Primer on Logic

Logic is the study of inferences. More precisely, logic is interested in deriving conclusions from permises.

All men are mortal. Socrate is a man ...
therefore Socrate is mortal.

In this derivation, we are interested in the conclusion "Socrate is mortal" drawn from the premises "All men are mortal" and "Socrate is a man". We are not interested in whether Socrate is really a man or if all men are really mortal. In the derivation we are just concerned on the validity of the argument. Since we don't really care about whether the premises are true or not, we can easily replace the sentences by variables. If I call SentenceA , SentenceB and SentenceC three different sentences, we can form this argument as :

SentenceA=> SentenceB (read sentenceA implies sentenceB)
SentenceA therefore SentenceB (read, if sentenceA is true therefore sentenceB must be true)

This says that if SentenceA is true whatever it is, we know that SentenceB is true as well (SentenceA=> SentenceB). Then anytime we have sentenceA we can derive SentenceB from it. This is one of the oldest and most basic derivation in logic. It is called "modus ponens". By accepting this rule we can by replacing any sentence in Sentence A,B make valid derivations. For example :

SentenceA is "It rains"
SentenceB is "I am sad"

From this we can conclude that : It rains therefore I am sad.What about the following :

SentenceA is "All horses have four legs"
SentenceB is "I am Donald Trump"

Can I say now using modus ponens that "all horses have five legs therefore I am Donald Trump"? This sound like an invalid argument and surely nobody would actually believe it. But if we are working in the system of natural deduction, this is a completely valid argument. Valid, because it is following the rule of modus ponens and we agree that  anything following the rule must be valid ! Remember, we are not talking about the truth of the statements themselves, we are talking about the validity of the argument !

The power of all this is that if we define a set of rules ( the rules of logic !! ) and we are handed a set of permises ( considered to be true ) we can derive new truth statements from them. Because we can prove that if an argument is valid and the premises are true, then the conclusion will also be true !

Validity: An argument is valid if the form of the argument, how it is constructed, is following the rules of logic.

Soundness: An argument is sound if the argument is Valid and the premises are true!

Let's look at some interesting examples :

1. Example of invalid argumentation : "The sky is blue => There is an atmosphere"
   "The sky is blue therefore I don't need an umbrella" is not a valid argument, because it is not following our (only) rule of logic. Nevertheless the conclusion is looking good !
2. Example of valid but not sound argumentation : "All horses have five legs => I am Donal trump"
   "All horses have five legs therefore I am Donal trump"Is a valid argument, but it is not sound because not all horses have five legs.
3. Example of valid and sound argumentation:"The sky is blue => There is an atmosphere"
   "The sky is blue therefore There is an atmosphere"Is valid and sound.

There are plenty of logical system one can adopt, but people usually use the one pretty close to natural deduction which include our modus ponens rule (don't worry if you can't read the symbols in the wiki page, intuition is enough here). An important aspect of using logic is to be logically consistent. Consistency means that you are not breaking your logical system. How can you break it ? Well you could derive a contradiction. Having a contradiction in your system means your system is not consistent. So if you managed to derive a contradiction, either :

• Your permises can't be true at the same time.
• You are not being consistent ( not applying the rule of logic correctly ).
• Your logic system is not sound.

What is a contradiction ?

Formally, a contradiction is the conjunction of a statement and it's negation. So the following is a contradiction :"The sky is blue and the sky is not blue"usually we denote the negation of a statement with the word "not". So :Not "The sky is blue" would be transformed to everyday langage to : "The sky is not blue".We use this denotation to be able to talk about a specific statement "p" and it's negation as "Not p". So If p is "The sky is blue" then Not p would be "The sky is not blue"

You are not being consistent !

Technically, if we are not consistent we are being inconsistent. Being consistent means to not hold a position and the opposite position at the same time ! This is rather difficult as most of human beings are being inconsistent all the time. If we hold two opposite position at the same time, we are in a state of cognitive dissonance. The brain has amazing tricks to help us resolve those dissonances. Why ? Because it helped our ancestors to survive !

Natural selection

We can easily make the case that our ancestors have been selected by nature depending on their degrees of consistency. If one of my ancestor would one day go pick some fresh fruits and managed to have his family survive, it would be consistent and logical to try to do the same the day after. Otherwise the family would die without having the chance to reproduce and passing the genes along. This would be a consistent behavior :

• (day 1) I will go find some food for my family so they can survive.
• (day 2) I will go find some food for my family so they can survive.

If this behavior stays consistent, the family has more chance of surviving to reproduce.