PHIL 4160: Symbolic Logic
Dr. Robert Lane
Lecture Notes: Tuesday January 7, 2014


[1.] Introduction: Arguments


Some of the material in this section of notes, as well as in the next few sections, should be familiar to you from Critical Thinking (PHIL 2020), which is a prerequisite for this class.


[1.1.] Logic.


The word “logic” has more than one meaning[1]:


1. the discipline of logic: the area of inquiry that studies reasoning and arguments; it is particularly concerned with what distinguishes good arguments from bad ones. (In this sense of the word “logic,” it refers to one of the four traditional branches of philosophy, along with metaphysics, epistemology and ethics.)


2. a symbolic logic = an artificial language (i.e., a language deliberately created by human beings, as opposed to “natural languages” like English, French, Japanese, Urdu, etc.) the purpose of which is to help us reason in a clearer and more precise way. (The creation and study of symbolic logics is a part of the discipline of logic.)


This course deals with logic in both senses of the word. We will use a symbolic logic—an artificial language—in order to study reasoning and arguments, with the goal of distinguishing between good arguments and bad arguments.



[1.2.] Arguments, Validity and Symbolism.


The discipline of logic, including the study of symbolic logics, involves the evaluation of arguments, i.e., examining arguments to distinguish the good ones from the bad ones.


Logicians do not use the word “argument” to mean a heated, angry exchange of words. Rather, they mean:

argument (df.): a set of statements some of which (the argument’s premises) are intended to serve as evidence or reasons for thinking that another statement (the argument’s conclusion) is true. In the textbook’s words: “a set of statements, one of which is the conclusion (the thing argued for) and the others are the premises (reasons for accepting the conclusion).” (p.1)


Here is a simple example of an argument:


All men are mortal
Socrates is a man.
Therefore, Socrates is mortal.


Here is another:


Some women are US Senators.

Elizabeth Warren is a woman.

Therefore, Elizabeth Warren is a US Senator.


The first argument has a desirable logical property that the second one lacks: validity.


validity (df.): A valid argument is one in which the truth of the premises would guarantee the truth of the conclusion [this is an incomplete definition; we'll get to a full definition in the next class.]


It is essential to understand that validity does not require true premises. An argument’s validity has nothing to do with whether its premises are actually true or false. As an illustration, consider the following valid argument:


All astronauts are cannibals. (false)
Lane is an astronaut. (false)
Therefore, Lane is a cannibal.


This argument is valid, even though it has two false premises. It is valid because, if both of its premises were true, then the conclusion would have to be true as well.


One of the central goals of this class is to help you distinguish arguments that are valid from arguments that are not.


You will learn how to do this by translating arguments into symbolic logic. For example, the Socrates argument translated into the system of symbolic logic contained in your textbook looks like this:

(x)(Hx É Mx)
Hs   /\ Ms

You will learn a system of rules for deriving conclusions from premises within this system of symbolic logic. If you can derive a given conclusion from given premises according to those rules, then the argument in question is valid. The conclusion of the Socrates argument can be derived from its premises by those rules:



1. (x)(Hx É Mx)                      p
2. Hs                                        p  / \Ms

3. Hs É Ms                              1 UI

4. Ms                                       2, 3 MP

[the left-hand column shows the premises and conclusion; the right-hand column indicates which rule has been applied to generate each line.]


…but the conclusion of the Elizabeth Warren argument cannot be.


You are not expected to understand any of this symbolism today. It is this language of symbolic logic that you will be learning in this course.



Here is another example of an argument, from p.1 of your textbook:


Identical twins often have different IQ test scores. Yet such twins inherit the same genes. So environment must play some part in determining IQ. (p.1)


The first two statements are the premises; the last is the conclusion. We can indicate the difference between premises and conclusion by exhibiting each statement of the argument on a different line, as your textbook does:


1.       Identical twins often have different IQ test scores.

2.       Identical twins inherit the same genes.

3.       So environment must play some part in determining IQ. (p.1)


Notice that one and the same argument can be formatted in different ways. This IQ argument was first formatted as prose (like you might find it in, e.g., a magazine article) and then formatted in a more formal way. Sometimes this more formal way of exhibiting an argument, with each statement numbered and on a separate line, premises first and conclusion last, is called standard form.[2]



[1.3.] Indicator Words.


Notice the word “so” in the IQ argument. This is one of many English language words that frequently (but not always) indicate what role a given statement plays in an argument. Some others are:[3]


premise indicators

conclusion indicators

because                        as indicated by

since                in that

for                   may be inferred from

as                     given that

seeing that        for the reason that

inasmuch as     owing to

therefore                      as a result

wherefore                    for this reason

thus                              it follows that

so                                 implies that

consequently               we may infer

accordingly                  we may conclude



The second argument featured on p.1 includes more of these “indicator words”:


Since it’s wrong to kill a human being, it follows that abortion is wrong, because abortion takes the life of (kills) a human being. (p.1)


Notice that the conclusion does not have to be the last statement to occur within the argument as the argument is stated or printed. When an argument appears in prose form (rather than in standard form), its conclusion can appear before, after, or among its premises. So do not assume that a statement appearing at the end of an argument in prose form is the conclusion.


Here is the abortion argument in standard form (p.2):


    1. It’s wrong to kill a human being.

    2. Abortion takes the life of (kills) a human being.

\ 3. Abortion is wrong.


Notice that \ is being used as a symbol for the word “therefore.”



[1.4.] Statements and Evidence.


The definition of “argument” given above specifies that all the sentences in an argument must express statements. In other words, they must be declarative sentences (not questions, commands, or exclamations) and thus eligible to be true or false.[4]


The definition of “argument” also specifies that even if every sentence in a group of sentences expresses a statement, that group of sentences does not necessarily constitute an argument.


For an argument, something else is needed: the sentences have to be connected in a specific way, viz. some of them must be intended as evidence for, or reasons for accepting, another.


This is the difference between arguments, on the one hand, and exposition and explanation, on the other.




EXAMPLES 1-4 (pp.2-3)

·         note that “because” indicates a premise in #2, but not in #1.

·         note that “therefore” indicates a conclusion in #4, but not in #3.


Exercise 1-1 (pp.3-4)

·         if it is an argument, then write it out in standard form, indicating which statements are premises and which is a conclusion (paraphrasing is OK); if it is not, then just write “NO”

·         check your answers to the even-numbered questions against the back of the book; we will go through the odd-numbered questions at the beginning of the next class.



Stopping point for Tuesday January 7. For next time, complete exercise 1-1 and read all of chapter 1.


[1] I adapt this distinction from Wilfrid Hodges, “Classical Logic I: First-Order Logic,” in The Blackwell Guide to Philosophical Logic, ed. Lou Goble, Malden, MA: Blackwell, 2001, 9-32, p.9.

[2] Your textbook uses the concept of standard form only in a narrower sense, in which it describes only syllogisms; see p.367.


[3] These lists are from Hurley, Concise Introduction to Logic, p.3.

[4] A more technical definition of “statement” is this: the content expressed by (i.e., what is said by) a declarative sentence. There has been a lot of debate among philosophers about what sorts of things are capable of truth and falsity: sentences, statements, or propositions. See for example Susan Haack, Philosophy of Logics (1978) ch.6.

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