LoGiC – some definitions =]

Hy boys and girls!! I’ll continue now my introduction to first order sentential logic.

We saw in the previous post on LoGiC what ‘being valid’ means for an argument. The notion of validity is basic for every deductive logic. Now, let me sum up some crucial definitions that will be used henceforth:

An argument is valid if, and only if, it is necessary that if all its premises are true, then the conclusion is true

If an argument is valid, then its premises logically imply the conclusion

If an argument is valid, then its premises entail the conclusion

An argument is invalid if, and only if, it is possible for its premises to be true while the conclusion is false

If an argument is invalid, then its premises do not logically imply the conclusion

If an argument is invalid, then its premises do not entail the conclusion

Also important is the notion of soundness. Being sound is ‘more’ than being valid, and it is also more important for theoretical purposes:

An argument is sound if, and only if, it is valid and all its premises are true.

If an argument is invalid, then it is unsound

If an argument has false premises, then it is unsound.

Well, thats it, clear and simple. In the next post I’ll talk about the also important notions of well formed formula (wwf) and logical implication.

See ya \o/

LoGiC: second steps

Hello fellow readers!

I’m now continuing with the main topic of this blog. Before we go through the Logic Land itself, in a big adventure, I would like to say something more general about this formal science. Logic is an abstract matter. That is to say: it supposedly deals with forms of reasoning, with structure of thinking. Form and content are different things. But, why should we be concerned with forms of reasoning? Well, our chief motivation seems to be preserving truth via valid reasoning. Once we have found something we can judge as true, we don’t want to infer from it something false. We want our arguments to be truth-preserving – and we wish not to be inconsistent.

But Sir! What is a VALID reasoning?

And what is an ARGUMENT?

Let us start with the second question: an argument is a complex composed by i) a set of sentences, which we call premises, and ii) a conclusion. And, to the first question: an argument is valid if, and only if, it is not possible for the conclusion to be false when its premises are all true. We could say the same in a different way: an argument is valid if it is necessary that, if the premises are true, then the conclusion is true. Accordingly, an argument is invalid if it is possible for the conclusion to be false while its premises are true.

Here is a very simple example of valid argument (it looks childish but, believe me, thing gets harder and harder here!):

1. If my video-game is turned off, no one is playing on it

2. Well, my video-game is turned off, and therefore:

3. No one is playing on it

As you can see, it is not possible that 1 and 2 are both true and 3 false. To sustain 1 and 2 altogether and attribute falsity to 3 is being incoherent. However, 3 could be false if only 1 were true, and 2 false: from the fact that if my video-game is turned off no one is playing on it, it does not follow that actually no one is playing on it. It could be the case that it is not turned off. But once we recognize the truth of 1 and 2, we can see that 3 must be the case.

Now an example of invalid argument:

1. If my video-game is turned off, no one is playing on it

2. No one is playing on my video-game, therefore:

3. My video-game is turned off.

This is an invalid argument because 3 can be false while 1 and 2 are true: from the fact that no one is playing on my video-game (and given 1) it does not follow that my video-game is turned off. It is possible, given the premises, that my video-game is turned on, and yet, no one is playing on it. It is an example of the fallacy of affirming the consequent.

Well, the notions of argument and validity are very basic to the study of logic. I gave you their definitions, and this concepts will return many times on the topic.

Now, a very brief (and certainly not exhaustive) overview of the field – the study of logic is divided grossly in the following way: i) classic logic, which comprehends propositional logic and predicate logic (first and second order), ii) intensional logic (or ‘extended’ logic), which comprehends modal logic, tense logic, deontic logic, epistemic logic, erotetic logic and still others, (iii) inductive logic, (iv) deviant logic, which comprehends many-valued logic (in contrast with 2-valued logic, or bivalent logic), intuitionist logic, quantum logic, free logic. This list is not complete. The field of logic is growing from time to time, and the logicians are applying their formal tools everywhere. Take your pick!

LoGiC – first steps

Howdy people!

I should have advised you that logic is the main topic on this blog. Why? Well, because I’ll turn to this topic from time to time, and I expect to give an overview of the field. The blog is supposed to have classical logic as well as modern symbolic logic.

And so, first of all, I would like to talk about some symbols that are going to be used along the whole treatment. As I said on the previous post on the topic, logic is guided by the truth-value of propositions in setting the rules and axioms for the deductive system. At least, that is the traditional and mostly accepted point of view. Classically, logic handles with binary functions – there are only two values of propositions: the truth and the false.

Well, this means that classical logic can only deal with propositions that can be true or false, and nothing more. Accordingly, we use ‘T’ to represent the truth-value truth, and ‘F’ to represent the truth-value false. If a proposition cannot be determined as true or false, it does not belongs to the range of binary logic.

But, someone could ask, what it means for a proposition to be true or false? Ok, curious people, I admire your knowing thirst, but it’s not a question that logicians must answer. Instead, it is a subject matter to philosophers that occupies with the meaning of the predicate ‘… is true’ and the predicate ‘… is false’. Philosophical semantics: that’s the name of the field you must enter to explore those questions concerning truth itself.

So, what are the questions of the logician if not that ones??? (watta heaven!) We could say that the logician answers the following questions: What are the deductive rules of thinking? If that proposition is true, what more is necessarily true? This proposition follows from that other, infallibly keeping the same truth-value?

Yeah buddy, that’s it!! Pretty cool questions.

Hãhãhãaããann.