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4.3: Propositional logic

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    4.3.1 Propositional operators

    In §4.1 we introduced the logical negation operator “¬”. (An alternate symbol for this is the tilde, “~”; so in logical notation, ‘not p’ can be written as either ¬p or ~p.) Logical negation is referred to as a “one-place” operator, because it combines with a single proposition to form a new proposition. The other basic operators of propositional logic are referred to as “two-place” operators, because they are used to combine two propositions to form a new complex proposition. The basic two-place operators include ∧ ‘and’, ∨ ‘or’, and the material implication operator → (generally read as ‘if…then…’). If p and q are well-formed propositions, then the formulae pq ‘p and q’, pq ‘p or q’, and pq ‘if p, (then) q’ are also well-formed propositions. (The p and q in these formulae are variables which represent propositions.)

    A word of caution is in order here. In reading logical formulae we use English words like not, and, or, and if to pronounce the logical operators, for convenience; but we cannot assume that the meanings of these English words are identical to the meanings of the corresponding operators. This turns out to be an interesting and somewhat controversial question, and we will return to it in chapters 9 and 19. For the purposes of this chapter, as a way to introduce the logical notation itself, we will use the English words as simple translation equivalents for the logical operators; but the reader should bear in mind that there is more to be said about this issue, and we will say some of it in later chapters.

    These four operators determine the “syntax” of the complex propositions that they are used to create. They specify, for example, that ¬p and pq are valid formulae but p¬ and pq∧ are not. These operators also determine certain aspects of the meaning of these complex propositions, specifically their truth values. For example, if we are told that proposition p is true in a given situation, we can be very sure that its negation (¬p) is false in that situation. Conversely, if p is false in a given situation, we know that its negation (¬p) must be true in that situation. We do not need to know what p actually means in order to make these predictions; all we need to know is its truth value.

    The other operators also specify the truth values of the complex propositions that they form based only on the truth values of the individual propositions that they combine with. For this reason, the meanings of these operators (i.e., their contribution to the meaning of a proposition) can be fully specified in terms of truth values. When we have said that p and ¬p must have opposite truth values in any possible situation, we have provided a definition of the negation operator; nothing needs to be known about the specific meaning of p. One common way of representing this kind of definition is through the use of a truth table, like that in (6). This table says that whenever p is true (T), not p must be false (F); and whenever p is false, not p must be true.

    (6)

    In the same way, the operator ∧ ‘and’ can be defined by the truth table in (7). This table says that pq (which is also sometimes written p&q) is true just in case both p and q are true, and false in all other situations.

    (7)

    Again, the truth value of the complex proposition does not depend on the meaning of the simpler propositions it contains, but only on their truth values and the meaning of ∧. Nevertheless, we can assign arbitrary meanings to the variables in order to illustrate the function of the operator. Suppose for example that p represents the proposition ‘It is raining,’ and q represents the proposition ‘The north wind is blowing.’ The formula pq would then represent the proposition ‘It is raining and the north wind is blowing.’ The truth table in (7) predicts that this proposition will only be true if, at the time of speaking, there is a north wind accompanied by rain; it will be false if the weather is different in either of these respects. This prediction seems to match our intuitions as speakers of English. We can see this by imagining someone saying to us, It is raining and the north wind is blowing. We would consider the speaker to have spoken truthfully just in case there was a north wind accompanied by rain, and falsely if the circumstances were otherwise.

    The operator ∨ ‘or’ is defined by the truth table in (8). This table says that p∨q is true whenever either p is true or q is true; it is only false when both p and q are false. Notice that this or of standard logic is the inclusive or, corresponding to the English phrase and/or, because it includes the case where both p and q are true. Suppose, for example, that p represents the proposition ‘It is raining,’ and q represents the proposition ‘It is snowing.’ Imagine a meteorologist looking at a radar display and, based on what he sees there, saying: ‘It is raining or it is snowing.’ This statement would be true if it was raining at the time of speaking, or if it was snowing, or if both things were happening at the same time. (This last possibility is rare but not impossible.)

    (8)

    In spoken English we often use the word or to mean ‘either … or … but not both’. For example, this is normally the usage that we intend when we ask, “Would you like white wine or red?” Table (9) shows how we would define this exclusive “sense” of or, abbreviated here as XOR. The table says that p XOR q will be true whenever either p or q is true, but not both; it is false whenever p and q have the same truth value. (We will return in Chapter 9 to the question of whether we should consider the English word or to have two distinct senses.)

    (9)

    The material implication operator (→) is defined by the truth table in (10). (The formula pq can be read as if p (then) q, p only if q, or q if p.) The truth table says that pq is defined to be false just in case p is true but q is false; it is true in all other situations.

    (10)

    In order to get an intuitive sense of what this definition means, suppose that a mother says to her children, If it rains this afternoon, I will take you to a movie. Under what circumstances would the mother be considered to have spoken falsely? In applying the truth table we let p represent it rains this afternoon and q represent I will take you to a movie. Now suppose that it does not rain. In that case p is false, and whether the family goes to a movie or not, no one would accuse the mother of lying or breaking her promise; and this is what the truth table predicts. If it does rain, then p is true; and if the mother takes her children to a movie, she has spoken the truth. Only if it rains but she does not take her children to a movie would her statement be considered false. Again, this is just what the truth table predicts. (It turns out that the material implication operator of standard logic does not always correspond to our intuitions about English if, and we will have much more to say about this in Chapter 19.)

    For convenience we will introduce one additional operator here, which is referred to as the biconditional operator (↔). The formula pq (read as ‘p if and only if q’) is a short-hand or abbreviation for: (pq) ∧ (qp). The biconditional operator is defined by the truth table in (11):

    (11)

    This table says that pq is true just in case p and q have the same truth value. Suppose the mother in our previous example had said I will take you to a movie if and only if it rains this afternoon. If it did not rain but she took her children to a movie anyway, the truth table says that she would have spoken falsely. This prediction seems linguistically correct, although her children would very likely have forgiven her in this case.

    Having introduced the basic operators of propositional logic, let us see how they can be used to identify certain kinds of tautologies and contradictions, and to account for certain kinds of meaning relations between propositions (entailment, paraphrase, and incompatibility), namely those that are the result of logical structure alone.

    4.3.2 Meaning relations and rules of inference

    In addition to using truth tables to define logical operators, we can also use them to evaluate more complex logical formulae. To begin with a very simple example, the formula p∨(¬p) represents the logical structure of sentences like Either you will graduate or you will not graduate. Sentences of this type are clearly tautologies, and we can show why using a truth table. We start by putting the basic proposition (p) at the top of the left column and the formula that we want to prove (p∨(¬p)) at the top of the last (right-most) right column, as shown in (12a). We can also fill in all the possible truth values for p in the left column.

    (12) a.

    The proposition we are trying to prove (p∨(¬p)) is an or statement; that is, the highest operator is ∨. The two propositions conjoined by ∨ are p and ¬p. We already have a column for the truth values of p, so the next step is to create a column for the corresponding truth values of ¬p, as shown in (12b).

    (12) b.

    The final step in the proof is to calculate the possible truth values of the proposition p∨(¬p), using the truth table in (8) which defines the ∨ operator. The result is shown in (12c).

    (12) c.

    Notice that both cells in the right-most column contain T. This means that the formula is always true, under any circumstances; in other words, it is a tautology. The truth of this tautology does not depend in any way on the meaning of p, but only on the definitions of the logical operators ∨ and ¬. Propositions which are necessarily true just because of their logical structure (regardless of the meanings of words they contain) are sometimes said to be “logically true”.

    Suppose we change the or in the previous example to and. This would produce the formula p∧(¬p), which corresponds to the logical structure of sentences like You will graduate and you will not graduate. It is hard to imagine any context where such a sentence could be true, and using the truth table in (13) we can show why this is impossible. Sentences of this type are contradictions; they are never true, under any possible circumstance, as reflected in the fact that both cells in the right-most column contain F.

    (13)

    Now let us consider a slightly more complex example: ((p∨q) ∧ (¬p)) → q. To construct a truth table which will allow us to evaluate this formula, we begin by putting the basic propositions p and q in the left-hand columns (1&2). We put the complete formula that we want to prove in the far right column (6). We introduce a new column for each constituent part of the complete formula and calculate truth values for each cell, building from left to right, as seen in (14). First, columns 1 & 2 are used to construct column 3, based on the truth table for ∨. Next, column 4 is calculated from column 1. Columns 3 & 4 are used to construct column 5, based on the truth table for ∧. Finally, columns 2 & 5 are used to construct column 6, based on the truth table for →.

    (14)

    Notice that every cell in the right-most column contains T. This means that the formula is always true, under any circumstances; in other words, it is a tautology. Furthermore, the truth of this tautology does not depend in any way on the meanings of p and q, but only on the definitions of the logical operators. This tautology predicts that whenever a proposition of the form ((p∨q) ∧ (¬p)) is true, the proposition q must also be true. For example, it explains why the sentence cited at the beginning of §4.2 (Either Joe is crazy or he is lying, and he is not crazy) must entail Joe is lying. A similar entailment relation will hold for any other pair of sentences that have the same logical structure.

    As mentioned above, it is helpful to check the predictions of the logical formalism against our intuition as speakers by “translating” the formulae into English or some other human language (i.e., replacing the variables p and q with sentences that express propositions). We noted at the beginning of §4.2 that when we hear the sentence Either Joe is crazy or he is lying, and he is not crazy, we seem to reach the conclusion Joe is lying automatically and without effort. It takes a bit more effort to process a formula like ((p∨q) ∧ (¬p)), but the table in (14) shows that the logical implication of this formula matches our intuition about the corresponding sentence.

    Now consider the biconditional formula (p∨q) ↔ ¬((¬p) ∧ (¬q)). Using the procedure outlined above, we can construct the truth table in (15). First, columns 1 & 2 are used to construct column 3, based on the truth table for ∨. Next, columns 4 & 5 are used to construct column 6, based on the truth table for ∧. Column 7 is calculated from column 6, and finally columns 3 & 7 are used to construct column 8, based on the truth table for ↔.

    (15)

    Once again we see that every cell in the right-most column contains T, which means that this formula must always be true, purely because of its logical form. The biconditional operator in this formula expresses mutual entailment, that is, a paraphrase relation. This formula explains why the sentence Either he is crazy or he is lying must always have the same truth value as It is not the case that he is both not crazy and not lying. The first sentence is a paraphrase of the second, simply because of the logical structures involved.

    As we noted in an earlier chapter, tautologies are not very informative because they make no claim about the world. But for that very reason, these logical tautologies can be extremely useful because they define logically valid rules of inference. A few tautologies are so famous as rules of inference that they are given Latin names. One of these is called Modus Ponens ‘method of positing/ affirming’, also called ‘affirming the antecedent’: ((p→q) ∧ p) → q. The proof of this tautology is presented in (16).

    (16)

    Modus Ponens defines one of the valid ways of deriving an inference from a conditional statement. It says that if we know that p→q is true, and in addition we know or assume that p is true, it is valid to infer that q is true. An illustration of this pattern of inference is presented as a syllogism in (17).

    (17) Premise 1: If John is Estonian, he will like this book. (p→q)
    Premise 2: John is Estonian. (p)
    Conclusion: He will like this book. (q)

    As we noted in §4.2, Modus Ponens guarantees a valid inference but does not guarantee a true conclusion. The conclusion will only be as reliable as the premises that we begin with. Suppose in this example it turns out that John is Estonian but hates the book. This does not disprove the rule of Modus Ponens; rather, it shows that the first premise is false, by providing a counter-example.

    Another valid rule for deriving an inference from a conditional statement is Modus Tollens ‘method of rejecting/denying’, also called ‘denying the consequent’: ((p→q) ∧ ¬q) → ¬p. This rule was illustrated in example (4a) above, repeated here as (18). It says that if we know that pq is true, and in addition we know or assume that q is false, it is valid to infer that p is also false.

    (18) Premise 1: If dolphins are fish, they are cold-blooded. (p→q)
    Premise 2: Dolphins are not cold-blooded. (¬q)
    Conclusion: Dolphins are not fish. (¬p)

    The tautology which we proved in (14) is known as the Disjunctive Syllogism: ((p∨q) ∧ (¬p)) → q. Another example which illustrates this pattern of inference is provided in (19).

    (19) Premise 1: Dolphins are either fish or mammals. (p∨q)
    Premise 2: Dolphins are not fish. (¬p)
    Conclusion: Dolphins are mammals. (q)

    Finally, the tautology known as the Hypothetical Syllogism is given in (20).

    (20) ((p→q) ∧ (q→r)) → (p→r)
    Premise 1: If Mickey is a rodent, he is a mammal. (p→q)
    Premise 2: If Mickey is a mammal, he is warm-blooded. (q→r)
    Conclusion: If Mickey is a rodent, he is warm-blooded. (p→r)

    The propositional logic outlined in this section is an important part of the logical metalanguage for semantic analysis, but it is not sufficient on its own because it is concerned only with truth values. We need a way to go beyond p and q, to represent the actual meanings of the basic propositions we are dealing with. Predicate logic gives us a way to include information about word meanings in logical expressions.


    This page titled 4.3: Propositional logic is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Paul Kroeger (Language Library Press) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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