Propositional Logic Symposia - [3] – Disjunction,Conditional,BiConditional,Negation

[VideCorSpoon]

In the previous post, symposium 2, I had shown the fundamentals of truth tables and introduced the conjunction.

The truth tables are essential to know when doing more advanced truth tables, and also in understanding some propositional proofs. If you do not like truth tables, don’t worry because they are not that essential when dealing with regular proofs, but they are still beneficial to know and make your understanding of logic more encapsulating.

The Four Basic Connectives and the Negation
So at this point I am going to provide the basic connectives and the negation. I do this in a constructed table in .jpeg form. The name of the connective is boxed in the top right. The symbolism as it is used in formal logic is provided (i.e. &,v, -->, <-->, ~). I then provide a sample sentence and a translated version of the sentence. This will come in handy when translating, which comes very soon. The rule is the important part, because it tells you the fundamental assumption you must understand when dealing with each connective. To the right of the main table, I have constructed a truth table that provides the definition of the connective in truth table form.









Understand these connectives and negation as they are essential elements of logical language. I know this seems rather meh, but it is important to reference later on.

IF YOU HAVE ANY QUESTIONS OR NEED FURTHER CLARIFICATION, DON’T HESITATE TO ASK, I’M HAPPY TO ANSWER ANY QUESTIONS!!!

[de_budding]

does 'if' get a symbol or have any impact on sentences?
Dan.

[VideCorSpoon]

Thats another really good point. Good eye!

The word "if" does have a symbol, which is the conditional (-->) symbol. It is an companion word with "then" Without the "if" in front of the fist part of the conditional sentence, the sentence would be hard to decipher becuase many times people would consider the first part of the conditional a simple sentence of its own. So if you think about it, "If" is kinda like the red flag saying "the conditional starts here" and "then" states that "the conditional ends with what follows."

Look at the conditional table above and look at the example sentence. "If" is a precursor to the word "then" to flag a conditional. You know its a conditional when you see "If x,then x."

[Zetetic11235]

I have seen & denoted with '^', why the disparity between symbolic notation? Is the ampersand a more modern or older symbol for and?

[VideCorSpoon]

You are right to wonder why a “formal” system has so many different variations for the same notation.

One would think that there would have been some sort of grand conference on that some time ago to come to some agreement on standard notation. Unfortunately, there really isn’t any consensus on standard notation. Suffice to say, different logicians use different symbols to convey the same meaning (although some over complicate at times).

Take this for example. I have six books that deal with predicate logic alone. In each book, the system and symbolizations they use are for the most part different from the other. Here are three different types of symbolizations.




Basically, you can choose which symbolizations to use and never be wrong about it. Historically, logical proofs used transferred and altered mathematical symbols which really complicated the whole system.

The symbolization process is now a little more streamlined, as the symbols look more like each other. But there is a reason for this. It’s simply because they symbolization could not be transferred to digital media by way of a universal font. It’s a matter of practicality. The older logicians (and by older, I mean age wise) aren’t that computer savvy and they won’t go to the hassle of downloading a special font, etc.

Personally, I use the Herrick system. But unfortunately, I honestly don’t know how to use the type of font required to make Herrick symbolizations quickly enough to make it practical. I’m not one of those older logicians (if I ever was one) but I don’t know how to do it either. Pospesel’s system uses simpler symbolizations that can be done right off the bat, so I use that system on the internet.

I have seen the “^” symbol used for conjunction before… although a lot of other people who use that symbolization use “/\” (two opposing slashmarks). I have also seen “.” (period) used for conjunction as well. ampersand just seems like the right symbolization to use though.

So the disparity between the symbols is just a matter of preference and practicality. Ampersand has always been around from what I understand, but there are other types of logical frameworks that meld with mathematical systems that would prefer using more complex symbols.

[Zetetic11235]

The thing which I find a bit distressing is the use of symbols such as the second biconditional notation which is the mathematical symbol for equivalence, however, the negation is also used in topology to denote equivalence and the inverted subset symbol doesn't help either especially when one wishes to use predicate logic in conjunction with set theory, which is not entirely unreasonable asthe two fields are quite intertwined.

[Zetetic11235]

Originally Posted by de_budding

does 'if' get a symbol or have any impact on sentences?
Dan.

It might be of worth to note that A only if B =!(does not equal) if B then A
that is only true of iff(<--->/the biconditional)
i.e. B is true iff A is true (B=T <--> A=T) is logically the same as A iff B, the variables are reversable. Also, A only if B is logically the same as if A then B. Check the TT's

|A|B|A|-->|B|
|T|T|T| T |T| :If A is true, B is true, thus A can be false& B true
|T|F|T| F |F|
|F|T|F| T |F|
|F|F|F| T |F| :A only if B, A is only but not necessarily true if B is

You can also derive a simple Tautology from the above symbols:
A|B|( A v B )| V| ~ ( A v B )| :Read A or B or Not A or B
F|F |F|F| F| T| T| F|F|F
T|T|T|T| T| T| F| T|T|T
T|F|T|T| F| T| F| T|T|F
F|T|F|T| F| T| F| F|T|T
The main connective (V) is true dispite the variable truth value