Direct inferences

Let's start with the simple ideas.  We know that the definition of a conjunction is that two elements are true at the same time.  This means that when we are given any two wff's as true, we may assert a new wff that uses the connective &.  It also means that when we start with a conjunction we may assert either of its parts separately.  We will give names to therse rules in terms of either introducing or eliminating a connective, and we will talk about what we are working with as an intelim system.

We will show the rules in the same manner in which later we will represent argument forms.  The |- is what we call an assertibility sign (and, no, no more than a comma, it is not a defined symbol in symbolic logic itself).  It cues us that whatever wff follows is said to be true, and ordinarily it is preceded by other wff's separated by commas that are already being assumed or supposed to be true.  More simply, we have premises in front of the sign and a conclusion afterward.

We can also work with disjunction by looking carefully at what we mean by it.  When we say P v Q we mean that at least one of these two is true (remember, this is a more rigid sense of "or" than we might have in ordinary conversation).  With this in mind we can always derive a disjunction that uses a wff we already know as true.

We can just as easily assert that one of the components is true if the other is known to be false.  If we say that someone would have either cream or sugar in her coffee (C v S) , and then we know she did not take sugar (~S), we can be sure she must have taken the cream (C).  This is often referred to as a disjunctive syllogism, but we will use it just as our technique for eliminating the connective v. Please note that this differs from the usage in the Schaum book.

Since conjunction and disjunction are commutative, it does not matter which wff we work with in these rules.  I am following a convention by which I use lower-case letters in place of any actual variables, but we can then see that in my earlier example we could put it either of two ways.

What we cannot do, though, is work backwards with these last four rules.  For instance, knowing that the person did take cream would not prove that she did not take sugar.

We can work back and forth with negation, however.  Since negation is a reciprocal action in a two-value logic, we have the idea that negating a negated wff takes us back to where we started.  ~~p |- p
and  having two connectives is a legitimate inference  p |- ~~p
so we will represent this as an equivalence    p =  ~~p   DN  (for double negation)

Perhaps the most vital rule is also the most venerable: we can set up a conditional, which says that the truth of the antecedent ensures the truth of the consequent.  Given this, we have the rule often known as modus ponens (the "putting" pattern).

We do not have a corresponding rule by which we can introduce the connective ->.  Later we will see another technique of derivation that in fact accomplishes just this.

With biconditionals we can see two things happening because of the meaning of an equivalence.

Finally, there are two rules that work with quantifiers.

If we have a universal quantifier, we are saying that anyone at all can be characterized in the way stated.  Consequently, we may eliminate the quantifier by using any name we choose in place of the variable.  (Often a text will use Greek letters to express the generalized rule.  We will be using ordinary capital letters as examples.)

Also, since we are not "using up" a wff if we posit a direct inference, we could repeat the process for as many names as we might want.  For instance, we might want to show that if everyone is funny, then Alva is funny and also Ben is funny.  Here we would instantiate (replace the quantified expression with a specific case)  with a and then again with b.

Given any named individual with a characteristic, we can generalize that there is at least one person with that characteristic, which is what we are saying when we use an existential quantifier.

Thought questions

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