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Logistics Of Logic

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1. Logic is the study of argument.

2. Logic uses notation (symbols to stand for aspects of our language) in order to break down elements within arguments.

3. An ‘argument’ in logic is a formal term, it is constituted by a sequence of propositions, and must have specific features. The most basic of these are two essential parts: a premise and a conclusion. The premise serves as a reason to accept the conclusion. The most basic argument is a simple tautology:

P, therefore P.

Or

Paul quit university after 2 years, therefore Paul quit university after 2 years.

This is also called ‘begging the question’.

4. A valid argument is one where the premises lead to the conclusion. Such arguments are divided into two kinds, deductive and inductive. If your argument is deductively valid, the premises necessarily lead to the conclusion. If it is inductive, it is merely probable that the conclusion follows.

5. A sound argument is one where, not only is the argument valid, but the premises are true, which means the conclusion is true as a result. Validity is a bit like hypothesis, it determines what would be the case were certain conditions to be manifested. Soundness is the actual manifestation of those conditions.

Logical languages

I learned particular logical languages, specifically ‘propositional logic’ and ‘predicate logic’, up to a certain (not very advanced) level. ‘Propositional logic’ is usually tackled first, as it involves simple statements and the relations between them. A predicate is a property belonging to a thing, so in ‘predicate logic’ a more complex notation is needed. Since we create a logical language to abbreviate the complex senses of our normal language, the choice of what stands for what is somewhat arbitrary. Clearly there are advantages in using symbols in ways which bear some connection to our everyday language.

Some logical notation:

P, Q, R, S... - names, such as ‘Pencil’, ‘Quentin’, ‘Rochester’, ‘Santa Claus’.

r, s, t, u... – predicates, such as ‘is red’, ‘sits down’, ‘takes a bath’, ‘uses a computer’.

x, y, z... – variables, which are similar to algebra, they can stand for an unknown, or as yet undetermined  name or predicate. All ‘x’s are red, for example, (whatever an ‘x’ might be.)

(x), (y), (z)... universal quantifiers. They allow you to say ‘all’ or ‘not all’ such and such. You might want to say ‘all pencils are red’. In which case you would have a universal quantifier at the beginning, and a symbol to stand for ‘pencils’ (e.g. ‘P’) and then an ‘r’ for ‘are red’ and a variable to associate them, ‘x’. It would look something like this:

(x) [for all x], if Px [if x is a pencil] then rx [then x is red].

Similarly, there is the ‘existential quantifier’ which looks like this: (Ex). It means that there exists at least one of something, or ‘there are some’ (e.g. ‘red pencils exist’.)

This may have you quite confused. Don't worry, the idea is that you will see what logic is. Why and how it works is secondary. That’s something I learned about maths. I’m no great mathematician, but I’ve been making a lot more progress in it since I concentrated on using the system and not enquiring too far into why it works. Logic isn’t a far cry from 0s and 1s in computing. It uses self-contained systems which attempt to represent ordinary language. In this way it can be more or less representative, and succeed or fail. Meanwhile it remains secure in its own self-contained system. You could feed the data into a computer and it would tell you whether an argument was valid...but that doesn’t necessarily mean the argument was representative of real life in the first place. Again, hypothesis is a good analogy – hypothesis is as relevant in logic as real life. Like a twin universe, logic applies to what might be the case equally as it applies to what is known to be the case. That’s why people talk about logical impossibility being the ultimate kind of impossibility...a physical impossibility may be overcome with advances in physics...but that which is logically impossible has never been, and never will be possible, because it is contradictory, like ‘all pencils are red and this one is not.’

Truth

Truth and falsity are factors closely associated with the boundlessness of logic’s application (across the real and the purely theoretical). Falsity is just as important as truth (again, like 0s and 1s). Don’t be fooled into thinking that truth makes an argument work. It could just as easily be falsity that is the missing element in your chain of reasoning. Consider a situation in which you are hoping to prove the falsity of a claim – in this case your argument may depend on demonstrating the falsity of someone else’s premise. Even more strangely, your argument may simply depend on the falsity of all its propositions, depending upon how they are organised in the sequence. This is difficult to get your head around but validity is to do with structure and shape. It is the shape of the argument that makes it valid. Whether or not the argument proves to be sound later is a matter for investigation into the real world. Logic is a tool. A method for understanding the essence of sense. A skeleton of language.

 

 
 

 

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