Fundamentals of Logic

Philosophy proceeds by reasoned discussion and debate. To do philosophy well, therefore, it is important to know some fundamental principles of logic. Logic is the study of reasoning. Reasoning is the process by which we use evidence to judge try to discover or persuade others of the truth. Reasoning is verbally expressed in arguments. An argument is a sequence of declarative sentences, one of which is intended as a conclusion; the remaining sentences, the premises, are intended to prove or at least provide some evidence for the conclusion. The premises and conclusion are declarative statements -- which may be true or false -- as opposed to questions, commands or exclamations. Nondeclarative sentences may sometimes suggest premises or conclusions, but they never are premises or conclusions.  Here is a very simple argument:

         All women are mortal.
         Tori Amos is a woman.
So     Tori Amos is mortal.

The first two statements are premises. The word 'So' is used to mark the conclusion.  Often three dots arranged in a triangle are used instead.  To analyze arguments, we generally write the premises first and the conclusion last as in this example, but in conversation or writing they may occur in any order. The two premises provide evidence -- in this case good evidence -- that the conclusion is true. Anyone who understands and agrees with the premises must, if she is rational, also agree with the conclusion.

Our definition of 'argument' stipulates that an argument's premises are intended to give evidence for the conclusion. But they need not actually give evidence. There are bad arguments as well as good ones. Consider:

        Humans are the only rational beings.
        Rationality alone enables a being to make moral judgments.
So   Only humans are ends-in-themselves.

Now this is an argument, but it's bad. The reason why it's bad is that we can't quite see what the capacity for moral judgment has to do with being an end-in-itself. (An end-in-itself is something that is valuable in and of itself, not merely as a means to someone else’s purposes.) Still, bad as it is, it's an argument; the author intended the first two propositions (sentences) to be taken as evidence for or proof of the third, and that's all that being an argument requires.

An argument succeeds if it meets the following two requirements:

Arguments which meet both requirements are called sound.  A sound argument, in other words, is a valid argument with true premises.  Sound arguments are the only kinds of arguments that prove their conclusions.  Any argument whose reasoning is not valid or which has at least one false premise is called unsound.  Unsound arguments fail to prove their conclusions.

Logic is concerned mainly with the validity of reasoning, not with the truth of the premises.  Determining whether or not the premises are true is the responsibility of other branches of knowledge (law, common sense, science, ethics, etc.).  What it means for the argument's reasoning to be valid is that there is no way for the conclusion not to be true while the premises are true. We'll sometimes put this in terms of "possible situations": there is no possible situation in which the premises are true but the conclusion isn't.

The Tori Amos argument is valid, for there is no possible situation in which all women are mortal, Tori Amos is a woman, and Tori Amos is not mortal; we can't even coherently think such a thing.  (This argument is also sound, since its premises are true.)

The end-in-itself argument is invalid (i.e., not valid), for there is a possible situation in which the premises are true and the conclusion isn't. That is, it is possible that humans are the only rational beings and that rationality alone enables a being to make moral judgments but that humans are not the only ends-in-themselves. One way in which this is possible is if being an end-in-itself has nothing to do with the ability to make moral judgements, but rather is linked to some more general capacity, such as sentience (the ability to feel) or the ability to live and flourish. Thus perhaps other critters are also ends-in-themselves even if the argument's premises are true.

A possible situation in which an argument's premises are true and its conclusion is not true, is called a counterexample to the argument. We may define validity more briefly simply by saying that a valid argument is one without a counterexample.
When we speak of possible situations, the term 'possible' is to be understood in a very wide sense. To be possible, a situation need not be something we can bring about; it doesn't even have to obey the laws of physics. It just has to be something we can coherently conceive -- that is, it has to be thinkable and describable without self-contradiction.

Thus, to tell whether or not an argument is valid, we try to conceive or imagine a possible situation in which its premises are true and conclusion is untrue. If we succeed (i.e., if we can describe a counterexample), the argument is invalid. If we fail, then either we have not been imaginative enough or the argument is valid. We appeal to counterexamples almost unconsciously in everyday life. Consider this mundane argument:

        They said on the radio that it's going to be a beautiful day today.
So    It is going to be beautiful today.

One natural (albeit cynical) reply is, "they could be wrong." This reply demonstrates the invalidity of the argument by describing a counterexample -- that is, a possible situation in which the conclusion ('It's going to be a beautiful day today') is untrue even though the premise ('They said so on the radio') is true: namely the situation in which the forecasters are simply wrong.

A counterexample need not be an actual situation, though it might; it is enough that the situation be conceptually possible. Thus, it need not be true that the forecasters are wrong; to see the invalidity of the argument, we need only realize that this is possible.

To give a counterexample, then, is merely to tell a kind of story. The story needn't be true, but it must be conceptually coherent. The cynical respondent to our argument above hints at such a story with the remark "they could be wrong."

That's enough for casual conversation. But for logical analysis it's useful to be a little more explicit. A well-stated description of a counterexample should contain three elements:

If we flesh out the cynic's counterexample to make all these elements explicit, the result might be something like this:

They said on the radio that it's going to be a beautiful day today. But they are wrong. A cold front is moving in unexpectedly and will bring rain instead of a beautiful day.

All three elements are now present. The first sentence of this "story" affirms the premise. The second denies the conclusion. The third explains how the conclusion could be untrue even though the premise is true.

This is not, of course, the only possible situation that would make the premises but not the conclusion true. I made up the idea of an unexpected cold front more or less arbitrarily. There are other counterexamples as well. Maybe an unexpected warm front will bring rain. Or maybe there will be an unexpected dust storm. Or maybe the radio announcer knew it was going to be an awful day and flat out lied. Each of these scenarios is a counterexample. This is typical; invalid arguments usually have indefinitely many counterexamples, each of which is by itself sufficient to show that the argument is invalid.

Let's consider another example. Is the following argument valid or invalid? (A freethinkiner is a person who does not automatically accept prevailing religious dogma.)

        All philosophers are freethinkers.
        Al is not a philosopher.
So    Al is not a freethinker.

To answer, we try to imagine a counterexample. Is there a way for the conclusion not to be true while the premises are true? (To say that the conclusion is not true, of course, is to say that Al is a freethinker.) A moment's thought should reveal that this is quite possible. Here's one counterexample:

All philosophers are freethinkers and Al is not a philosopher, but Al is nevertheless a freethinker, because there are some freethinking bricklayers who are not philosophers, and Al is one of these.

Again all three elements of a well-described counterexample are present. The statement 'All philosophers are freethinkers and Al is not a philosopher' affirms both the premises. The statement 'Al is nevertheless a freethinker' denies the conclusion. And the remainder of the story explains how this can be so. The story is perfectly coherent, and thus it shows us how the conclusion could be untrue even if the premises were true.

Notice again that the counterexample need not be an actual situation. It's just a story, a scenario, a fiction. In fact, it isn't true that all philosophers are freethinkers, and maybe it isn't true that Al (whoever Al is) is a freethinker, either. That doesn't matter; our story still provides a counterexample, and it shows that the argument is invalid, by showing how it could be that the conclusion is untrue while the premises are true.

Notice, further, that we needn't have said that Al is a bricklayer; for purposes of the example, he could have been an anarcho-communist or some other species of freethinker -- or an unspecified kind of freethinker. The details are flexible; what counts, however we formulate the details, is that our "story" is coherent and that it makes the premises true and the conclusion untrue.

Let's consider another argument:

        Anything boring is worthless.
        All philosophy classes are boring.
        Our class is a philosophy class.
So    Our class is worthless.

This has no counterexample. If we affirm the premises, then we cannot without lapsing into incoherence deny the conclusion. So the argument is valid. That, of course, doesn't mean it's a good argument in all respects. A sound argument must not only have valid reasoning but also true premises, and I hope that you will find the second premise of this argument to be false (the first is false as well).

Sometimes what appears to be a counterexample turns out on closer examination not to be. Unless the mistake is trivial (e.g., the story fails to make all the premises true or fails to make the conclusion untrue) the problem is often that the alleged counterexample is subtly incoherent and hence impossible. To return to the argument about Tori Amos, suppose someone said:

The argument is invalid because we can envision a situation in which all women are mortal and Tori Amos is a woman, but Tori Amos is nevertheless immortal because she has an immortal soul.

This story does seem to make the premises of the argument true and the conclusion false. But is it really intelligible? If having an immortal soul makes one immortal and Tori Amos has an immortal soul, then not all women are mortal. The story is simply incoherent; it contradicts itself. It is therefore not a genuine counterexample, since a counterexample is a possible situation; that is, its description must be conceptually coherent.

To summarize:  an argument succeeds in proving its conclusion if and only if it is sound--that is, it has true premises and valid reasoning.  To prove that an argument's reasoning is invalid, we give a counterexample.  If there is no counterexample, the reasoning is valid.  But keep in mind that valid reasoning alone is not enough to prove the conclusion.  To prove its conclusion, an argument must also have true premises.

Some additional invalid arguments with accompanying counterexamples are listed below. Keep in mind that invalid arguments generally have many counterexamples, so that the counterexamples presented here are not the only ones. Note also that each counterexample contains all three elements (though sometimes more than one element may be expressed by the same sentence). The three elements, once again, are:

In each case, the counterexample is a logically coherent story (not an argument) that shows how the conclusion could be untrue while the premises are true, thus proving that the argument is invalid.

INVALID ARGUMENT:
        Sandy is not a man.
So    Sandy is a woman

COUNTEREXAMPLE: Sandy is neither a man nor a woman but a hamster.
 

INVALID ARGUMENT:
        If the TV is unplugged, it doesn't work.
        The TV is not working.
So    It's unplugged.

COUNTEREXAMPLE: If the TV is unplugged it doesn't work, and it's not working. However, it is plugged in. The reason it's
not working is that there's a short in the circuitry.
 

INVALID ARGUMENT:
        All charged particles have mass.
        Neutrons are particles that have mass.
So    Neutrons are charged particles.

COUNTEREXAMPLE: All charged particles have mass, but so do some uncharged particles, including neutrons.
 

INVALID ARGUMENT:
        The winning ticket is number 540.
        Beth holds ticket number 539.
So    Beth does not hold the winning ticket.

COUNTEREXAMPLE: The winning ticket is number 540; Beth is holding both ticket 539 and ticket 540.

 
INVALID ARGUMENT:
        There is nobody in this room taller than Amy.
        Bill is in this room.
So    Bill is shorter than Amy.

COUNTEREXAMPLE: Bill and Amy are the only ones in this room and they are the same height.

 
INVALID ARGUMENT:
        Sally does not believe that Eve ate the apple.
So    Sally believes that Eve did not eat the apple.

COUNTEREXAMPLE: Sally has no opinion about the story of Eve. She doesn't believe that Eve ate the apple, but she doesn't disbelieve it either.

 
INVALID ARGUMENT:
        Some people smoke cigars.
        Some people smoke pipes.
So    Some people smoke both cigars and pipes.

COUNTEREXAMPLE: There are pipe-smokers and cigar-smokers, but nobody smokes both pipes and cigars, so that the two groups don't have any members in common.

 
INVALID ARGUMENT:
        Some people smoke cigars.
So    Some people do not smoke cigars.

COUNTEREXAMPLE: There are people, and all of them smoke cigars. (If everybody does, then some people do and so the premise is true!)
 

INVALID ARGUMENT:
        We need to raise some money for our club.
        Having a bake sale would raise money.
So    We should have a bake sale.

COUNTEREXAMPLE We need to raise money for the club, and having a bake sale would raise money, but so would other kinds of events, like a holding a car wash or a telethon. Some of these alternative fund-raising ideas better suit the needs of the club and the abilities of its members, and so they are what should be done instead of a bake sale.
 

INVALID ARGUMENT:
        Kate hit me first.
So    I had to hit her back.

COUNTEREXAMPLE: Kate hit the (obviously immature) arguer first. But the arguer could have turned the other cheek, or
simply walked away; there was no need to hit back.
 

EXERCISE: Classify the following arguments as valid or invalid. For those that are invalid, describe a counterexample, making sure that your description includes all three elements of a well-described counterexample. Take each argument as it stands; that is, don't alter the problem by, for example, adding premises.

1      No plants are sentient.
        All morally considerable things are sentient.
So    No plants are morally considerable.

2      All mathematical truths are knowable.
        All mathematical truths are eternal.
So    All that is knowable is eternal.

3      Most geniuses have been close to madness.
        Blake was a genius.
So    Blake was close to madness.

4      A high gasoline tax is the most effective way to reduce the trade deficit.
        We need to reduce the trade deficit.
So    We need a high gasoline tax.

5      Some angels are fallen.
So    Some angels are not fallen.

6      To know something is to be certain of it.
        We cannot be certain of anything.
So    We cannot know anything.

7      The surface area of China is smaller than the surface area of Russia.
So    The surface area of Russia is larger than the surface area of China.

8      Some men are mortal.
So    Some mortals are men.

9      The witnesses said that either one or two shots were fired at the vicitm.
        Two bullets were found in the victim's body.
So    Two shots were fired at the victim.

10     People do climb Mount Everest without oxygen tanks.
So    It is possible to climb Mount Everest without oxygen tanks.
 
11     Some fools are greedy.
        Some fools are lecherous.
So    There are some fools who are both lecherous and greedy.
 
12     No one has ever lived for 200 years.
So    No one ever will.

13     DNA contains the code of life.
        Life is sacred.
So    It is wrong to manipulate DNA.

14     There are fewer than a billion people in the whole United States.
        New York is only a part of the United States.
So    There aren't a billion people in New York.

ARGUMENT INDICATORS
            We have defined an argument as a sequence of declarative sentences, one of which is intended as a conclusion which the others, the premises, are intended to support.  We now consider the grammatical cues by which speakers of English communicate such intentions.  The most important of these are argument indicators, words or phrases that signal the presence and communicate the structure of arguments.  These fall into two classes:  premise indicators and conclusion indicators.  A premise indicator is an expression such as 'for', 'since' and 'because' which connects two statements, signifying that the one to which it is immediately attached is a premise from which the other is inferred as a conclusion.  So, for example, in the sentence:

                        The soul is indestructible because it is indivisible.

the premise indicator 'because' signals that the statement 'it is indivisible' (where 'it' refers to the soul) is a premise supporting the conclusion 'the soul is indestructible'.  Premise indicators can also occur at the beginnings of sentences, but the rule still holds:  the statement to which the premise indicator is attached is the premise; the other is the conclusion.  Hence, for example, in the sentence:

                        Since numbers are nonphysical, nonphysical objects exist

the word 'since' shows that the statement 'numbers are nonphysical' is a premise leading to the conclusion 'nonphysical objects exist'.
            Conclusion indicators are words or phrases that signify that the statement to which they are attached is a conclusion that follows from previously stated premises.  English is rich in conclusion indicators.  Some of the most common are 'therefore', 'thus', 'so', 'hence', 'then', 'it follows that', 'in conclusion', 'accordingly', and 'consequently'.  In the following argument, for example, 'hence' indicates that the third statement 'God exists' is a conclusion from the first two:

                                    Without God, there can be no morality.  Yet
                                    morality exists.  Hence God exists.

But the same thing can be signaled by a premise indicator:

                                    God exists, for without God there can be no morality,
                                    and morality exists.

or by a mix of premise and conclusion indicators:

                                    Without God there can be no morality.  Then God
                                    exists, since morality exists.

These are three different expressions of the same argument.  There are many others.  Notice that the conclusion (in this case 'God exists') may occur either at the end, at the beginning, or in the middle of the argument, depending on the arrangement of argument indicators.  All three positions are common in ordinary speech and writing.  But for logical analysis it is customary to list the premises first and the conclusion, prefixed by '...', last, as we have been doing.  This is called standard form.
            Arguments may also be stated without indicators, in which case we must rely on subtler clues of context, intonation, or order to discern their structure.  Most often when argument indicators are lacking the conclusion is given first, followed by the premises.  Here is an example:

                                    There is no truth without thought.  Truth is a
                                    correspondence between thought and reality.
                                    And a correspondence between two things cannot

                                    exist unless the things themselves exist.

Here the first statement is a conclusion from the remaining two.
            Like most English words, many of the terms we use for argument indicators have more than one meaning.  So not every occurrence of 'since', 'because', 'thus', and so on, is an argument indicator.  If someone says "I got down on my hand and knees and thus I escaped beneath the smoke," it is unlikely that they are offering an argument.  'Thus' here means "in this way", not "it follows that."  The speaker is not attempting to prove that she escaped.  Similarly, in the sentence 'Since the summer began, there hasn't been a drop of rain', the word 'since' indicates temporal duration, not a logical relationship between premise and conclusion.  Neither sentence is an argument.

            Sometimes arguments are not completely stated.  A premise may be omitted because it is so obvious that it need not be stated (or, more sinisterly, because the arguer is trying to get listeners to take it for granted without thinking).  Likewise, a conclusion may be omitted because it is very obvious, or because the arguer wants listeners to draw it for themselves, and thus perhaps be more inclined to accept it.  This argument, for example, has an implicit premise:

 

                        The moon has no atmosphere and therefore cannot support life.

 

The unstated premise is, of course, that an atmosphere is needed to support life.  (Notice also that the conclusion is only partly stated; its subject, having been mentioned already in the premise, is not repeated.)  The argument, stated in full, is:

 

                        An atmosphere is needed to support life.

                        The moon has no atmosphere.

            So        The moon cannot support life.

 

Here is an argument with an implicit conclusion:

 

                        Ailanthus trees have smooth bark, but the bark of this tree is rough.

 

The full argument is:

 

                        Ailanthus trees have smooth bark.

                        The bark of this tree is rough.

            So        This tree is not an ailanthus tree.

 

            Arguments are sometimes confused with conditional statements.  A conditional statement is an assertion that one thing is the case if another thing is, for example: 

 

                        If three is an even number, then it is divisible by two.

 

A conditional asserts neither of its components.  This statement, for example, asserts neither that three is even nor that it is divisible by two, but the latter is the case if the former is.  In this way it differs significantly from this argument, which is formed from the same components:

 

                        Since three is an even number, it is divisible by two.

 

In an argument, both the premises and the conclusion are categorically asserted.  A person who utters these words is saying (absurdly) that three is even and that three is divisible by two.

            The point here is that 'if' and certain related terms, such as 'unless' and 'only if', are not premise indicators.  Instead, they form compounds which function as single statements. 

 

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