the simple statement A & B, you are implying that there are other operators and variables to consider. So in the example { [ ( A & B ) ] }, all of the parentheses and brackets are redundant because there is nothing to go in-between those divisions.

Now say you have the compound statement; (A & B) v C

You need those parentheses there to individuate the compound sentence “A & B” from “v C” to satisfy logical grammar. Now if you had all those brackets; { [ (A & B) v C ] }, you don’t need either bracket because they are redundant.

Now say you have the compounded statement; [ (A & B) v C ] --> D

The brackets are necessary because that left antecedent has to be completely solved before the conditional can be solved. The brackets in this instance are necessary. But if we added another bracket; { [ (A & B) v C) --> D }, we wouldn’t need it because it is redundant.

Now say you had the compounded statement { [ (A & B) v C ] --> D } <--> E

Then you would need all the parentheses and brackets.

[Holiday20310401]

{ [ (A & B) v C ] --> D }

So if I imply truth variables to the variables, and simplify it to just T or F, what am I proving when I get to T and F. What do they mean? Is T just saying that the truth value is true, and F saying that the truth value is false?

Will I always be able to simplify it to just T or F, or by virtue of being able to do so, I am proving that the function is valid. And if the function cannot be simplified it is invalid?

Lemme just try this one and see if I'm on right track here.
I will say that A=T B=T C=F D=F
Therefore, (is there a bb script for therefore where you get those 3 dots by any chance) { [ (T & T) v F ] --> F }
Since T & T = T I get... { [ T v F ] --> F }
Since T v F = T I get... { T --> F }
Therefore I get F, since T--> F = F. Did I do ok? Basically, I have to look at the charts you gave us at the beginning to discern with what connectives do I arrive at what truth value, or intuitively grasp it.

[VideCorSpoon]

The whole problem when worked out will show what specific truth configuration will work. There may be none, some, or all. This is useful in its own particular way. You are right in assuming that T means that the truth value is true and F is the truth value is false. We are using a truth-functional system under a deductive system.

Will you always be able to simplify to T or F? In propositional logic, yes. But you do not really encounter it in many of the other logical sub-systems. Modal logic uses truth values out the wazzoo. The modal logical subsystem is basically the study of the modes of truth and their relationship to argument and reasoning. This is where you knowledge of truth tables in propositional logic comes into play. To tell the truth, this is very useful to know, but it requires propositional and predicate logic to make any sense. But anyway, back to your question. If the function cannot be simplified in a variable sense, it is indeed invalid. In a truth value sense, you will always get a truth value… it all depends on how correctly you did the problem.

In your example, I would point out that variables cannot have just one truth value. But besides this, looking at what you did in the truth value proof, yes! …you did it correctly. Now the trick comes when you get that kind of reasoning down, applying it to arguments in the real world. You can, when acquainted, see through the most confounding degrees of BS in anyone’s argument and see whether or not it is logical or not. It is a very useful skill to know.