A Concise Introduction to Logic

1. Developing a Precise Language

1.1 Starting with sentences

We begin the study of logic by building a precise logical language. This will allow us to do at least two things: first, to say some things more precisely than we otherwise would be able to do; second, to study reasoning. We will use a natural language - English - as our guide, but our logical language will be far simpler, far weaker, but more rigorous than English.

We must decide where to start. We could pick just about any part of English to try to emulate: names, adjectives, prepositions, general nouns, and so on. But it is traditional, and as we will see, quite handy, to begin with whole sentences. For this reason, the first language we will develop is called "the propositional logic". It is also sometimes called "the sentential logic" or even "the sentential calculus". These all mean the same thing: the logic of sentences. In this propositional logic, the smallest independent parts of the language are sentences (throughout this book, I will assume that sentences and propositions are the same thing in our logic, and I will use the terms "sentence" and "proposition" interchangeably).

There are of course many kinds of sentences. To take examples from our natural language, these include:

What time is it?

Open the window.

Damn you!

I promise to pay you back.

It rained in Central Park on June 26, 2015.

We could multiply such examples. Sentences in English can be used to ask questions, give commands, curse or insult, form contracts, and express emotions. But, the last example above is of special interest because it aims to describe the world. Such sentences, which are sometimes called "declarative sentences", will be our model sentences for our logical language. We know a declarative sentence when we encounter it because it can be either true or false.

 

1.2 Precision in sentences

We want our logic of declarative sentences to be precise. But what does this mean? We can help clarify how we might pursue this by looking at sentences in a natural language that are perplexing, apparently because they are not precise. Here are three.

Tom is kind of tall.

When Karen had a baby, her mother gave her a pen.

This sentence is false.

We have already observed that an important feature of our declarative sentences is that they can be true or false. We call this the "truth value" of the sentence. These three sentences are perplexing because their truth values are unclear. The first sentence is vague, it is not clear under what conditions it would be true, and under what conditions it would be false. If Tom is six feet tall, is he kind of tall? There is no clear answer. The second sentence is ambiguous. If "pen" means writing implement, and Karen's mother bought a playpen for the baby, then the sentence is false. But until we know what "pen" means in this sentence, we cannot tell if the sentence is true.

The third sentence is strange. Many logicians have spent many years studying this sentence, which is traditionally called "the Liar". It is related to an old paradox about a Cretan who said, "All Cretans are liars". The strange thing about the Liar is that its truth value seems to explode. If it is true, then it is false. If it is false, then it is true. Some philosophers think this sentence is, therefore, neither true nor false; some philosophers think it is both true and false. In either case, it is confusing. How could a sentence that looks like a declarative sentence have both or no truth value?

Since ancient times, philosophers have believed that we will deceive ourselves, and come to believe untruths, if we do not accept a principle sometimes called "bivalence", or a related principle called "the principle of non-contradiction". Bivalence is the view that there are only two truth values (true and false) and that they exclude each other. The principle of non-contradiction states that you have made a mistake if you both assert and deny a claim. One or the other of these principles seems to be violated by the Liar.

We can take these observations for our guide: we want our language to have no vagueness and no ambiguity. In our propositional logic, this means we want it to be the case that each sentence is either true or false. It will not be kind of true, or partially true, or true from one perspective and not true from another. We also want to avoid things like the Liar. We do not need to agree on whether the Liar is both true and false, or neither true nor false. Either would be unfortunate. So, we will specify that our sentences have neither vice.

We can formulate our own revised version of the principle of bivalence, which states that:

Principle of Bivalence: Each sentence of our language must be either true or false, not both, not neither.

This requirement may sound trivial, but in fact it constrains what we do from now on in interesting and even surprising ways. Even as we build more complex logical languages later, this principle will be fundamental.

Some readers may be thinking: what if I reject bivalence, or the principle of non-contradiction? There is a long line of philosophers who would like to argue with you, and propose that either move would be a mistake, and perhaps even incoherent. Set those arguments aside. If you have doubts about bivalence, or the principle of non-contradiction, stick with logic. That is because we could develop a logic in which there were more than two truth values. Logics have been created and studied in which we allow for three truth values, or continuous truth values, or stranger possibilities. The issue for us is that we must start somewhere, and the principle of bivalence is an intuitive way and - it would seem - the simplest way to start with respect to truth values. Learn basic logic first, and then you can explore these alternatives.

This points us to an important feature, and perhaps a mystery, of logic. In part, what a logical language shows us is the consequences of our assumptions. That might sound trivial, but, in fact, it is anything but. From very simple assumptions, we will discover new, and ultimately shocking, facts. So, if someone wants to study a logical language where we reject the principle of bivalence, they can do so. The difference between what they are doing, and what we will do in the following chapters, is that they will discover the consequences of rejecting the principle of bivalence, whereas we will discover the consequences of adhering to it. In either case, it would be wise to learn traditional logic first, before attempting to study or develop an alternative logic.

We should note at this point that we are not going to try to explain what "true" and "false" mean, other than saying that "false" means not true. When we add something to our language without explaining its meaning, we call it a "primitive". Philosophers have done much to try to understand what truth is, but it remains quite difficult to define truth in any way that is not controversial. Fortunately, taking true as a primitive will not get us into trouble, and it appears unlikely to make logic mysterious. We all have some grasp of what "true" means, and this grasp will be sufficient for our development of propositional logic.

 

1.3 Atomic sentences

Our language will be concerned with declarative sentences, sentences that are either true or false, never both, and never neither. Here are some example sentences.

2+2=4.

Malcolm Little is tall.

If Lincoln wins the election, then Lincoln will be President.

The Earth is not the center of the universe.

These are all declarative sentences. These all appear to satisfy our principle of bivalence. But they differ in important ways. The first two sentences do not have sentences as parts. For example, try to break up the first sentence. "2+2" is a function. "4" is a name. "=4" is a meaningless fragment, as is "2+". Only the whole expression, "2+2=4", is a sentence with a truth value. The second sentence is similar in this regard. "Malcolm Little" is a name. "is tall" is an adjective phrase (we will discover later that logicians call this a "predicate"). "Malcolm Little is" or "is tall" are fragments, they have no truth value.^[2] Only "Malcolm Little is tall" is a complete sentence.

The first two example sentences above are of a kind we call "atomic sentences". The word "atom" comes from the ancient Greek word "atomos", meaning cannot be cut. When the ancient Greeks reasoned about matter, for example, some of them believed that if you took some substance, say a rock, and cut it into pieces, then cut the pieces into pieces, and so on, eventually you would get to something that could not be cut. This would be the smallest possible thing. (The fact that we now talk of having "split the atom" just goes to show that we changed the meaning of the word "atom". We came to use it as a name for a particular kind of thing, which then turned out to have parts, such as electrons, protons, and neutrons.) In logic, the idea of an atomic sentence is of a sentence that can have no parts that are sentences.

In reasoning about these atomic sentences, we could continue to use English. But for reasons that become clear as we proceed, there are many advantages to coming up with our own way of writing our sentences. It is traditional in logic to use upper case letters from P on (P, Q, R, S….) to stand for atomic sentences. Thus, instead of writing

Malcolm Little is tall.

We could write

P

If we want to know how to translate P to English, we can provide a translation key. Similarly, instead of writing

Malcolm Little is a great orator.

We could write

Q

And so on. Of course, written in this way, all we can see about such a sentence is that it is a sentence, and that perhaps P and Q are different sentences. But for now, these will be sufficient.

Note that not all sentences are atomic. The third sentence in our four examples above contains parts that are sentences. It contains the atomic sentence, "Lincoln wins the election" and also the atomic sentence, "Lincoln will be President". We could represent this whole sentence with a single letter. That is, we could let

If Lincoln wins the election, Lincoln will be president.

be represented in our logical language by

S

However, this would have the disadvantage that it would hide some of the sentences that are inside this sentence, and also it would hide their relationship. Our language would tell us more if we could capture the relation between the parts of this sentence, instead of hiding them. We will do this in chapter 2.

 

1.4 Syntax and semantics

An important and useful principle for understanding a language is the difference between syntax and semantics. "Syntax" refers to the "shape" of an expression in our language. It does not concern itself with what the elements of the language mean, but just specifies how they can be written out.

We can make a similar distinction (though not exactly the same) in a natural language. This expression in English has an uncertain meaning, but it has the right "shape" to be a sentence:

Colorless green ideas sleep furiously.

In other words, in English, this sentence is syntactically correct, although it may express some kind of meaning error.

An expression made with the parts of our language must have correct syntax in order for it to be a sentence. Sometimes, we also call an expression with the right syntactic form a "well-formed formula".

We contrast syntax with semantics. "Semantics" refers to the meaning of an expression of our language. Semantics depends upon the relation of that element of the language to something else. For example, the truth value of the sentence, "The Earth has one moon" depends not upon the English language, but upon something exterior to the language. Since the self-standing elements of our propositional logic are sentences, and the most important property of these is their truth value, the only semantic feature of sentences that will concern us in our propositional logic is their truth value.

Whenever we introduce a new element into the propositional logic, we will specify its syntax and its semantics. In propositional logic, the syntax is generally trivial, but the semantics is less so. We have so far introduced atomic sentences. The syntax for an atomic sentence is trivial. If P is an atomic sentence, then it is syntactically correct to write down

P

By saying that this is syntactically correct, we are not saying that P is true. Rather, we are saying that P is a sentence.

If semantics in the propositional logic concerns only truth value, then we know that there are only two possible semantic values for P; it can be either true or false. We have a way of writing this that will later prove helpful. It is called a "truth table". For an atomic sentence, the truth table is trivial, but when we look at other kinds of sentences their truth tables will be more complex.

The idea of a truth table is to describe the conditions in which a sentence is true or false. We do this by identifying all the atomic sentences that compose that sentence. Then, on the left side, we stipulate all the possible truth values of these atomic sentences and write these out. On the right side, we then identify under what conditions the sentence (that is composed of the other atomic sentences) is true or false.

The idea is that the sentence on the right is dependent on the sentence(s) on the left. So the truth table is filled in like this:

Atomic sentence(s) that compose the dependent sentence on the right	Dependent sentence composed of the atomic sentences on the left
All possible combinations of truth values of the composing atomic sentences	Resulting truth values for each possible combination of truth values of the composing atomic sentences

We stipulate all the possible truth values on the bottom left because the propositional logic alone will not determine whether an atomic sentence is true or false; thus, we will simply have to consider both possibilities. Note that there are many ways that an atomic sentence can be true, and there are many ways that it can be false. For example, the sentence, "Tom is American" might be true if Tom was born in New York, in Texas, in Ohio, and so on. The sentence might be false because Tom was born to Italian parents in Italy, to French parents in France, and so on. So, we group all these cases together into two kinds of cases.

These are two rows of the truth table for an atomic sentence. Each row of the truth table represents a kind of way that the world could be. So here is the left side of a truth table with only a single atomic sentence, P. We will write "T" for true and "F" for false.

P
T
F

There are only two relevant kinds of ways that the world can be, when we are considering the semantics of an atomic sentence. The world can be one of the many conditions such that P is true, or it can be one of the many conditions such that P is false.

To complete the truth table, we place the dependent sentence on the top right side, and describe its truth value in relation to the truth value of its parts. We want to identify the semantics of P, which has only one part, P. The truth table thus has the final form:

P	P
T	T
F	F

This truth table tells us the meaning of P, as far as our propositional logic can tell us about it. Thus, it gives us the complete semantics for P. (As we will see later, truth tables have three uses: to provide the semantics for a kind of sentence; to determine under what conditions a complex sentence is true or false; and to determine if an argument is good. Here we are describing only this first use.)

In this truth table, the first row combined together all the kinds of ways the world could be in which P is true. In the second column we see that for all of these kinds of ways the world could be in which P is true, unsurprisingly, P is true. The second row combines together all the kinds of ways the world could be in which P is false. In those, P is false. As we noted above, in the case of an atomic sentence, the truth table is trivial. Nonetheless, the basic concept is very useful, as we will begin to see in the next chapter.

One last tool will be helpful to us. Strictly speaking, what we have done above is give the syntax and semantics for a particular atomic sentence, P. We need a way to make general claims about all the sentences of our language, and then give the syntax and semantics for any atomic sentences. We do this using variables, and here we will use Greek letters for those variables, such as Φ and Ψ. Things said using these variables are called our "metalanguage", which literally means "the after language", but which we take to mean, our language about our language. The particular propositional logic that we create is called our "object language". P and Q are sentences of our object language. Φ and Ψ are elements of our metalanguage. To specify now the syntax of atomic sentences (that is, of all atomic sentences) we can say: If Φ is an atomic sentence, then

Φ

is a sentence. This tells us that simply writing Φ down (whatever atomic sentence it may be), as we have just done, is to write down something that is syntactically correct.

To specify now the semantics of atomic sentences (that is, of all atomic sentences) we can say: If Φ is an atomic sentence, then the semantics of Φ is given by

Φ	Φ
T	T
F	F

Note an important and subtle point. The atomic sentences of our propositional logic will be what we call "contingent" sentences. A contingent sentence can be either true or false. We will see later that some complex sentences of our propositional logic must be true, and some complex sentences of our propositional logic must be false. But for the propositional logic, every atomic sentence is (as far as we can tell using the propositional logic alone) contingent. This observation matters because it greatly helps to clarify where logic begins, and where the methods of another discipline ends. For example, suppose we have an atomic sentence like:

Force is equal to mass times acceleration.

Igneous rocks formed under pressure.

Germany inflated its currency in 1923 in order to reduce its reparations debt.

Logic cannot tell us whether these are true or false. We will turn to physicists, and use their methods, to evaluate the first claim. We will turn to geologists, and use their methods, to evaluate the second claim. We will turn to historians, and use their methods, to evaluate the third claim. But the logician can tell the physicist, geologist, and historian what follows from their claims.

 

1.5 Problems

1. Vagueness arises when the conditions under which a sentence might be true are "fuzzy". That is, in some cases, we cannot identify if the sentence is true or false. If we say, "Tom is tall", this sentence is certainly true if Tom is the tallest person in the world, but it is not clear whether it is true if Tom is 185 centimeters tall. Identify or create five declarative sentences in English that are vague.
2. Ambiguity usually arises when a word or phrase has several distinct possible interpretations. In our example above, the word "pen" could mean either a writing implement or a structure to hold a child. A sentence that includes "pen" could be ambiguous, in which case it might be true for one interpretation and false for another. Identify or create five declarative sentences in English that are ambiguous. (This will probably require you to identify a homonym, a word that has more than one meaning but sounds or is written the same. If you are stumped, consider slang: many slang terms are ambiguous because they redefine existing words. For example, in the 1980s, in some communities and contexts, to say something was "bad" meant that it was good; this obviously can create ambiguous sentences.)
3. Often we can make a vague sentence precise by defining a specific interpretation of the meaning of an adjective, term, or other element of the language. For example, we could make the sentence "Tom is tall" precise by specifying one person referred to by "Tom", and also by defining "…is tall" as true of anyone 180 centimeters tall or taller. For each of the five vague sentences that you identified or created for problem 1, describe how the interpretation of certain elements of the sentence could make the sentence no longer vague.
4. Often we can make an ambiguous sentence precise by specifying which of the possible meanings we intend to use. We could make the sentence, "Tom is by the pen" unambiguous by specifying which Tom we mean, and also defining "pen" to mean an infant playpen. For each of the five ambiguous sentences that you identified or created for problem 2, identify and describe how the interpretation of certain elements of the sentence could make the sentence no longer ambiguous.
5. Come up with five examples of your own of English sentences that are not declarative sentences. (Examples can include commands, exclamations, and promises.)

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Source: Craig DeLancey, https://milnepublishing.geneseo.edu/concise-introduction-to-logic/chapter/1-developing-a-precise-language/
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License.

2. "If…then…" and "It is not the case that…"

2.1 The Conditional

As we noted in chapter 1, there are sentences of a natural language, like English, that are not atomic sentences. Our examples included

If Lincoln wins the election, then Lincoln will be President.

The Earth is not the center of the universe.

We could treat these like atomic sentences, but then we would lose a great deal of important information. For example, the first sentence tells us something about the relationship between the atomic sentences "Lincoln wins the election" and "Lincoln will be President". And the second sentence above will, one supposes, have an interesting relationship to the sentence, "The Earth is the center of the universe". To make these relations explicit, we will have to understand what "if…then…" and "not" mean. Thus, it would be useful if our logical language was able to express these kinds of sentences in a way that made these elements explicit. Let us start with the first one.

The sentence, "If Lincoln wins the election, then Lincoln will be President" contains two atomic sentences, "Lincoln wins the election" and "Lincoln will be President". We could thus represent this sentence by letting

Lincoln wins the election

be represented in our logical language by

P

And by letting

Lincoln will be president

be represented by

Q

Then, the whole expression could be represented by writing

If P then Q

It will be useful, however, to replace the English phrase "if…then…" by a single symbol in our language. The most commonly used such symbol is "→". Thus, we would write

P→Q

One last thing needs to be observed, however. We might want to combine this complex sentence with other sentences. In that case, we need a way to identify that this is a single sentence when it is combined with other sentences. There are several ways to do this, but the most familiar (although not the most elegant) is to use parentheses. Thus, we will write our expression

(P→Q)

This kind of sentence is called a "conditional". It is also sometimes called a "material conditional". The first constituent sentence (the one before the arrow, which in this example is "P") is called the "antecedent". The second sentence (the one after the arrow, which in this example is "Q") is called the "consequent".

We know how to write the conditional, but what does it mean? As before, we will take the meaning to be given by the truth conditions—that is, a description of when the sentence is either true or false. We do this with a truth table. But now, our sentence has two parts that are atomic sentences, P and Q. Note that either atomic sentence could be true or false. That means, we have to consider four possible kinds of situations. We must consider when P is true and when it is false, but then we need to consider those two kinds of situations twice: once for when Q is true and once for when Q is false. Thus, the left hand side of our truth table will look like this:

P	Q
T	T
T	F
F	T
F	F

There are four kinds of ways the world could be that we must consider.

Note that, since there are two possible truth values (true and false), whenever we consider another atomic sentence, there are twice as many ways the world could be that we should consider. Thus, for n atomic sentences, our truth table must have 2n rows. In the case of a conditional formed out of two atomic sentences, like our example of (P→Q), our truth table will have 22 rows, which is 4 rows. We see this is the case above.

Now, we must decide upon what the conditional means. To some degree this is up to us. What matters is that once we define the semantics of the conditional, we stick to our definition. But we want to capture as much of the meaning of the English "if…then…" as we can, while remaining absolutely precise in our language.

Let us consider each kind of way the world could be. For the first row of the truth table, we have that P is true and Q is true. Suppose the world is such that Lincoln wins the election, and also Lincoln will be President. Then, would I have spoken truly if I said, "If Lincoln wins the election, then Lincoln will be President"? Most people agree that I would have. Similarly, suppose that Lincoln wins the election, but Lincoln will not be President. Would the sentence "If Lincoln wins the election, then Lincoln will be President" still be true? Most agree that it would be false now. So the first rows of our truth table are uncontroversial.

P	Q	(P→Q)
T	T	T
T	F	F
F	T
F	F

Some students, however, find it hard to determine what truth values should go in the next two rows. Note now that our principle of bivalence requires us to fill in these rows. We cannot leave them blank. If we did, we would be saying that sometimes a conditional can have no truth value; that is, we would be saying that sometimes, some sentences have no truth value. But our principle of bivalence requires that—in all kinds of situations—every sentence is either true or false, never both, never neither. So, if we are going to respect the principle of bivalence, then we have to put either T or F in for each of the last two rows.

It is helpful at this point to change our example. Let us consider two different examples to illustrate how best to fill out the remainder of the truth table for the conditional.

First, suppose I say the following to you: "If you give me $50, then I will buy you a ticket to the concert tonight." Let

You give me $50

be represented in our logic by

R

and let

I will buy you a ticket to the concert tonight.

be represented by

S

Our sentence then is

(R→S)

And its truth table—as far as we understand right now—is:

That is, if you give me the money and I buy you the ticket, my claim that "If you give me $50, then I will buy you a ticket to the concert tonight" is true. And, if you give me the money and I don't buy you the ticket, I lied, and my claim is false. But now, suppose you do not give me $50, but I buy you a ticket for the concert as a gift. Was my claim false? No. I simply bought you the ticket as a gift, but, presumably would have bought it if you gave me the money, also. Similarly, if you don't give me money, and I do not buy you a ticket, that seems perfectly consistent with my claim.

So, the best way to fill out the truth table is as follows.

R	S	(R→S)
T	T	T
T	F	F
F	T	T
F	F	T

Second, consider another sentence, which has the advantage that it is very clear with respect to these last two rows. Assume that a is a particular natural number, only you and I don't know what number it is (the natural numbers are the whole positive numbers: 1, 2, 3, 4…). Consider now the following sentence.

If a is evenly divisible by 4, then a is evenly divisible by 2.

(By "evenly divisible," I mean divisible without remainder.) The first thing to ask yourself is: is this sentence true? I hope we can all agree that it is—even though we do not know what a is. Let

a is evenly divisible by 4

be represented in our logic by

U

and let

a is evenly divisible by 2

be represented by

V

Our sentence then is

(U→V)

And its truth table—as far as we understand right now—is:

U	V	(U→V)
T	T	T
T	F	F
F	T
F	F

Now consider a case in which a is 6. This is like the third row of the truth table. It is not the case that 6 is evenly divisible by 4, but it is the case that 6 is evenly divisible by 2. And consider the case in which a is 7. This is like the fourth row of the truth table; 7 would be evenly divisible by neither 4 nor 2. But we agreed that the conditional is true—regardless of the value of a! So, the truth table must be:^[3]

U	V	(U→V)
T	T	T
T	F	F
F	T	T
F	F	T

Following this pattern, we should also fill out our table about the election with:

P	Q	(P→Q)
T	T	T
T	F	F
F	T	T
F	F	T

If you are dissatisfied by this, it might be helpful to think of these last two rows as vacuous cases. A conditional tells us about what happens if the antecedent is true. But when the antecedent is false, we simply default to true.

We are now ready to offer, in a more formal way, the syntax and semantics for the conditional.

The syntax of the conditional is that, if Φ and Ψ are sentences, then

(Φ→Ψ)

is a sentence.

The semantics of the conditional are given by a truth table. For any sentences Φ and Ψ:

Φ	Ψ	(Φ→Ψ)
T	T	T
T	F	F
F	T	T
F	F	T

Remember that this truth table is now a definition. It defines the meaning of "→". We are agreeing to use the symbol "→" to mean this from here on out.

The elements of the propositional logic, like "→", that we add to our language in order to form more complex sentences, are called "truth functional connectives". I hope it is clear why: the meaning of this symbol is given in a truth function. (If you are unfamiliar or uncertain about the idea of a function, think of a function as like a machine that takes in one or more inputs, and always then gives exactly one output. For the conditional, the inputs are two truth values; and the output is one truth value. For example, put T F into the truth function called "→", and you get out F.)

 

2.2 Alternative phrasings in English for the conditional. Only if.

English includes many alternative phrasings that appear to be equivalent to the conditional. Furthermore, in English and other natural languages, the order of the conditional will sometimes be reversed. We can capture the general sense of these cases by recognizing that each of the following phrasings would be translated as (P→Q). (In these examples, we mix English and our propositional logic, in order to illustrate the variations succinctly.)

If P, then Q.

Q, if P.

On the condition that P, Q.

Q, on the condition that P.

Given that P, Q.

Q, given that P.

Provided that P, Q.

Q, provided that P.

When P, then Q.

Q, when P.

P implies Q.

Q is implied by P.

P is sufficient for Q.

Q is necessary for P.

An oddity of English is that the word "only" changes the meaning of "if". You can see this if you consider the following two sentences.

Fifi is a cat, if Fifi is a mammal.

Fifi is a cat only if Fifi is a mammal.

Suppose we know Fifi is an organism, but, we don't know what kind of organism Fifi is. Fifi could be a dog, a cat, a gray whale, a ladybug, a sponge. It seems clear that the first sentence is not necessarily true. If Fifi is a gray whale, for example, then it is true that Fifi is a mammal, but false that Fifi is a cat; and so, the first sentence would be false. But the second sentence looks like it must be true (given what you and I know about cats and mammals).

We should thus be careful to recognize that "only if" does not mean the same thing as "if". (If it did, these two sentences would have the same truth value in all situations.) In fact, it seems that "only if" can best be expressed by a conditional where the "only if" appears before the consequent (remember, the consequent is the second part of the conditional—the part that the arrows points at). Thus, sentences of this form:

P only if Q.

Only if Q, P.

are best expressed by the formula

(P→Q)

 

2.3 Test your understanding of the conditional

People sometimes find conditionals confusing. In part, this seems to be because some people confuse them with another kind of truth-functional connective, which we will learn about later, called the "biconditional". Also, sometimes "if…then…" is used in English in a different way (see section 17.7 if you are curious about alternative possible meanings). But from now on, we will understand the conditional as described above. To test whether you have properly grasped the conditional, consider the following puzzle.^[4]

We have a set of four cards in figure 2.1. Each card has the following property: it has a shape on one side, and a letter on the other side. We shuffle and mix the cards, flipping some over while we shuffle. Then, we lay out the four cards:

Given our constraint that each card has a letter on one side and a shape on the other, we know that card 1 has a shape on the unseen side; card 2 has a letter on the unseen side; and so on.

Consider now the following claim:

For each of these four cards, if the card has a Q on the letter side of the card, then it has a square on the shape side of the card.

Here is our puzzle: what is the minimum number of cards that we must turn over to test whether this claim is true of all four cards; and which cards are they that we must turn over? Of course we could turn them all over, but the puzzle asks you to identify all and only the cards that will test the claim.

Stop reading now, and see if you can decide on the answer. Be warned, people generally perform poorly on this puzzle. Think about it for a while. The answer is given below in problem 1.

 

2.4 Alternative symbolizations for the conditional

Some logic books, and some logicians, use alternative symbolizations for the various truth-functional connectives. The meanings (that is, the truth tables) are always the same, but the symbol used may be different. For this reason, we will take the time in this text to briefly recognize alternative symbolizations.

The conditional is sometimes represented with the following symbol: "⊃". Thus, in such a case, (P→Q) would be written

(P⊃Q)

 

2.5 Negation

In chapter 1, we considered as an example the sentence,

The Earth is not the center of the universe.

At first glance, such a sentence might appear to be fundamentally unlike a conditional. It does not contain two sentences, but only one. There is a "not" in the sentence, but it is not connecting two sentences. However, we can still think of this sentence as being constructed with a truth functional connective, if we are willing to accept that this sentence is equivalent to the following sentence.

It is not the case that the Earth is the center of the universe.

If this sentence is equivalent to the one above, then we can treat "It is not the case" as a truth functional connective. It is traditional to replace this cumbersome English phrase with a single symbol, "¬". Then, mixing our propositional logic with English, we would have

¬The Earth is the center of the universe.

And if we let W be a sentence in our language that has the meaning The Earth is the center of the universe, we would write

¬W

This connective is called "negation". Its syntax is: if Φis a sentence, then

¬Φ

is a sentence. We call such a sentence a "negation sentence".

The semantics of a negation sentence is also obvious, and is given by the following truth table.

Φ	¬Φ
T	F
F	T

To deny a true sentence is to speak a falsehood. To deny a false sentence is to say something true.

Our syntax always is recursive. This means that syntactic rules can be applied repeatedly, to the product of the rule. In other words, our syntax tells us that if P is a sentence, then ¬P is a sentence. But now note that the same rule applies again: if ¬P is a sentence, then ¬¬P is a sentence. And so on. Similarly, if P and Q are sentences, the syntax for the conditional tells us that (P→Q) is a sentence. But then so is ¬(P→Q), and so is (¬(P→Q) → (P→Q)). And so on. If we have just a single atomic sentence, our recursive syntax will allow us to form infinitely many different sentences with negation and the conditional.

 

2.6 Alternative symbolizations for negation

Some texts may use "~" for negation. Thus, ¬P would be expressed with

~P

 

2.7 Problems

1. The answer to our card game was: you need only turn over cards 3 and 4. This might seem confusing to many people at first. But remember the meaning of the conditional: it can only be false if the first part is true and the second part is false. The sentence we want to test is "For each of these four cards, if the card has a Q on the letter side of the card, then it has a square on the shape side of the card". Let Q stand for "the card has a Q on the letter side of the card." Let S stand for "the card has a square on the shape side of the card." Then we could make a truth table to express the meaning of the claim being tested:

Q	S	(Q→S)
T	T	T
T	F	F
F	T	T
F	F	T

Look back at the cards. The first card has an R on the letter side. So, sentence Q is false. But then we are in a situation like the last two rows of the truth table, and the conditional cannot be false. We do not need to check that card. The second card has a square on it. That means S is true for that card. But then we are in a situation represented by either the first or third row of the truth table. Again, the claim that (Q→S) cannot be false in either case with respect to that card, so there is no point in checking that card. The third card shows a Q. It corresponds to a situation that is like either the first or second row of the truth table. We cannot tell then whether (Q→S) is true or false of that card, without turning the card over. Similarly, the last card shows a situation where S is false, so we are in a kind of situation represented by either the second or last row of the truth table. We must turn the card over to determine if (Q→S) is true or false of that card.

Try this puzzle again. Consider the following claim about those same four cards: If there is a star on the shape side of the card, then there is an R on the letter side of the card. What is the minimum number of cards that you must turn over to check this claim? What cards are they?

2. Consider the following four cards in figure 2.2. Each card has a letter on one side, and a shape on the other side.

For each of the following claims, in order to determine if the claim is true of all four cards, describe (1) The minimum number of cards you must turn over to check the claim, and (2) what those cards are.

1. There is not a Q on the letter side of the card.
2. There is not an octagon on the shape side of the card.
3. If there is a triangle on the shape side of the card, then there is a P on the letter side of the card.
4. There is an R on the letter side of the card only if there is a diamond on the shape side of the card.
5. There is a hexagon on the shape side of the card, on the condition that there is a P on the letter side of the card.
6. There is a diamond on the shape side of the card only if there is a P on the letter side of the card.

3. Which of the following have correct syntax? Which have incorrect syntax?

1. P→Q
2. ¬(P→Q)
3. (¬P→Q)
4. (P¬→Q)
5. (P→¬Q)

4. Use the following translation key to translate the following sentences into a propositional logic.

Translation Key
Logic	English
P	Abe is able.
Q	Abe is honest.

1. If Abe is honest, Abe is able.
2. Abe is not able.
3. Abe is not able only if Abe is not honest.
4. Abe is able, provided that Abe is not honest.
5. If Abe is not able then Abe is not honest.

5. Make up your own translation key to translate the following sentences into a propositional logic. Then, use your key to translate the sentences into the propositional logic. Your translation key should contain only atomic sentences. These should be all and only the atomic sentences needed to translate the following sentences of English. Don't let it bother you that some of the sentences must be false.

1. Josie is a cat.
2. Josie is a mammal.
3. Josie is not a mammal.
4. If Josie is not a cat, then Josie is not a mammal.
5. Josie is a fish.
6. Provided that Josie is a mammal, Josie is not a fish.
7. Josie is a cat only if Josie is a mammal.
8. Josie is a fish only if Josie is not a mammal.
9. It's not the case that Josie is not a mammal.
10. Josie is not a cat, if Josie is a fish.

6. This problem will make use of the principle that our syntax is recursive. Translating these sentences is more challenging. Make up your own translation key to translate the following sentences into a propositional logic. Your translation key should contain only atomic sentences; these should be all and only the atomic sentences needed to translate the following sentences of English.

1. It is not the case that Tom won't pass the exam.
2. If Tom studies, Tom will pass the exam.
3. It is not the case that if Tom studies, then Tom will pass the exam.
4. If Tom does not study, then Tom will not pass the exam.
5. If Tom studies, Tom will pass the exam—provided that he wakes in time.
6. If Tom passes the exam, then if Steve studies, Steve will pass the exam.
7. It is not the case that if Tom passes the exam, then if Steve studies, Steve will pass the exam.
8. If Tom does not pass the exam, then if Steve studies, Steve will pass the exam.
9. If Tom does not pass the exam, then it is not the case that if Steve studies, Steve will pass the exam.
10. If Tom does not pass the exam, then if Steve does not study, Steve won't pass the exam.

7. Make up your own translation key in order to translate the following sentences into English. Write out the English equivalents in English sentences that seem (as much as is possible) natural.

1. (R→S)
2. ¬¬R
3. (S→R)
4. (¬S→¬R)
5. ¬(R→S)

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Source: Craig DeLancey, https://milnepublishing.geneseo.edu/concise-introduction-to-logic/chapter/2-ifthen-and-it-is-not-the-case-that/
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License.