Propositional Logic

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Introduction

Note: This article presumes that you have read the following topic articles:

Logical Structure in Deductive and Inductive Reasoning,

Deductive Argument Forms, and

Categorical Logic.

While Aristotle's categorical logic was based on the logical relationships that hold among categories, propositional logic focuses on relationships among propositions -- claims that are either true or false. Because we are still doing logic, we need to abstract from the actual content of arguments to see logical structure. Therefore, we will take a passage such as this

If Saskwatch is real, then Peter Pan is too. Peter Pan isn't real. Therefore I do not think Saskwatch is real.

And symbolize it like this:

Premise 1: If S, then P.

Premise 2: Not P.

Conclusion: Not S.

In this symbolization, S stands for "Saskwatch is real," and P stands for "Saskwatch is real"

We will discuss the logic of some natural language expressions -- more than we did with categorical logic -- but our general goal is the same. We want to refine our ability to recognize valid argument patterns. In the pattern above, the truth of the premises guarantees the truth of the conclusion, so the pattern is valid. Whether you are arguing about Sasquatch or taxes, if your argument fits that pattern and the premises you offer are true, then the conclusion given in the pattern will be true.

Before we discuss more argument patterns, we should introduce the basic statement types in our propositional logic system. The statement types are determined by a few different types of logical connectives:

Statement Type	Primary Natural Language Expression	Logical Symbol
Conjunction	"and"	.
Disjunction	"or"	v
Conditional	"if ... then"	->
Negation	"not"	"~"

All of these connectives join two propositions, usually symbolized by P, Q, R, and so on, except the negation symbol, called a "tilde," which simply negates a single expression. Since the propositions are either true or false, the expressions created by the connectives For example, "P . Q" can be understood as a "function" of the various combinations of truth and falsity of the individual propositions. We call these the "truth table" or "truth functional" definitions for each connective and we will explain them as we proceed.

Standard Form and Natural Language

Because language is so much richer and more complex than terms in a logic (logicians would say that it is more vague and ill-defined), the correspondence between, for example, the connective "v" and the natural language expression "or" is not precise, so we need to discuss the differences for each of the connectives. In general, logical systems work by having clear and precise definitions for terms. Becoming more aware of the difference between the connectives in natural language and logical systems can help you avoid ambiguities in your speech and writing that might cause confusion. In some cases, however, it is precisely the richness and diversity of meaning in natural language that allow reflection to go forward, so we will not give up language for logic, even if we have much to learn from it.

The conjunction is the simplest connective and it maps onto English in the most straightforward way. Conjunctions, in which we assert two propositions with an "and" between them are true as long as both propositions are true. If I say "I'm going to the river, and I'm going swimming" and then you later find out that I went to the mall or I went to the river, but did not swim, or I went swimming but not in the river, you would say that what I said earlier was not true. The truth value of the "." ("and")is defined in the table below for all possible combinations of true and false propositions, P and Q.

Truth Table Definition of Conjunction

P	Q	P.Q
T	T	T
T	F	F
F	T	F
F	F	F

Negations are also pretty straightforward. If the proposition "P" is true, then the proposition "not-P" is false. Of course, confusions are still possible. They usually occur when people try to get you to assign a "true or false" label to a complex statement, one that may not be simply true or false. Imagine that you're trying to choose a leader for an organization and someone ask you, "Do you think Mary would make a good president?" This sounds like a simple yes/no or true/false question. If you say yes, then you must think the proposition, "Mary would make a good president" is true. Then the negation of that proposition is false. But suppose you respond by saying "She wouldn't be a bad president," or "I'm not sure," or "Mary would be a good president in three years, after Ed serves a term." Now you are not simply saying "yes" or "no." These responses involve contexts in which a "bivalent" -- yes/no or true/false -¬assumption is intended in the question, but rejected in the answer. Logical systems, especially simple logical systems, are more effective as formal systems if they can abstract from complexities like these.

As long as you have genuinely true/false propositions, then the truth value of a negation is the opposite of the true value of the proposition. Unlike the other connectives, negation only takes one expression, so its truth table is shorter.

Truth Table Definition of Negation

P	~P
T	F
F	T

With disjunction, matters become more complex. Consider the circumstances under which the following statements, in the situations described, are true:

1. On a menu: "All gut-buster entrees come with your choice of soup or salad."
2. On your university's website: "Students who are planning to graduate in May or think they may be qualified to graduate in May should contact the Registrar for degree evaluation by January 15th."
3. On a sign in front of a swimming area of a lake: "Pets or glass containers are not allowed in the swimming area."

In the first statement, the meaning is clearly "one or the other but not both." But the meaning of "or" in the second example is not quite the same. It is a safe bet that many students planning to graduate in May also think they are qualified to graduate. If you said to yourself that the rule was for people who met one or the other condition but not both, and therefore you didn't contact the registrar (because both disjuncts were true for you), most people would say that you just misread the rule. That is because in this context "or" means "one or the other or both". The first example uses what is called "exclusive or" and the second uses "inclusive or," because it involves the possibility that this proposition is true if both conditions are met. Example number 3 is inclusive as well, but because it involves a negation ("not permitted"), it is an inclusive "neither . . . nor". If someone asked, "Are you following the rule?," you could only answer "yes" if you had neither glass nor pets with you. These meanings of "or" produce propositions that are true or false under different circumstances. Usually the context of communication makes the meaning clear, but in logic, where we abstract from context, we need to pick one meaning of "or" to work with. In our treatment of disjunction in propositional logic we will use "inclusive or." In a formal system of logic, we can express "exclusive or" by a more complicated expression, so we use the more basic meaning (inclusive) as the basis for the truth table.

Truth Table Definition of Disjunction

P	Q	P v Q
T	T	T
T	F	T
F	T	T
F	F	F

The logical pattern "If P, then Q" is sometimes referred to as the "conditional" or "hypothetical." Its relationship to natural language meaning is the most complex of the four connectives. Consider some of the following examples:

1. If you get the EZ-Ski package, you get three ski lessons, equipment rental and three ski passes for $90. (definitional)
2. If you fire your clay in a hot kiln, it will harden. (prediction, causal)
3. If Sally pitches, we will win. (prediction, causal)
4. If you fail to respond to this traffic summons, a warrant will be issued for your arrest. (general consequential, causal)
5. If you say you will help her, you should do it. (moral, decisional, definitional)
6. If you do that one more time, I'll hit you. (threat, decisional)
7. If she goes on a date with you, I'll eat my hat. (expression of incredulity, negative prediction)
8. If it makes you happy, why the hell are you so sad? (challenge, conceptual)

Our goal here is the same as it was for the other connectives -- to understand how we should think about the relationship between the truth or falsity of each part of the connective (in this case, the "if" clause taken separately from the "then" clause) and the truth or falsity of the whole. In conditionals, we call the "if" clause the antecedent and the "then" clause the consequent.

In the case of number 8, the popular lyric, it is not clear there is any truth value for the whole. Questions are moves in a conversational strategy. They are neither true nor false. Of course, that is not to deny that questions convey propositional content. You could represent it as a statement: "If something makes you happy, then it shouldn't make you sad."

A core meaning of many uses of "if ... then" is that the whole expression is true as a result of a connection between the truth of the "if" clause and the "then" clause. The hard thing to figure out is how we should evaluate the conditional if the antecedent does not obtain or if the consequent turns out to be true, but not as a result of the truth of the antecedent. Most people would say, after reading number 1, that if you don't get the EZ-Ski package, you won't get the lessons, etc, and that would be a reasonable inference in this context. We will see later that from a purely formal standpoint, this is a fallacious way to reason. Perhaps there is a conversational implication in number 1 that that is the only way to get everything it promises for that price. Notice that that isn't implied by our causal predictions. If you don't fire your clay, it might still harden for some other reason. Would you evaluate the whole expression as true in the case of someone who found another means of hardening their pot (maybe by freezing it!), or indeterminate? It might still be true that if you had fired it in a hot kiln, it would have hardened.

Even if you respond to the traffic summons mentioned in item #4, you might be the subject of a warrant for arrest for some other reason. Would you consider conditional number 4 true in that case? Or would you say that the truth value cannot be determined because you did not test the conditional by failing to reply to the traffic summons?

Suppose you make the prediction, "If Sally pitches, we will win the baseball game" and then she pitches very poorly, but the team wins anyway. Someone could reasonably say that your prediction was defective even though both the antecedent and consequent were true.

As confusing as this topic can be, a couple of points can be drawn out from the discussion. First, some conditionals in everyday discourse do not really mean to implicate an "if ... then" logic. That's what we saw in numbers 7 and 8. Second, some conditions really involve a stronger set of conditions than the simple "if ... then" structure suggests. If you corrrectly infer that the conditional goes in both directions (If P, then Q AND If Q, then P), and this often happens with definitions and exclusive ski packages, then you really have a "biconditional" on your hands. We say the truth conditions for bi-conditionals are "stronger" than for conditionals because more conditions must be satisfied for the conclusion to be true.

The big remaining problem is what to do with cases in which the conditional as a whole seems "undetermined" either because the antecedent did not obtain or because both parts were true, but not related to each other. . The key thing in these cases is to keep in mind the difference between actual reasoning in real situations, which is governing by the logic implicit in the actual meanings of our communication, and the needs of a logician creating a logical system, in which all of the possible truth values for an expression must have a determinate outcome. For a logician, the relationships among propositions have to be investigated without any reference to the content of the propositions. The logician cannot leave any lines of the truth table blank, but real reflective practitioners, working in natural language, can. With this in mind, we can define the logical connective "if... then" in terms of its truth table. The truth functional definition of the conditional is arrived at by finding the weakest or minimal logical meaning of "if ... then" and then relying on more elaborate translations of everyday meanings that say more or something more subtle. Following this way of thinking, the truth table for conditional is filled in as follows

Truth Table Definition of Conditional

P	Q	P -> Q
T	T	T
T	F	F
F	T	T
F	F	T

We are almost ready to introduce the valid patterns of reasoning for propositional logic, but we do need to comment on the logic of a few specific English language terms first.

"Implies" or "provided that"

"You may park in this lot provided that you have a green parking sticker," translates as "G -> P."

"Only if"

The translation of "only if" is somewhat more confusing. To translate "Your car will run only if it has gas," you might think that because the "if" appears right before "it has gas" the conditional is "G 4 R", roughly, "if it has gas, it will run". But that would be saying that "having gas" is a sufficient condition for the car to run, and that's probably not what a speaker of this sentence means. After all, any number of mechanical problems can keep a car with a full tank from running. The sentence meaning is "If it doesn't have gas, then it won't run," which is logically equivalent to "If it runs, then it has gas." ("~Q -> ~P" is logically equivalent to "P -> Q").

"Necessary and Sufficient"

We will talk more about "necessary conditions" and "sufficient conditions" when we discuss causality, but since they relate to the conditional, we should also explain their translation. If some claim "P" is a "sufficient condition" for "Q", then it is translated "If P is true, then Q is true," which is "P -> Q". But if something is a "necessary condition" then we are saying that if it doesn't obtain, then other claims won't be true. And that is what "~P -> Q" says, "If ~P, then ~Q." For the reason given above, this could be written as "Q -> P".

"Unless"

Another common, and confusing, English expression that relates to the hypothetical is "unless". Consider the following two examples

1. We will go out Tuesday night unless Professor Timberlake sets the exam for Wednesday.
2. You are eligible for the deduction unless you are over 65 years of age.

The first example seems to most naturally suggest that "if Professor Timberlake sets the exam for Wednesday then we will not go out," which translates as "E -> ~O", but the context of communication would also usually support this interpretation: "If he doesn't set the exam, then we will go out," which translates as "~E -> O." These are not logically equivalent, but they might both be implied by a speaker. When both of these conditions are implied, "unless" means the same thing as "exclusive or". So the first example also translates as "We will go out Tuesday or the exam is on Wednesday."

The second example could also be translated, "If you are over 65, then you are not eligible for the deduction," and it might also mean, "If you are not over 65, then you are eligible." However, the context for "eligibility" discussions, especially involving taxes, is often complex. There is usually an understanding that there are other eligibility requirements. Depending upon the context, "P unless Q may or may not mean both "Q -> ~P" and "-Q -> P".

Valid Argument Patterns for Propositional Logic

As with categorical logic, the valid argument patterns in propositonal logic are syllogisms (arguments with two premises) which reflect basic structures of reasoning. Five of the basic patterns are Modus Ponens, Modus Tollens, Disjunctive Syllogism, Hypothetical Syllogism and Double Negation. The patterns are summarized in the table below. We use lower case letters to signify that the propositions in these patterns are "sentential variables" and can be replaced by any particular sentence using an upper case letter to represent a particular sentence). The particular sentence is called a "sentential constant":

Pattern Name	Pattern	Logical Features of the Pattern
Modus Ponens	p -> q p q	From any conditional, and the antecedent of the conditional, you may infer the consequent.
Modus Tollens	p -> q ~q ~p	From any conditional, and the negation of its consequent, you may infer the negation of its antecedent.
Disjunctive Syllogism	p v q ~p q	From any disjunction, and the negation of either disjunct, you may infer the other q disjunct.
Hypothetical Syllogism	p -> q q -> r p -> r	From two conditionals in which the consequent of one occurs as the antecedent of the other, you may infer a conditional with the corresponding antecedent and consequent.
Double Negation	~~p p	From any double negated expression, you may infer an unnegated expression. (And vice versa.)

Notice that the third column of the table gives a natural language explication of the pattern. This statement of the patterns may seem cumbersome because it uses the vocabulary of "antecedent" and "disjunct", etc., however, this explanation is clearer and more complete than the pattern expressed symbolically. You should read through these descriptions of our patterns carefully, paying attention to how the pattern should be applied to specific instances. For example, it is important to know that in a disjunctive syllogism, "either disjunct" may be negated and you can infer "the other". This is actually harder to show with the symbols, which necessarily pick one disjunct for the negation symbol.

Many of these details are more relevant if you plan to learn the proof technique for propositional logic, which is explained below. Doing proofs in propositional logic is a good way to sharpen your eye for logical structure and validity, but it also takes us away from natural language arguments and the broader task of reflection, so we present it as an attractive detour, rather than as part of the main path.

There are some general comments to make about the patterns, however. Notice that three of the five patterns concern "if. . .then" or conditional statements. If you know that the antecedent of a conditional is true, you can infer the consequent. If you know that the consequent is false, you can infer the negation of the antecedent. Finally, hypothetical syllogism establishes something like a "transitive property" for conditionals. just as you know that if x is greater than y, and y is greater than z, then x is greater than z, so too, if you know that the consequent of one conditional is linked to the antecedent of another, you can make a new conditional with the other two parts.

Knowing the patterns should help you spot instances of them in natural language. Try to identify the pattern at work in the following premises. Then identify the conclusion that the pattern allows you to draw:

1. A: You can have strawberry or vanilla ice cream. B: I cannot have vanilla.
2. If wishes were horses, pigs would fly. Pigs can't fly.
3. If the train arrives at seven, we will have time for dinner. The train arrived at seven.
4. If you mow the lawn, I'll do the laundry. If I do the laundry done, then we can go swimming.

Try a few more in which two patterns are combined:

1. Shirley either loves me or she doesn't. If she loved me, she would call me on my birthday. But she didn't.
2. If astrology is a real science, we can find the tendencies of human behavior in the movements of the planets and stars. If people could find the tendencies of human behavior in the movements of the planets and starts, astrologers would be successful in predicting the future. Astrologers are not very successful in predicting the future.

With this knowledge of propositional logic you can work on the skill of recognizing logical structure in short, semi-formalized passages. Untangling the premises and conclusions of these arguments and formalizing them can help you see logical structure in real natural language arguments.

Exercise: Formalizing Propositional Arguments

The arguments in the following exercise set can be formalized into standard form propositional logic, but some of them require the addition of implied premises. Supply missing premises, rephrase claims and label them with individual letters, then write out the form of the argument. Then see if you can recognize the pattern or patterns at work in each deduction.

Example:

Unless he is a saint, a used car salesman will give in to the temptation to withhold information about the defects of the cars he sells. If he withholds information about used cars for sale, he will cause people to spend more for the cars than they are worth. Used car salesmen aren't saints.

H - Used Car salemen are saints. T - Used Car salesmen withold information about the defects of the cars they sell. S - Used car salesmen cause people to spend more for the cars they buy.

Premise 1: H v T

Premise 2: T -> S

Premise 3: ~H

Conclusion: S

1. If the city should approve more funds for the garage, it will go bankrupt. If the garage goes bankrupt, businesses will close. If businesses close, the tax revenue for the city will go down. No one wants the city tax revenue to go down. The city should approve more funds for the garage.
2. "It is only about the things that do not interest one, that one can give a really unbiassed opinion; and this is no doubt the reason why an unbiassed opinion is always absolutely valueless." Missing Premise: If you are not interested in something, your opinion doesn't have value. [More Letters of Oscar Wilde, ed. Rupert Hart¬Davis. Cited in Kelly, The Art o f Reasoning]
3. Either the war on Iraq was justified or it wasn't. If it was, then we should continue to work for a stable democratic Iraq. If it wasn't, then we should pull out and apologize. If Iraq didn't have weapons of mass destruction, then the war wasn't justified. Iraq didn't have weapons of mass destruction. Therefore, we should pull out and apologize.
4. If people do not receive good information about sexually transmitted diseases, then they will not make good choices about their sexual health. If people do not make good choices about sex, them may come to negative associations with it. If people have negative associations with sex, them may become repressed. Therefore, if people do not receive good information about sexually transmitted deseases, they may become repressed.

On a practical level, deductive reasoning is especially noticeable when you have a number of facts with a very clear logical structure. While you probably are not frequently in the position of Sherlock Holmes, Angela Landsbury (in "Murder She Wrote") or Peter Faulk (in "Columbo"), you should be able to recall the classic scence of a "who dunnit" in which the investigator puts all the pieces together. Suppose, for example, you know the following:

1. Smith was killed at his home with a large heavy object on the night of the 25th.
2. The cook, the butler, and the gardener are the only people who stay at the house with Smith.
3. The gardener takes the last week of the month off to visit his sister.
4. The cook burned his hands making creme broulee for a party on the 24"'.
5. The butler had been working out and bragging to people that Smith had left him a million dollars in his will.

Well, things don't look good for the butler, but murder mysteries are not solved exclusively by deductive logic; or even by logic alone. You have to know something about human psychology and, in this case, the means of committing the murder. There are, however, deductive structures at work in our thinking....