Propositional Logic: Some Intuitive Ideas

On seeing a film with Steve McQueen after he had died, "He must have made that movie before he died."...Yogi Berra It ain't over 'til its over.....Yogi Berra

One of the questions we might ask about human reasoning is whether or not it is logical. One answer to this question is that: "Human reasoning is either logical or it is not logical." On the face of it, this is not a very interesting answer to what seems like a profound question. But notice the form of this "answer"...Either it is or it isn't." Now this form is interesting precisely because we recognize that any answer of the form 'p or not p' is not a very interesting answer. Why do we know this? Is this something we must be taught or does the human mind have built into it certain assumptions such as this one that provide the basis for what we often refer to as logic?        Consider next the value assigned to some statement p. If the value assigned to p is something like 'true' or 'false,' then most of us would, I hope, agree that if things are either true or false; and, if something is not true, then it is false. That is, we would agree that if we are speaking of truth or falsity of a statement, then negation of a statement should reverse these values. The little table to the left reflects this assumption. This table is referred to as the truth table for Negation. What it tells us is that if we have some proposition, p, then if it is true (T), the entry on the left, then the negation of p (the funny little hyphen-like sign, ¬) shown on the right is false (F). And, row 2 says that if p is F then the negation of p is T.

Two things can be noted at this point. First, in order to think of something as being true or false, that something can't be the thing itself! If I point to an apple, the apple itself and the fact that-it-is-red, that-it-is-on-the-counter, etc. are not things that are true or false. A statement about that apple; for example; that 'the apple is red' may be true or false. It is because our mind can represent beliefs about things that we can talk about a relation between those beliefs and the things themselves. And, one such relation is whether the belief is true or false. Second, note that we have made a distinction here between some statement about something, p and an operation on the statement, namely, negation. Negation is a unary operator, it takes a single proposition, and it maps the truth value of p to the opposite truth value. That is what the truth table above says and you can think of it as analogous to the minus sign which is a unary operator on a number. That is, if I have some number n, then -n is also a number. For example, -(3) is -3 and -(-3) is 3.      Now a question. This operation of negation, is it something that happens in the world; or is it something that only our mind can carry out? If it is not something that is in the world, then our mind may already have the idea "built-in" so to speak.      Recall that we distinguished above between a representation about something and the thing itself. This distinction is, roughly speaking, the distinction between what is termed syntax and semantics. Recall that so far we have spoken of some proposition which we refer to as p, a unary operation, '¬' on propositions which we refer to as negation, and two values, true and false. Our language so far consists of names for statements where the names are things like p, q, p1, p2, ... and the ¬ operator. The syntax of this language is quite simple. We can either say a name, or a name preceded by any number of ¬ symbols and nothing else. That is, we can not say ¬p¬, or p¬, or pqppr¬, and so on. And, we assume that every p has associated with it one and only one value from the set {T,F} or {True,False}. This truth value is what tells us about the relation between the proposition and "the world". This is the semantics....a mapping of each p, q, ...to one and only one truth value. Thus, all we have are two sets, a set of names of propositions and the set of truth values, {T,F}. And there is a mapping from each proposition name to a single truth value. What could be simpler?

Well things are going to become a bit trickier. There are other operations that we feel are appropriate operations to carry out on propositions. These latter operations are all binary operators, they operate on two propositions rather than one. For example, let p be "the apple is red" and q be "the apple is on the counter" then it seems reasonable to be able to say that "the apple is red" and "the apple is on the counter" or in our shorthand for these propositions, p and q. Now just as in the case of negation, the intuition is that the truth value of this conjunctive statement should be a function of the truth values of each component. Now, what should be the function between the truth values of 'p' and of 'q' and the truth value of 'p and q'?      The truth table on the left provides the standard answer. For example the top row says that if p is T and q is T then the conjunctive statement is T. For all other combinations, the table says that the truth value of the conjunctive statement is F.

Again, as in the case of 'not', it does not seem that this sense of 'and' is something that is a property of the world. Rather, it appears to be a property of the mind...one of the operations that the mind feels comfortable doing (on at least some but maybe all) propositions!      Our language provides many words that can be used to put two statements together. For example, or, if...then, ...because...., ....therefore..., and so on. In the development of "formal logic" one of the decisions that must be made is to decide how many logical operators are required and the truth table that define the meaning of the operator.      To the left is shown the truth table of the logical operator known as disjunction or inclusive 'or'.

This sometimes causes confusion because we intuitively recognize another sense of 'or' which is referred to as exclusive 'or'. The truth table for this operator is shown to the right.        There are a few more logical operators that are typically defined. The most important is the conditional or " If ...then" operator which is shown below.

This is probably the one that causes the most confusion to persons when they study propositional logic. Part of the difficulty may be that we don't really have an unambiguous natural language term for this operator. For example: "If p, then q"; "p only if q"; "p is a necessary condition for q"; "q is a sufficient condition for p"; "q if p"; "q follows from p"; "q provided p"; "q is a logical consequence of p"; and "q whenever p" have all been suggested as appropriate readings for this logical operator.

(As you read the chapter in the text you will find that psychologists have done a good deal of research on people's ability to interpret conditional statements. It isn't always clear exactly why this has generated so much interest.)      One final truth table and we will have looked at all of the standardly defined logical operators. This final operator is referred to as the biconditional and is often read as "If and only if".      It turns out that for purposes of developing propositional logic, you don't really need all of these logical operators because you can define some in terms of others. This is perhaps one of the points at which our intuitions and the development of propositional logic part company. For example, an equivalence for the conditional is shown below. This means that we can avoid using this connective if we wish.

Now you may have to think a long time before this equivalence seems intuitively obvious. But if you refer to the truth table above for the conditional and work out the truth values for 'not p or q', then you will see that you end up with the same truth table.

Another, logical operation that can be "defined away" is the biconditional. Conjunction and the conditional can be used to define an identity for the biconditional. A more extensive list of some of the more important logical identities is provided for your reference.

Now, if you are blurring over a bit at this point, not to worry...that is entirely appropriate if you haven't spent much time pondering truth tables. So let us step away from this detail and recall that the basic ideas are: there are a set of basic propositions; each proposition has a truth value associated with it; complex propositions can be formed using the logical operators the truth value of the complex proposition should be a function of the truth values of their constituent basic propositions. Each of these basic ideas seems quite reasonable. And, each seems like something that could well be true about the way our mind works.      Well, it is a pretty sure bet that our mind doesn't work exactly this way. But is that because it "tries" but can only approximate this way of reasoning, or, is there a totally different "logic" that is used by the mind, or have our intuitions totally misled us and each of the ideas above are totally foreign to the working of the mind?

	Deduction
© Charles F. Schmidt