Introduction to Logic
8 Symbolic Logic
The starting point for appreciating symbolic logic is the appreciation of the difference between simple statements and compound statements. You might have thought it would be some symbols, but symbols are only going to be useful once we are clear on what we are symbolizing.
This is more important psychologically than it may appear, because some people who think that mathematics is scary think that symbolic logic is scary too. That’s because they conflate math and logic. But in Logic we are using symbols not to identify numbers or operations done to numbers, but to identify meanings, words, statements; things that everyone of us deals with, uses, and mentions every day.
“Symbolic logic” is just an extension of the kinds of abbreviations we already learned to use in dealing with Categorical logic: we stopped saying “All S is P,” and started saying “ASP,” letting the “A” symbolize the fact that “S” and “P” were pieces of a universal affirmative proposition. Abbreviations are not so scary.
So we’ll be using symbols to refer to elements of meaning, and the big difference here is that we’re now dealing with statements rather than with the elements of statements (categories and quantifying words). Here at the outset, the smallest meaningful elements will either be simple statements or what we call “operators,” which means meanings that can be applied to statements, usually to join them to other ones.
A simple statement is a single assertion or declaration. “Washington is dead” is a simple statement. “Washington was assassinated” is another one. These two simple statements can be connected by the operator or meaning “not,” or by “if…then,” or by the meaning “either…or,” or by the meaning “…and…,” for instance.
The convention in Logic is that simple statements are affirmative statements. Any statement that strikes us as making a denial of something (like “No Popes are Hindus”) will be understood to be a compound statement rather than a simple one; we’ll say that it consists of two elements: the simple affirmative statement plus a negation or denial. We’d represent it as having two parts: “not” and “P” (where “P” would stand for “Popes are Hindus”).
Capital letters, by conventions that logicians uphold universally, stand for complete but simple statements.
It’s raining: R
Colbert’s a gifted comedian: C
I got a Les Paul for my birthday: L
We then introduce five symbols to act upon or connect these capital letters, to represent more complex statements, such as statements made up of multiple statements.
The mark “ ~ ” known as “tilde,” stands for “not,” or for any form of negation in general. We put it in front of a capital letter to exhibit that the simple statement has been negated. For instance:
Colbert was not very funny tonight
(as long as we agreed to represent “Colbert was funny tonight” by “F”). The tilde will not stand only for “not,” it will also stand for “it’s false that,” “it is not the case that,” “no,” and any number of other ways that a denial or negation may be expressed.
Washington’s not dead
By convention, a tilde affects the unit it comes before. We’ll see some significance to this in a moment. All the other symbols connect one statement to another.
The simplest connecting symbol is the one for Conjunction, which means “adding one thing to another.” Now, remember that this is not about mathematics, so this sense of “adding” is not the same as finding a sum.
Consider this statement:
Martin Luther King was a civil rights leaders and James Farmer was too.
This takes two individual statements (1. “Martin Luther King was a civil rights leader”; 2. “James Farmer was a civil rights leader”) and links them together, saying that both statements are true.
The sense of “and” we are talking about here in Logic is pretty broad. What we mean by it here is that as long as each of the two distinct statements correctly claims to be true, the statement that results from linking them together with a word like “and” will also be true. So:
The US and Mexico are in the western hemisphere
is certainly a true conjunction.
“And” is the paradigm for conjunction in English, but it is not the only word that operates this way, i.e., there are other terms that can join two true statements into an assertion of both and still be true. “But,” “although,” “however” also do this. So does “also.” So does the semi-colon. Consider these compounds:
New York is on the east coast, but so is Georgia.
New York is on the east coast; Georgia is too.
New York is on the east coast, however, so is Georgia.
New York is on the east coast, although Georgia is as well.
Logically, these are all equivalent to
New York and Georgia are on the east coast.
We’d symbolize all of them like this:
N ∙ G
That symbol between the “N” and the “G” is called a dot; it’s a period, but raised above the line. (Unfortunately, standard keyboards do not have this raised period, so in typed text, you’re going to see a dot on the line, but when you or I write it by hand, we’ll raise it above the line.) Over the course of the history of symbolic logic, which is only about a century long, there have been a variety of symbols used by different logicians. When Bertrand Russell and Alfred North Whitehead introduced symbols for Logic, they used an upside down “v” to stand for “and,” so they might have written “N ^ G.” The dot has come to be the norm today, however. If you work on the Bluestorm/ TheLogicCourse site, you’ll see they use an “&” for conjunction. That should not give you pause.
One more comment on “and” is that in everyday life it doesn’t really always mean “and,” i.e., conjunction. Consider the everyday expression “You do that again and you’ll be sorry.” You have to be attentive to what’s going on in these little words; this statement’s not conjoining two statements, it’s asserting a conditional. It means “If you do that again, you’ll be sorry.”
Unlike that example, the way we’ll be using it in Logic, “and” is what’s called a truth-functional operator. That means that the truth-value of an “and” statement is a function of the truth-value of the pieces that make it up. The rule for “and” is quite intuitive: an “and” statement is true only if both of the statements joined by “and” are true. If either one (or both) of them is false, then the conjunction itself is false.
If we put parentheses around “N ∙ G,” this will indicate that that statement is being treated as a unit, and so the following symbolizations will not mean the same thing:
a) ~N ∙ G
b) ~ (N ∙ G)
a) says New York’s not on the east coast but Georgia is, and
b) says Not both New York and Georgia are on the east coast.
The tilde affects the unit that follows it immediately, so the parentheses matter: they determine, in this case, the size of the unit that is being negated. These two statements don’t mean the same thing, but they are false for different reasons. The important thing to see is that in “a,” the tilde affects the statement “N,” whereas in “b,” it affects the dot, the word “and.”
The upside-up “v” (i.e., the regular old “v”), known in Logic as a “wedge,” has come to be standard to represent the meaning of “either…or…,” which is also known as Disjunction. You’ve heard of “driving a wedge between things,” I bet; so you can draw on that to remember the name of the symbol for the Disjunction. When you drive a wedge between things or people, you split them off from each other, and that is also the function of “or” when you are told by your mother:
You can either have Thai food or Chinese.
That means “not both.” “One or the other” means “not both.”
This is actually only one of two meanings that “or” has in English (in fact, we’ll see a third soon enough). This meaning is called the strong or exclusive “or,” because it sets two things completely apart. But there is also a very common weaker meaning of “or,” one that allows for the possibility of both things being acceptable or true, and we often enough use the expression “and/or” to make this clear. An example would be
You can take Logic or Statistics for the Quantitative Reasoning requirement.
Taking Logic is an option; taking Stats is also an option; you can even do both. The “or” here is inclusive of both possibilities.
Absences are excused in case of illness or emergency
is not meant to exclude the possibility that someone might fall ill suddenly and violently; she would still be excused.
So these are different “or’s” from the “or” of
Everyone here is either a Democrat or a Republican,
which means that no one here is both.
Now let’s put these last four statements into symbols:
T v C You can either have Thai or Chinese.
L v S You can take Logic or Stats.
I v E Absences are excused for illness or emergency.
D v R Everyone is either a Democrat or a Republican
You cannot tell from the symbolization that some of these are weak “or’s” and some are strong “or’s.” So we seem to have a dilemma: if we don’t want to treat a weak “or” as the same as a strong “or,” we shouldn’t use the wedge for both.
The solution will be to remember what ~(N . G) meant above: “not both.” With “not both,” we can clear up any ambiguity that “either…or” might create, i.e., we can make a strong “or” explicit by tacking “and not both” onto a weak one.
So if the context required that we clear away the ambiguity, we could leave the “Logic” and “emergency” statements as they are, but change the others to look like this:
(T v C) ∙ ~(T ∙ C)
You can have either Thai or Chinese, but not both Thai and Chinese.
(D v R) ∙ ~(D ∙ R)
Everyone here is either a Democrat or a Republican, but not both a Democrat and a Republican.
It should be clear that the truth-functional definition of “or” will be the one that fits the weak or inclusive “or.” A disjunction is false if and only if both of its disjuncts are false; otherwise it is true (i.e., as long as at least one disjunct is true). This is why some tautologies (we’ll talk about them soon enough) are true, since they say things like “Either it’s too late or it’s not too late.”
Let’s consider another example:
Neither Arthur nor Chloe can tie their own shoes.
There are two ways you might approach understanding this. You might emphasize the role of “neither,” and write it as a negated disjunction: ~ ( A v C ). But you might also break it down into two negations, and understand it as
Arthur can’t tie his own shoes and neither can Chloe.
Which is the same as:
Arthur can’t tie his own shoes and Chloe can’t either.
~A ∙ ~C
These are strictly equivalent, and we’ll prove it to you in a little while with a truth table. For now, let’s describe what each is in its own right, and appreciate that they say the same thing differently: a negative disjunction says the same thing as a conjunction of negatives.
The main operator of the two statements is different:
~ (A v C) the main operator is the tilde; it’s a negative disjunction: “neither…nor.”
~A ∙ ~C the main operator is the dot; it’s a conjunction of negatives: “not the one and not the other.”
Although in English most disjunction is expressed with either “either…or” or just plain “or,” it can also be expressed by “unless.”
It will rain unless it gets a lot colder.
In this statement, it may not appear that an “either…or” is at work; it may appear that an “if not…” is:
It will rain if it does not get a lot colder.
And that really means just the same thing:
Either it will rain or it will get a lot colder.
So “or” has three distinguishable meanings: exclusive “or,” inclusive “or,” and “if not.”
And that brings us to “if.”
“If,” of course, is really shorthand for “if…then,” which we know is the form of the conditional statement. We call the “if” clause the antecedent, and the “then” clause the consequent; and we know from earlier in this course that the antecedent can be read as expressing a sufficient condition for the consequent, while the consequent can be read as expressing a necessary condition for the antecedent.
The standard written symbol for the conditional is a horseshoe pointing to the right. But keyboards have arrows (>) on them rather than horseshoes, and they are close enough that we can tell they mean the same thing. The horseshoe or arrow goes between the antecedent and the consequent, where the word “then” would be expected. It’s important to get clear on this, because in the symbolization of a conditional, the antecedent has to come first, but in ordinary language, we don’t always have antecedents preceding consequents. Like this:
Scott can take the trip if Betty lends him some cash.
This would have to be symbolized as B > S, not S > B, because the statement about Scott is really the consequent, even though it came first.
“Given that,” “on condition that,” and “in case” are three other ways to express the conditional relationship between two statements in English. “If” is certainly the most common, but given that there are others, you can’t count on it.
Note that in that last sentence, “given that” didn’t mean “if,” it meant “since.” “Given that” can indicate either an antecedent or it can indicate a premise –determining which is a matter of understanding the statement in context. “Given that there are others, you can’t count on it” is an argument. Probably the best way to identify this argument would be as what we’ve been calling a hypothetical syllogism up to this point, by spelling it out like this:
If a word has more than one meaning, you can’t count on it meaning the same thing every time; “given that” has more than one meaning, so you can’t count on it meaning the same thing every time.
Rather few people ever consider the meaning of “if,” but if you do, you’ll find that it’s quite a tricky word. “If” sets up a relation between statements which can be intended and understood in quite different ways. For instance, these conditionals don’t mean the same thing by “if,” if you stop and think about it:
If the Pope’s not Catholic, I’m a monkey’s uncle.
If Watson has chalk on his fingers, he played billiards.
The first of these is the kind of conditional we can deal with in Logic; maybe you can see how it’s different from the second. The second is about cause and effect: such-and-such conditions will bring about such-and-such an effect. The first is about the compatibility of statements; the point of the conditional is that the Pope is Catholic. The person who says “If the Pope’s not Catholic I’m a monkey’s uncle,” is just trying to make it clear that he thinks the Pope is Catholic. This conditional implies an argument we’ll be calling “modus tollens,” and which is quite common and obvious: If he’s not Catholic, I’m a monkey’s uncle. But I’m not a monkey’s uncle, so obviously he’s a Catholic.
Another way to show the difference would be by appreciating the logical equivalence of a conditional with a disjunction in which one statement is negated: you could rewrite If the Pope’s not Catholic, I’m a monkey’s uncle as Either the Pope’s Catholic or else I’m a monkey’s uncle.
But a similar rewrite of the causal conditional would fail to capture its point: Either Watson doesn’t have chalk on his fingers or else he played billiards just doesn’t capture the sense of If Watson has chalk on his fingers, he played billiards. Either Watson didn’t play billiards or else he has chalk on his hands comes a bit closer, but is still much more difficult to understand than the “if…then…” construction.
What we deal with in Logic are called “material conditionals,” and they are different from causal ones.
The specific nature of the material conditional can be brought out convincingly by the following example, which will also provide us with the truth-functional definition of conditionals. (I owe this example to Hurley; it’s one of the best things in his Concise Introduction to Logic.) If you can come up with another example that’s as good as this, it’s worth extra credit.
Let’s say this statement appears on the syllabus for one of your classes:
If you get an A on the final, you’ll get an A in the course.
There are four possibilities, right?
1. You get an A on the final, and you get an A in the course. You’re happy.
2. You get an A on the final, but you don’t get an A in the course. You’re not happy; in fact you feel betrayed: “he lied!”
3. You don’t get an A on the final, but you do get an A in the course. You’re happy.
4. You don’t get an A on the final, and you don’t get an A in the course. That’s ok; what did you expect?
The moral of this example is that a Material Conditional statement is true unless its antecedent is true while its consequent is false. That is, it’s true even when its antecedent is false. That’s why “If the Pope’s not Catholic I’m a monkey’s uncle” works: even though both antecedent and consequent are false, it has the effect of being a true conditional. Another good example: “If I’m 25 feet tall I can touch the ceiling in 106A.” I’m not that tall, and I cannot touch the ceiling in 106A, but the statement is clearly true.
But some conditionals are glaringly inappropriate to treat or interpret this way. For instance, “If Chicago is in Florida, it’s near San Diego.” Obviously this is a false statement, and both its antecedent and consequent are false. It would not do to claim this is a true statement. This is not a Material Conditional.
IF AND ONLY IF
So that’s part of the can of worms that “if” offers. Another is its tempting similarity to “only if.” Most people probably think, when they hear “only if,” that they are just hearing “if.” But they are mistaken. “If” and “only if” mean two distinct things (and “if and only if” means something else again –please be careful with the quotation marks!)
This pair of statements might make the point visible:
The car runs if there’s gas in the tank.
The car runs only if there’s gas in the tank.
If you can tell intuitively which of these is “more true,” that’s the start.
It would be the second one.
“Only if” is saying that gas is a necessary condition for the car to run. The first sentence might “make do” as we say in everyday life, but to really be adequate, it would have to be continued with a series of other conditions “…and the alternator belt is on, and the spark plugs are attached, and the battery is charged, etc., etc.” This is because (as we said in the beginning of the course when we identified conditionals as non-arguments) the antecedent of a conditional expresses a sufficient condition, and much more than just gas in the tank makes the car run. But you can be certain that there’s gas in the tank if the car is running.
So The car runs ONLY IF there’s gas in the tank means the same thing as IF the car runs, there’s gas in the tank.
These are equivalent statements, but notice that in the one, “only if” is introducing the consequent, and in the other, “if” is introducing the antecedent. That’s because “if” and “only if” really do not mean the same thing.
“If” introduces a sufficient condition, but “only if” introduces a necessary one. It’s very important to keep this straight. As a practical piece of advice, remember this: always look to see if “if” is accompanied by “only.” If it is, then it is not really “if,” it is “only if,” and must be treated as such.
This leads naturally enough to the question “What do people mean when they say “if and only if”?
Well, that depends on them, but what they should mean can be made clear by the following:
Someone might use the expression like this: I’ll buy you that new guitar if and only if you promise to practice every single day.
I doubt that the speaker of this statement meant: If I buy you that guitar you promise to practice every day, and if you promise to practice every day I will buy you that guitar. All he or she really meant was “only if,” and he or she added the “if and” as a way of being emphatic.
A correct use of the expression is something like
You are President if and only if you are commander-in-chief
A conjunction is true if and only if both of its conjuncts are true.
These are called genuine bi-conditionals, and they each assert that one thing is both a necessary and a sufficient condition for the other:
If you are President then you are Commander-in-Chief, and if you are Commander-in-Chief, then you are President.
If the conjunction is true, then both its conjuncts are true, and if both its conjuncts are true, then the conjunction is true.
The symbol for a bi-conditional is a triple bar, ≡. It looks like an equal sign with a third line. Since our keyboards don’t have the triple bar symbol, when you are working on a computer, just use the equal sign, “=.” But when you are writing by hand, remember to add the third bar to it.
That triple bar is an abbreviation for a conjunction of conditionals (hence the name “bi-conditional”).
(P > C) ∙ (C > P) is the conjunction of conditionals that P ≡ C says in one symbol.
Given that we’ve introduced the truth-functional definitions of the operators as we’ve introduced the operators themselves, we can use them to determine what the truth-functional definition of the triple bar is, since after all, the triple bar is just a way of symbolizing in one symbol what can be symbolized using a dot and two horseshoes.
Truth-Table Definitions of Truth-Functional Operators
The tables below sum up the truth functional definitions so far. As you review their meanings in the following presentation of the truth tables, see if you can anticipate how to use them to build up the table that will provide the truth-functional definition of the triple bar.
Tilde/ Negation. This table says that the negation of any statement has the opposite truth-value of that statement. It says this under the tilde.
Dot/ Conjunction. This one says that a conjunction of statements is true if and only if both statements are true. Look under the dot to see this. If you were filling in this table, you would fill in the T T F F under the “p” first; then the T F T F under the “q,” and then you would fill in the values for the dot.
Wedge/ Disjunction. This one says that a disjunction is true unless (strong “or”) both disjuncts are false. Another way to say that is: a disjunction is false if and only if both of its parts are false.
Horseshoe (arrow)/ Conditional. This one says that a conditional statement is true unless its antecedent is true but its consequent is false.
Triple bar/ Bi-conditional. And this one says that triple bar statement is true if and only if its parts have the same truth value. They don’t have to both be true, they have to either both be true or both be false, for the triple bar itself to be true. The triple bar is another way of saying two statements are equivalent in truth value to each other. With the information from these tables (actually only from two of them) you should be able to fill in the boxes on the table below, which is the biconditional equivalent to the triple bar.
Begin by writing T T F F under “p,” and T F T F under “q.” Since the table has four lines or rows, the left-most letter gets 2 “t’s” and then2 “f’s.” If the table had 8 rows, this left-most letter would get 4 of each; if the table had 16 rows, the left-most letter would get 8 “trues” and 8 “falses.” The next letter in gets half that amount, and the next one again gets half of that amount. Of course for these latter letters you would have to repeat the pattern until you filled in the table to the bottom.
Fill in the values under the arrows/ horseshoes first, then the value under the dot (which joins the values of the horseshoes).
These links will open powerpoints that review this material:
This version is accompanied by narration: