Propositional Logic Symposia - [2] – Truth Funct. Logic, Truth Tables, Conjunctions

[VideCorSpoon]

Think of logic as a different type of language, but not in the sense of Italian or Spanish, but more like music or math. You have to understand the special subject matter in order to “speak” it. So this thread is meant to introduce you to the special logical language that composes propositional logic.

Truth Functionality
But more importantly, the type of logical language we have to speak has to be truth functional, that is, it has to be distinguished in its truth or falsity. The basic thing you need to understand with truth and falsity in truth functional logic is;

a)If some sentence expresses a truth, it’s taken to be true.
b)If some sentence expresses a falsity, it’s taken to be false.

But this is a key concept to understand. If a sentence is taken to be true or false, it will have a truth value of either true or false. This comes in later though.

Logical Sentences
This is the tip of the iceberg for truth functional logic, so It’s probably easiest to understand it in terms of a common sentence and extrapolate from that.

Take this example sentence that we will break down;

Dan is at home and Ann is at home

Simple, right? But that one sentence has three constituent sentences and structures within it, of which understanding this will come in handy when translating sentences to logical proofs. It may seem obvious, but this is just to make it obvious.

The whole sentence is a compound sentence, which basically means that it is composed of two simple sentences joined by a connective sentence (or word in this instance).

[Dan is home] simple sentence
[and] connective
[Ann is home] simple sentence

So…

[Dan is home] [and] [Ann is home]
Why is this important? It is important because of that connective that’s wedged in between the simple sentences.

The main topic has to deal with the connective sentence between the two simple sentences. This is where it starts getting tricky. From here on, reliance on common knowledge will be tough to incorporate.
Before I go any further, I want to show you what a basic truth table looks like, because that’s how we are going to classify truth values for connectives.

Solving a Conjunction truth table
This is what a basic truth table for a conjunction looks like.


This looks rather confusing, but if you take it in five simple steps, you can draw the truth tables as well.

This is the five step process…




Step 1 – Draw a basic truth table, which is basically a cross. In the top right quadrant, insert the compound (conjunction) sentence A & B. Now notice the top left quadrant, A B. The top left quadrant contains all the variables in the compound sentence A & B. There can be more or less variables, but for now let’s stick to two.

Step 2 – Now that you have put down the compound sentence in the top left and all the variables in the top, you can write down the possible truth combinations for the variables. Now this is how it goes in a nutshell. If there is one variable in a sentence, there are two horizontal rows one on top of the other of possible truth combinations. If there is two variables, like the example has, there are four possible truth combinations. If there is three variables, there are eight, etc, etc.

The easiest way to do it is to write in the first column under A the truth variables as followed; T,F,T,F,T,F, etc. Also, it doesn’t matter how many variables you have in the top left, you keep going T, F, T,F, etc. continually. The second variable B has a different order. This order is T, T, F, F, T,T, F, F, etc. Basically, there are groups of two for each truth value.

Keep in mind, this bottom left matrix is meant to help you reference truth value combinations. It does not factor into the truth table calculation. That comes next.

Step 3 – Now that you have the truth values written down in your bottom left matrix, you simply transfer what is under A in the bottom left matrix to the bottom right quadrant under A. Make sure they are in line, this is important.

Step 4 – Transfer the B values from bottom left to bottom right, making sure to keep them in line.

Step 5 – This is where the trouble starts. Up until now you have been figuring out the truth value possibilities. Now here is the solution part. I’ll go into more detail with truth tables later on, but suffice to say for now, the solution for a truth table like this rests under the connective (remember from earlier in the post?) It is obvious which connective it is now, but later there may be dozens of connectives, but for now there is only one.

So how do you solve a truth table? This is where the fundamental rules come in. For a conjunction, the rule is, and keep this in mind, a conjunction is true if the left conjunct is true and the right conjunct is true. This is the rule. No Bull S.’ery can change that fundamental rule. THIS IS IMPORTANT!!!! REMEMBER IT!!!! I’ll write down a whole diagram of all the connectives later, but this is a conjunction.


So now that you have that fundamental rule in mind, you can now solve the basic truth table. If a conjunction is true only if the left conjunct is true and the right conjunct is true, the only possible combination that could be true is the first row, as it equates to the rule. The other combinations are false.

RECAP!!!
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CONJUNCTION
Formal Logic : & (aka ampersand)

This could be translated from the sentence “Alan is hairy and Barry is not,” but translations come later.

Rule : A conjunction is true only if the left conjunct is true and the right conjunct is true.
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First, sorry for the quality of the examples, I did it in paint, so its no masterpiece. Tomorrow, I’ll do the disjunction, conditional, bi-conditional, and negation. But first acquaint yourself with the truth table format and the qualities of a conjunction. I’m trying to figure out how I can do answer keys for these tables as they are jpeg, so if you have any suggestions, let me know.

IF YOU HAVE ANY QUESTIONS, PLEASE ASK AWAY!!!

[VideCorSpoon]

Here is a more elaborated version of the truth table steps. also, observe the artistic modern grunge arrows... yeah, thats right... I meant for them to look like that, LOL.

[chad3006]

So, if I understand it correctly, a False always trumps a True, correct?

[VideCorSpoon]

Actually, you relatively correct, but it is really difficult and abstract to rationalize it that way, but good observation.

Think of it like this, “a true always trumps a false” and here’s why.

We want to find the instances in which a conjunction can be true because we are concerned only with finding valid logical answers to questions or problems.

We want to weed out the instances in which a conjunction can be false. We want to do this because it is easier to say “this one thing is true” instead of saying “this is false” “and this is also false” and “this is false as well.” Its just more convenient that way, plus it will get confusing later on.

In step four, after we had transferred the possible combinations to the bottom right area, we have to remember the rule of a conjunction

A conjunction is true if the left conjunct is true and the right conjunct is true

The matrix that we put down reveals to us that there is only one possible combination in four which gives us a true value based on that rule, which I put a check mark next to. The other three are instance in which a conjunction is not true.



If I didn’t properly answer your question, just tell me because I think your question was very relevant.

[VideCorSpoon]

Here is an example of a four variable truth probability matrix.



The thing to simply remember is that the initial "trues" grow exponentially.

[de_budding]

I don't get why in the diagram at step 2 that 'A' goes TFTF but 'B' goes TTFF... whats the difference?
Dan.

[VideCorSpoon]

That’s actually a good question, because I never explained why the potential truth possibilities are grouped and written in that way.

When I first learned how to do truth tables, the professor just told us “this is the way to do it.” But it makes sense when you think of it like this.

Say you flip a coin twice, and on the first try you can either get a head or a tail, and on the second try you can get either a head or a tail again. Think of head and tail as true and false. Now think of all the possible ways you could flip a coin twice. You can get a head and a head, a tail and a tail, a head and a tail, a tail and a head. {HH,TT,HT,TH}, or as we know it {TT,FF,TF,FT}.

Also, the order in which they occur doesn’t even matter.

Basically, they are all the possible combinations of true and false presented to you in a logical, orderly manner so your thought processes order them in that particular way. But it just so happens that if you put under the first variable T,F,T,F and under the second variable T,T,F,F … you get the four possibilities two variables can have. So the difference is because they need to be ordered like that to reveal the possible combinations.

[de_budding]

I get it know thank you!

Dan.

[Holiday20310401]

So these are a lot like punnett squares then, and true is the recessive while false is the dominant

[VideCorSpoon]

Yeah, I never really thought about it like that! The truth table is basically an account of what the different combinations of truth values would be for a given argument. I guess a comparison could be made between the dual truth values and the punnett square alleles. But the dominant and recessive traits of truth values comes only from the inference rules.