Lecture 5

Philosophy 404/English 501/EDTE 404 & 504
June 22, 1999

  1. Administrative

    1. Homework #3: Due today, except for Ex. XXIII, which is now due on Thursday.

    2. Reserves: I have annotated most of the list of reserve books. Please start looking at those with an eye to your projects. It would be worth your while to check out a few and scan them for ideas.

    3. Distribute Handouts

    4. Questions?

  2. Circularity and Begging the Question --- See Lecture 4

  3. Argument Analysis #3

  4. Propositional Logic: A Study of the Structure of Arguments

    1. Arguments consist of sequences of claims that are related in certain ways. One mark of a good argument is that it is valid. In Chapter 5, we will look more closely at validity, but to do this, we must first become more familiar with the nature of the constituent claims, or propositions.

    2. Propositions constitute arguments, and so one obvious place to begin a close analysis of arguments is with the proposition. This is what we will do.

      1. Propositions are claims that can be evaluated for truth and falsity. (This is a simplification that rules out imperatives and interrogatives, but they are hard cases anyway.)

      2. These claims can either be simple or complex.

        1. A simple proposition is one that cannot be divided into more than one part that is itself a claim.

        2. A complex proposition is one that can be divided into more than one claim, and so is a combination of at least two claims.

      3. At this point, we will treat simple propositions like atoms that have truth values. We will symbolize them with propositional variables, such as p, q, r, s, ... . This symbolization will help us avoid being distracted by content so that we can hone in on the structural characteristics of propositions and the arguments they produce.

      4. Complex propositions are combinations of simple propositions, but what can we say about the nature of these combinations?

        1. Consider the following complex proposition: "I am from Kansas and you are from Idaho."

        2. This is a complex proposition because it contains two simple propositions in combination: "I am from Kansas" and "You are from Idaho."

        3. What is the nature of the combination? Well, it is called a conjunction and it is signaled by the word "and".

      5. Indeed, many combinations will be signaled by words they contain. Thus, there are more words that we need to keep in mind when analyzing arguments, in addition to guarding words, assuring words, etc. These we will call logical particles or logical connectives, since they connect propositions up together. We will now turn our attention to these words and their logical characteristics.

  5. The Logical Connectives I: Syntax

    1. We will focus on four words that function as logical connectives: "and", "or", "not", "if ... then."

      1. "And"

        1. Type of Combination: conjunction.

        2. Symbol: "&"; E.g., (p & q)

        3. English Equivalents: "and", "but", "moreover", "still", "although". (Be careful, though.)

      2. "Or"

        1. Type of Combination: disjunction.

        2. Symbol: ; E.g., (p v q)

        3. English Equivalents: "or"

      3. "Not"

        1. Type of Combination: negation.

        2. Symbol: ~ ; E.g., ~p

        3. English Equivalents: "not", "no", "un-", "non-"

      4. "If ... then ..."

        1. Type of Complex Proposition Formed: conditional.

        2. Symbol: the horseshoe.

        3. English Equivalent: "if ... then ...", "provided", "when"

        4. The proposition that appears after the "if" is called the antecedent, while the one appearing after the "then" is called the consequent.

      5. These connectives can appear many times in a single sentence. Thus, we can take simple propositions and build highly complex propositions up out of them by applying the connectives in succession. (Example)

  6. The Logical Connectives II: Semantics

    1. What meaning do we assign to these words? They don't seem to refer to anything, like "Eric" or "red", or even "run". Dictionaries might help here, but those definitions are usually circular. How do get an independent read on the meanings of these terms?

    2. The key is to notice that while they may not have a readily discernable independent meaning, they do shape the meanings of the complex propositions into which they figure. This suggests that we should look for their meaning by examining their combinatory function; that is, by examining the type of propositions they help create. But how are we to do this?

    3. At this point, it is good to pry our eyes away from this tree for a moment and gaze a bit at the forest. We are engaged in this analysis so that we can understand arguments, and in particular, the validity of arguments. Recall that validity was defined in terms of the truth values of sentences, and this was so because truth is inextricably tied to the intuitive notion of following. Given the centrality of truth in this investigation, it only makes sense to begin with it at this level in making out the semantic nature of the logical connectives.

    4. The idea, then, is to get at the combinatory function of logical connectives by examining their impact on the truth conditions of sentences. Once again, we will turn to the specific connectives.

      1. "And"

        1. A conjunction is true just in case both of the sentences conjoined are true.

        2. (p & q) is true just in case p is true and q is true.

      2. "Or"

        1. A disjunction is true just in case one or both of the sentences disjoined are true.

        2. (p v q) is true just in case either p is true or q is true or both are true. (Inclusive disjunction.)

      3. "Not"

        1. A negation is true just in case the sentence negated is false.

        2. ~p is true just in case p is false.

      4. "If ... then"

        1. This connective creates a relationship between the two propositions connected, such that the truth of the first is sufficient for the truth of the second.

        2. As with the other connectives, the meaning of this connective is to be identified with the impact it has on the meaning of the complex propositions into which it figures. This connective creates a relationship between the two propositions connected, such that the truth of the first is sufficient for the truth of the second.

        3. In the case of the conditional, we have the following: "If p then q" is true just in case p is false or q is true. Put another way: "If p then q" is true if and only if it's not the case that p is true and q is false.

        4. The meaning of this connective is not as intuitive as the meanings of the others, but we can expand on it a bit. First, it is intuitive that "If p then q" cannot be true in a circumstance where p is true but q is false. Thus, if the conditional is true, then so is the sentence, "It is not the case that both p is true and q is false" (call this "S")---that is, the first proposition implies the second one. Second, let's begin with S: for this to be true, it must be the case that the sentence "p is true and q is false" is false; thus, S is true and, furthermore, p is true, then we know that "q is false" must not be true, which is to say that q must be true. Thus, against the assumption that S is true, we have determined that if p is true, then q must also be true. That is, we have shown that S entails the sentence, "If p then q". Therefore, these two sentences are equivalent, or what is the same, have the same truth condition.

        5. If S and our conditional are equivalent, we can use S to get at the meaning of our conditional. If S is true, then we know that the sentence "p is true and q is false" must be false. But for this to be false, it must either be the case that p is false or q is true. That is, if S is true, then either p is false or q is true, and this is precisely the meaning that we started with above.

    5. These are called truth-functional connectives because the truth values of the complex sentences they create are functions of the truth values of their parts; that is, they take the truth values of the connected sentences as arguments and spit out the truth value of the complex sentence as a value. (Exercise VII)

      1. Review truth conditions.

      2. The truth condition of the constructed sentence can be seen as a function of the connectives that constitute it.

    6. These can be represented more clearly if we use truth tables ...

  7. Truth Tables

    1. The truth tables for the connectives.

    2. The rows of a truth table give you the possible alternatives, in terms of the truth values of the constituent parts, and the columns give you the truth conditions of the sentences that appear at their head.

    3. Constructing a truth table. (See Handout)

    4. Do truth tables for several sentences.

    5. Do a couple of validity exercises involving truth tables. (Exercise VIII.)

  8. Arguments and Validity

    1. We have defined validity as a property arguments have just in case there conclusion is true whenever all their premises are true.

    2. The logical connectives play a very important role in many arguments, and the validity of arguments can often be determined by closely inspecting the roles played by these connectives. Define valid inferences involving disjunction, conjunction, and negation. (Exercise IV.)

    3. This close inspection can be done with the help of truth tables. How to test for validity using truth tables. (This may involve placing symbols where you find sentences.) (Exercise VIII.)

    4. Work some examples from Exercise XV.