SOME SIMPLE LOGIC

>> Rabu, 02 Desember 2009

Idris PBI
Logic is a word that means many things to different people. Many everyday uses of the words logic and logical could be replaced by expressions such us reasonable behavior. For instance, “Sue acted logically in locking the door”, meaning that Sue had good, well-thought- out reasons for doing what she did. We shall use the words logic and logical in a narrower sense, familiar to semanticists. We give a partial definition of our sense of logic below.


Logic deals with meaning in a language system, not with actual behavior of any sort. Logic deals most centrally with proposition. The term “logic” and “logical” do not apply directly to utterances (which are instance of behavior).
There is an important connection between logic (even in our narrow sense) and rational action, but it is wrong to equate the two. Logic is just one contributing factor in rational behavior. Rational behavior involves :
1) Goals, assumption and knowledge about existing states of affairs
2) Calculations, base on these assumptions and knowledge, leading to ways of achieving of the goals.
Example :
Goal : to alleviate my hunger
Assumption and knowledge : a). Hunger is alleviated by eating food.
b). Cheese is food.
c). There is a piece of cheese in front of me.
d). I am able to eat this piece of cheese.
Word such as and, or, and note are not predicates and cannot be use as referring expressions. Logic calls such words connectives. The kind of meaning that is involved is structural, i.e. it deals with the whole structures, rather than with individual items within proposition, such as names and predicates. It is possible to talk of the extensions (or, more loosely, the denotation) of names and predicates take in isolation but it is not possible to imagine extensions or denotations for words such as and, or, if, and not taken isolation. It follow from the special structural nature of the meaning of connectives that they are topic-Free and hence more basic, or general. A topic free meaning is one that can be involved in discourse or conversation on any topic whatever, without restriction.
Generic sentence have a different a logical form from non generic sentences. The two sentence type express logically different types of proposition. They would therefore be represented by different types of formula in logical notation and the logical rules of inference working on these formula would arrive at different conclusion in the two cases, as is appropriate.
A system of logic, like a system of arithmetic, consist of two things , (1) a notation (in effect, a definition of all possible proper formulae in the system) and (2) a set of rules (for calculation with the formulae in various ways).
We will give some informal examples of the kind of rules of calculation that is necessary (or to exclude) in a logical system which captures the essence of rational human thought.
The case of valid argument here are example of basic rules of logical inference. The case of invalid argument example of some well-known logical fallacies. Obviously, a logical system should not permit any fallacious argument. The first example make use of a logical rule generally known by the Latin name modus ponens.
Modus ponens is a rule of stating that if a proposition P entails a proposition Q, and P is true, than Q is true. Put in the form of a diagram, modus ponens look like this :
P Q
P
Q

The formulae above the line in this diagram represent the proppsition which are the premise of the argument, and the letter below the line represent the proposition which is the conclusion of the argument. Note that is logical rule only mentions whole proposition. It is does not go into details concerning the various parts of proposition, e.g. it is does not mention name of predicates. It is a very simple rule.




Notation for Simple Proposition
Logic provides a notation for ambiguously representing the essential of proposition. Logic has in fact beeb extremely selective in the part of language it has dealt with have been treated in great depth.
The notation we adopt here is closer to English, and therefore easier for beginners to handle, than the rather more technical notation found in some logic books and generally in the advanced literature of logic.
We assume that simple proposition, like simple sentences, have just one predicator, which we write in CAPITAL LETTERS. The arguments of the predicator we presented by single lowercase letters, putting one of these letters before the predicator ( like the subject of an English subject) and the others (if there are others) after the predicator, usually in the preferred English word order. Anything that is not a predicator or of referring expression is simply omitted.
Example : Abraham died would be represented by the formula a DIE.
Fido is a dog by f DOG
Idris loves Aisyah by i LOVE a
Irfun introduced Khmaerah to Idris by i INTRODUCE k i.
These formulae are very bare stripped down to nothing but names and predicators. The reasons for eliminating elements such as forms of the verb be, articles (a, the, tense markers (past, present), and certain proposition are partly a matter of convenience. These most serious principle involved is the traditional concentration of logic of truth.
Logic, then, tell us nothing about goals, or assumptions, or action in themselves. It simply provides rules (or calculation which may be used to get a rational being from goals and assumption to action. There is a close analogy between logic and arithmetic (which why we have used the word calculations).
‘arithmetical fact’ does not mean just fact involving numbers in some way, but rather fact arising from the system of rules defining addition, subtraction, multiplication, and division. A similarity between arithmetic and logic is the unthinkability of alternatives.
Example : ‘2 + 2 = 5’ is unthinkable. We can say the word easily enough, there is no way that we can put together the concepts behind ‘2’, ‘+’, ‘=’, and ‘5’ so that they fit what ‘2 + 2 =5’ seem to mean. This is an a arithmetical contradiction.
The concept of contradiction and analyticity are fundamental in logic, so that logic and the study of sense relations to a large extent share the same outlook and goals. But there is a small different of emphasis. The above example are all centered around a small set of words, namely and, or, not, and some. It is the concepts behind these words the logician have singled out of special attention. These words are thought of as belonging to a small set constituting the logical vocabulary.
Although one may, of course, take a legitimate interest in the meaning of individual predicates such as red, round, or ruthless (as linguists and dictionary writers do), and understanding of the meaning of such as basic words as and, if, or, and not is more central to the enterprise of semantics, the study of meaning. An early book on logic was called The Lows of Thought, and thi is the view we take of the subject. Logic analyzes the basis of so-called logical thought. (Proposition can be grasped by the mind, i.e. they can be the object of thought). Thought are notoriously difficult things to talk about, since we can physically experience them, correspondingly, it is difficult to talk clearly and systematically about proposition, as the logician tries to do.
In the units the follow, we will introduce a logical notation, a specially developed way of representing proposition unambiguously. The notation will include a few special symbols, for example, &, V, ~ , and will learn some rules for putting logical formulae together correctly, using these symbols.
Example : Idris Abdullahi and Inayah Khumaerah are married is ambiguous, being paraphrasable either as :
Idris Abdullahi and Inayah Khumaerah is married to someone
In logical notation, the first interpretation (proposition) here could be represented by the formula:
( I MARRIED TO I ) & ( I MARRIED TO I)

In addition to providing a means for representing the various meaning of ambiguous sentences, logical notation brings another advantage, that is formulae can be used much more systematically than ordinary language sentences for making the calculations that we mentioned at the beginning of this unit. We illustrate below some of the difficulties that arise when trying to state the rules for logical calculations in terms of ordinary language sentences.
The problem is that pairs of sentences with similar or identical grammatical forms may sometimes have different logical forms. In order to state rules of calculations, or ‘rules inference’ completely systematically, these rules have to work on representation of the logical form of sentences, rather than on the grammatical form of the sentences themselves. Here are some examples :
1. A plant normally gives off oxygen
A geranium is plant therefore
A geranium normally gives off oxygen
The truth of the third sentence follows necessarily from the truth of the first two.
2. A plant suddenly fell off the window-sill
A geranium is a plant therefore
A geranium suddenly fell of the window sill
The truth of the third sentence above does not follow the truth of the first two.
The two trios of sentence above are of similar grammatical form.
The crucial difference between the two cases above lies in their first sentences. A plant gives off oxygen is a generic sentence and A plant fell off the window sill is neither a generic nor an equities sentence.

Generic sentence have a different logical form from non-generic sentences. The two sentence types express logically different types of proposition. They would there be represented by different types of formulae in formulae in logical notation and the logical rules of inference working on these formulae would arrive at different conclusions in the two cases, as is appropriate.
An analogy may again be made between logic and arithmetic. The Arabic notation used in arithmetic is simple, useful, and familiar. Logical notation is equally simple, equally useful in its own sphere, and can become equally familiar with relatively little effort or difficulty. As with arithmetic, learning to use the system sharpens up the mind. In particular, learning to translate ordinary language sentences into appropriate logical formulae is a very good exercise to develop precise thinking about the meanings of sentences. (Of course, logic does not involve any specific numerical ability).
A system logic, like a system of arithmetic, consist of two things: a notation (in effect, a definition of all possible proper formulae in various ways). To conclude this unit, we will give some informal examples of the kind f rules of calculation that it is necessary to include (or to exclude) in a logical system which captures the essence of rational thought.
The cases valid of argument here are examples of logical inference. The cases of invalid argument are examples of some well-known logical fallacies. Obviously, a logical system should not permit any fallacious arguments.
Rule : Modus Ponens is a rule stating that if a proposition P entails a proposition Q, and P is true, then Q is true. Put in the form a the diagram, Modus Ponens looks like this:
P Q
P
---------------
Q


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