What is logic? - By Ellese Elliott

What is logic?

Logic is hidden behind every conversation, joke, story or argument.  Logic is necessary in order to be able to have an understandable form of communication.  Without it, conversation, jokes, stories or arguments could not take place.

Aristotle was a famous old bloke from Ancient Greece who thought a lot about a lot of stuff, but most importantly, for this subject, he thought about thought; what it was and how it takes place. He said that thought has laws. There are rules which enable the process of thinking to happen. The most crucial law of thought is the law of identity. The law of identity states that:

                                                     A = A. 

The equals sign is another way of saying the same as.  Anything can be put in the place of ‘A’.  For example, 'A = A' can stand for ‘that orange’ is ‘that orange’,’ you’ are ‘you’ and ‘logic’ is ‘logic’. This means that a thing is identical with itself.

 
The other thing that Aristotle said was that if it follows that if everything is the same with itself and that this is a law, then it cannot be a contradiction. This thought is expressed in the law of non-contradiction.  It cannot be that:

                                      

    ORANGE = NOT ORANGE

 

We cannot think this orange is both an orange and not an orange. Let’s put this in a scenario:  let’s say you and your mate Kinglsy arranged to go out on the pull. You both agreed to meet outside Tottenham Court road tube station at 10:00pm.

You are standing there, it’s pouring down with rain and Kingsly isn’t there. You try calling him, but it’s forwarding to 02 voicemail and taking your last bit of credit and your thinking, ‘I hope this twat has got a good reason why I’ve been standing here for half an hour like a mug in the rain.’  He turns up causally at 11.00pm. You’re very angry, so you say, “Where the hell have you been?”

Kingsly says, “Mate, by 10.00 pm I did mean 10.00 pm, but I didn’t mean 10.00 pm and I meant 11.00pm. Do you know what I mean? “ 

This is an example of a contradiction and is not a reason at all, let alone a good one.  When we are met with a contradiction we are just utterly confused. We have to ask, “Kingsly what the hell are you on about?” Kingsly then goes on to say some other nonsense because he has taken acid. Kingsly is no longer using the laws of identity to express himself. This raises an important point as it doesn’t matter how logical you are being if the other person is being illogical, a conversation cannot take place.

Now, it seems pretty obvious that things are what they are and are not what they are not. Therefore, when we talk about whether an object is either an orange or not orange, whether it is either 10.00pm or not 10.00pm, we are referring to the last law of thought called the law of excluded middle.  An example is as follows:

I am a human being

Or

I am not a human being.

One of these statements has to be true. However, they cannot both be true or they cannot both be false.  When Aristotle thought about the laws of thought thousands of years ago, he realised that human beings cannot think outside of these laws. As a last exercise, try it yourself. Can you think of an orange and not an orange, which are both properties of the same thing? Can you understand what Kingsly means by 10.00pm and not 10.00pm? Or do you think it is true that you are a human and not a human?

If you could think of any of these things you may have a hard time explaining it.....

Ellese Elliott

The Philosophy Takeaway 'Logic' Issue 45

One axiom in logic for Aristotle’s three laws of thought

Aristotle has propounded three basic laws of thought:

(i)             Law of Identity,

(ii)           Law of Non-Contradiction,

(iii)          Law of Excluded middle.

To my mind these are in fact one law, when expressed in terms of formal logic.  But firstly let me return to my contribution to the previous issue (‘Axioms or Circularity’), where I argued that we needed axioms to avoid arguments which went ‘round and round in circles’.  I stated that we need at least one new axiom if we add a new concept.  Let us informally try this out:

- Negation.  Two negatives make an affirmative.

- One proposition.  p= p, p implies itself, p and ~ p (not p) cannot be true at the same time.

- Two propositions.  If p implies q, then if q is untrue, then p must also be untrue.

- Three propositions.  If p implies q, and q implies r, then p implies r. (transitivity)

My main concern here is with item1 – one proposition – but I would to look a little at items 2 and 3 first. If we look at the question of two propositions, an important concept is that a&b is equivalent to (and indeed implies) b&a.  This is known as commutativity, and of course goes on the tree of logic.  But this is a bit of a red herring: we know that meeting the love of your life and getting married is not the same thing as getting married and meeting the love of your life.  But logicians can show that this is saying the same thing as “If p implies q, then if q is untrue, then p must also be untrue.”  And this is the important thing for reasoning.

Likewise transitivity is important for reasoning:  “If p implies q, and q implies r, then p implies r.”  But associativity is also important for logicians, and also for mathematicians (along with commutativity).  But associativity is likewise a red herring for the layman – and perhaps the non-logician philosopher – since there’s always degrees of association!  Which of course is important for ethics in matters that are not black and white.

 

So both associativity, along with double negation, go on the Tree of Logic.  … :] Well let us now get back to the question of item 1, one proposition, where a proposition is equivalent to itself, implies itself, and cannot be true at the same time as its opposite. Let us look more closely at Aristotle’s three basic laws of thought:

(i)             Law of Identity,

(ii)           Law of Non-Contradiction,

(iii)          Law of Excluded Middle.

The Law of Identity simply states that a=a, where to say p=q means that to say anything about p will always be equally true of q.  Aristotle argues in Metaphysics, Book VII, Part 17, that “the fact that a thing is itself is the single reason and the single cause to be given in answer to all such questions as why the man is man, or the musician musical…”

The Law of Non-Contradiction states, according to Aristotle, that “one cannot say of something that it is and that it is not in the same respect and at the same time”, and relates to what I stated above, namely that p and ~ p (not p) cannot be true at the same time.  In formal logic this is represented by

~ (p & ~p)

But Aristotle is talking about what things are, so that we consider object x, and property A, we can never have a proposition of the form

~ (Ax & ~Ax)

i.e. x is both A and not A.

Now the Law of the Excluded Middle deals with exactly the same issue as above, but makes clear that if a reference is ambiguous, a proposition may appear to be both true and false at the same time, but there can be no contradiction in addressing the facts themselves.

But in all of this, the axiom ~ (p & ~p) is ever present, either in this form, or in the form  (p É p), p implies p.  For logicians (p É q) is by definition saying the same thing as ~ (p & ~p).  This is not so clear in natural language, since if I consider whether p implies q, then if p is in fact false, our immediate reaction is that we cannot tell.

And “p implies p” is the same thing as “p is equivalent to p” - since equivalence here means that p implies p and  vice versa!

As I said earlier, if we look at formalisms, it is useful to see how they apply to robust argumentation, and here the idea that p implies p can be expanded to the following three principles:

(i)             p is equal to and equivalent to itself,

(ii)           if we say that x is A, x cannot also be not A at the same time,

(iii)          but just to be precise, if we say that x is A, it cannot also be not A at the same time.

Note the very subtle difference between (ii) and (iii).

(iota being the affirmative operator ! )

Martin Prior

The Philosophy Takeaway 'Open Topic' Issue 34