2207-103 Philosophy & Logic, Week Three

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Dedutive Arguments of Sentential Connectives

In the previous chapter we have seen one type of deductive argument, classical syllogism. Actually that is just one small part of deductive argument, so this chapter deals with another part, which is the logic of sentential connectives. Actually you have studied some basic ideas of this part before when you were in high school doing mathematics. Remember the p and the q and how they are joined together in various ways according to truth tables? Well, that is about what we are doing in this chapter. What we want to emphasize, though, is not how to do logic mathematically, but to study how valid deductive goes, and how we can separate valid from invalid arguments of this type.

Classical syllogisms deal with the relations among logical terms. And we have seen that terms work to group things together. Thus syllogism actually is about how groups of things are related to one another. One group may belong to another group, or some part of one group may be included in another group but not all of it, or at least one member of one group is also a member of another group, and so on. However, sentential logic has none of these. Instead it deals with the relations between propositions. And here propositions are related through the work of the wonderful little things called 'logical connectives.' There are a few of these, and different logic textbooks give slightly different versions of them. However, here we are going to talk about only four, namely 'and', 'or', 'if...then', and 'not'. What these connectives do is that they take on one or a pair of propositions and then manipulate their truth values. For example, 'not' will change the truth value of any proposition it is attached to. But what are truth values? Well, it's best not to give a formal definition of them. Let us agree that there are only two of them -- the True and the False. We will have a special symbol for each of them -- T for the True, and F for the False.

In sentential logic, we employ logical symbols as a shortcut to our discussion of propositions and the connectives. Thus we use letters p, q, r, s and t to stand for propositions (or more precisely for formulae of propositions, but at this stage we need not be that precise), and the symbols, '&', 'v', '-->', and '~' for 'and', 'or', 'if...then', and 'not' respectively.

Thus, very intuitively, these symbols represent propositions:

     p v q
     (p v q) --> r
     [(p v q) --> r] & s

You are all familiar with truth tables, but here since we are going to discuss this issue in some detail, it's best to remind ourselves of the tables once again:

Here is the truth table for 'and':

     p  q  | p & q
     T  T  |   T
     T  F  |   F
     F  T  |   F
     F  F  |   F

Here is the truth table for 'or':

     p  q  | p v q
     T  T  |   T
     T  F  |   T
     F  T  |   T
     F  F  |   F

Here is the truth table for 'if...then':

     p  q  | p --> q
     T  T  |   T
     T  F  |   F
     F  T  |   T
     F  F  |   T
Here is the truth table for 'not' (easiest):
     p  | ~p
     T  |  F
     F  |  T

Before we get on to use these symbols to help us decide the validity of the deductive arguments of sentential connectives, some concepts and terminologies need to be spelled out. The first thing you need to know is that of tautology. A tautology is a proposition which is true no matter what. It is true simply because of its propositional form. Here is the classic example of a tautology: p v ~p. Next is satisfiability. A proposition or a group of propositions are said to be satisfiable when all of them can be true together. For example, the proposition, p, is satisfiable because its form does not preclude its being true. The propositions, p and q are satisfiable together (why?), so are the propositions in this group -- p, q, and p v q. These three propositions can all be true together. (You can see why, can't you?) If a group of propositions is not satisfiable, then they are inconsistent. That is, there is no way to interpret them to be all true. The proposition, p, and ~p, together form an inconsisten group because they obviously cannot be true together.

Now we are in the position to examine arguments of sentential logic and find a means to test their validity. Let us pay attention to this argument closely:

If it is raining, then Samorn must be wet.
But Samorn is not wet.
Therefore it is not raining.

If this argument is valid, then if we take all the premises to be true then there must be no way at all for the conclusion to be false. So if it is true that, if it is raining, then Samorn is wet, and true that Samorn is not wet, a conclusion can be conclusively drawn is that it is not raining. How is this argument valid? We can show that this argument is really valid by utilizing the symbols we have just seen. So let 'p' be 'It is raining' and 'q' be 'Samorn is wet'. Then tbe whole argument can be racast in symbolic form thus:

p --> q
Therefore, ~p

Now when the argument is cast in symbolic form, it is straightforward to see whether it is valid or not. Remember that an argument is valid if and only if if all of its premises are true, then the conclusion is true. That is to say, if we take all the premises and hook them up together with the symbol '&', and join that large conjunction with the conclusion with the symbol '-->', then we will have a large proposition. This large proposition must be a tautology if the argument is valid, and is not a tautology if the argument is not valid. However, even though this method is straightforward, it is very cumbersome because you can see that if you have a number of premises then the whole resulting proposition becomes very long and it is tedious to test whether it is a tautology using the truth table method. Thus we will find a way to make testing easier and less involved. We know that, if an argument is valid, it is not possible for the conclusion to be false if all the premises are true. Thus, if we can find a way for the conclusion to be false although the premises are all true, then we can show that the argument is invalid. On the other hand, if we take the negation of the conclusion (the proposition of the conclusion [enclosed in a pair of parenthesis if needs be] attached in front by the symbol '~') and consider it with the premises and find that the group is not satisfiable, then we have to conclude that the argument is valid. (Why?)

Back to the argument above. Its two premises are 'p --> q' and '~q'. Let the negation of the conclusion be true. That is, let 'p' be true. Now we have three propositions -- 'p --> q', '~q', and 'p'. But if it is true that if p, then q, then if 'p' is true, then 'q' must be true. But if 'q' is true it will contradict with '~q'. So we have a contradiction and indeed the argument is valid.

Here is another example:

     If Indonesia persists in defying the IMF and go ahead with setting up
       a Currency Board, then the IMF will withdraw its loan.
     Indonesia does persist in defying the IMF and ...,
     Therefore, the IMF will withdraw its loan.

This argument can be case in symbolic form as follows:

     p --> q
     Therefore, q

This argument is also valid. If we suppose that the conclusion is false, that is, if we suppose '~q' to be true, then we have a group consisting of . However, according to the truth table, in the propositional form 'p --> q', if 'p' is true, then 'q' is true also; otherwise the whole propositional form is false. Thus it is not possible for 'p --> q', 'p' and '~q' to be all true together, and hence the argument is valid.

Here is another example:

     Indonesia will follow the IMF's directions or risk massive 
       political instability.
     Indonesia is being in a risk of massive political instability right now.
     Therefore, Indonesia will follow the IMF's directions.

Recast in symbolic form, this argument becomes:

     p v q
     Therefore, p

This is an invalid argument, as you can see. From the premises that p or q and that q, it cannot be validly inferred that p. If 'p v q' is true, 'p' can be true or false. Thus there is no guarantee that p must be true if 'p v q' and 'q' are true.

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Some Validity Rules of Sentential Connective Logic

Here are some rules of valid arguments, according to most logic textbooks. Note that these rules act rather as a shortcut when you try to prove arguments valid. They are not substitutes for the thinking that you have to do in order to find out whether an argument you are considering is valid or not. In other words, these rules are not given, but they themselves must be shown to be true too. Here we will focus on only a few such rules:

Modus Ponens
The rule of modus ponens says that, if 'p' and 'p --> q' are true, then q must be true. We have already shown in the text above hos this is a form of a valid argument.

Modus Tollens
Modus tollens works in the opposite direction from Modus Ponens. It says that any argument of the form 'p --> q, ~q, therefore ~p' is valid. You can see why.

Disjunctive Syllogism
Disjunctive Syllogism says that any argument of the form 'p v q, ~p, therefore q' is valid, so is 'p v q, ~q, therefore ~p.'

Simplification is what it says. The conclusion "simplifies" the premise(s). More precisely it is the rule saying that if 'p & q' is true, so is 'p' and 'q'. Example: Chavalit is a soldier and a politician; therefore, he is a politician. Usually this kind of argument is not often found in real life because why would one want to make it? However, it is useful in constructing proofs.

This is another simple rule. If you have any proposition, say q, as a premise, then you can always conclude 'q v p' where 'p' is any proposition you like. It does not matter whether 'p' is true or false here. (Why?)

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