What
we have covered in this first part of the course is how we symbolize
and evaluate the kinds of statements that can be called definitely true
or false. Symbolic logic, at least as we have seen it so far,
will not apply to statements that express only possibility or
probability (for instance, "it might rain tomorrow," since we could say
without contradicting ourselves that it might not) or value judgments
("Alice ought to study tonight," since whether this is "true" or not
depends on the way in which we look at a situation: from one angle she
should if she wants to pass, but from another angle it's not true that
she should if it means staying home and not going to her job).
We saw that we can take whole statements and use single letters to represent them. S could stand for the whole idea that Alice studies hard, for instance, and P could stand for the whole idea that Alice passes her test. What we then do is combine these atomic expressions in any of several ways to form new expressions.
We then learned how we would use truth tables to decide the value of the main connective in any of these new expressions. The table below indicates the possibilities for the five examples above.
Our first use of truth tables was to test whether any two expressions could be said to have the same logical relationship (meaning, that we say they are equivalent and so one could be substituted for the other without changing the information in a story. For instance, if you look at the table above you see that S v ~P and P->S do tell the same story, in effect letting us know that study is a requirement for passing.
The one operator that gives students the most trouble is
the
arrow. In expressing a conditional relationship we cannot just go
by the English word order but we have to think about the exact
relationship expressed. "Logic is easy if it is fun" and "if
logic is fun then it is easy" both would be expressed as F->E so that we see that
being easy is the result and being fun is the sufficient condition
that, if met, gives us that result. In the same way "Only if
Alice studies hard will she pass" and "Alice will pass only if she
studies hard" are both P->S,
with study a necessary condition (and, yes, if we negate the antecedent
and use a wedge to get ~P v S--"Alice
will not pass unless she studies hard"--we have another way of
expressing that condition).
We also saw that we can atomic statements that express something we say is true about an individual when we use a capital letter for the predicate and a small letter for the name. We can have Sa for "Alice studies hard" and Pa for "Alice passes" and we could have the same truth table as we did before.
We could mix and match these two types of notation, but ordinarily we don't. The reason that we do have these predicate expressions is that for much of our reasoning we are interested in statements such as "everyone studies" or "everyone who studies will pass." We use what we call quantifiers to express the idea that we are talking about everyone in a group or just some (at least one individual) in a group. We can then reason from generalizations such as "Everyone who studies will pass" (x)(Sx->Px) to a particular statement about Alice (Sa -> Pa) and if we know it's true that Alice does study we can then reach the conclusion that she indeed will pass. How we do this will be a major part of the course still ahead.
For what you need to know now, make sure you understand how to symbolize using both universal and existential quantifiers. The following examples are models for what you need to do on the first review test.
Keep in mind that x (or the other variables y and z) is not actually a name in the same way as the a in Sa. What these letters do is allow us to point to a group and indicate we are talking about potentially anyone at all or just about some individual(s) not identified. A universally quantified expression will be be true when there are no exceptions, and an existentially quantified expression is true when there is at least one instance where it applies. We could, of course, set up a truth table if we have only a few individuals in a group to talk about, but, as we will see later, there are other ways to test whether expressions are equivalent or whether one expression has to follow from one or more others.
A final bit of symbolization, which we will make use of much later, involves setting up relationships among individuals by using more than one name.
Fab could symbolize "Al is Bob's father" and ~Fba could symbolize "Bob is not Al's father."
~(x)(Ey)Fxy could symbolize "Not everyone is someone's father." Later we will see how we turn this into the expression (Ex)(y)~Fxy, telling us that there is someone who is no one's father.
By now you should be ready to do three things: (1) symbolize various English sentences using the suggested letters, (2) decide the truth value of any given expression, and (3) use truth tables to decide whether or not two expressions are equivalent.
We saw that we can take whole statements and use single letters to represent them. S could stand for the whole idea that Alice studies hard, for instance, and P could stand for the whole idea that Alice passes her test. What we then do is combine these atomic expressions in any of several ways to form new expressions.
S & P would
represent the statement "Alice does study hard and passes her
test." conjunction
S v ~P would represent the statement "Either Alice studies hard or she will not pass." what we call inclusive disjunction, allowing both to be true
S->P would represent the statement "If Alice studies hard she will pass" (making study a sufficient condition for passing, although the door is open to the possibiity that she could also pass some other way). implication ( a conditional)
P->S would represent the statement "Alice will pass only if she studies hard" (making study a necessary condition for passing, although it could be true that study alone might not be enough). implication (a conditional)
P<->S would represent the statement "Alice will pass if and only if she studies hard" (making study both a necessary and a sufficient condition for passing). equivalence (a biconditional)
S v ~P would represent the statement "Either Alice studies hard or she will not pass." what we call inclusive disjunction, allowing both to be true
S->P would represent the statement "If Alice studies hard she will pass" (making study a sufficient condition for passing, although the door is open to the possibiity that she could also pass some other way). implication ( a conditional)
P->S would represent the statement "Alice will pass only if she studies hard" (making study a necessary condition for passing, although it could be true that study alone might not be enough). implication (a conditional)
P<->S would represent the statement "Alice will pass if and only if she studies hard" (making study both a necessary and a sufficient condition for passing). equivalence (a biconditional)
We then learned how we would use truth tables to decide the value of the main connective in any of these new expressions. The table below indicates the possibilities for the five examples above.
| P |
S |
S
& P |
S
v ~P |
S->P |
P->S |
P<->S |
| 1 |
1 |
1 |
1 |
1 |
1 |
1 |
| 1 |
0 |
0 |
0 |
1 |
0 |
0 |
| 0 |
1 |
0 |
1 |
0 |
1 |
0 |
| 0 |
0 |
0 |
1 |
1 |
1 |
1 |
Our first use of truth tables was to test whether any two expressions could be said to have the same logical relationship (meaning, that we say they are equivalent and so one could be substituted for the other without changing the information in a story. For instance, if you look at the table above you see that S v ~P and P->S do tell the same story, in effect letting us know that study is a requirement for passing.
The one operator that gives students the most trouble is
the
arrow. In expressing a conditional relationship we cannot just go
by the English word order but we have to think about the exact
relationship expressed. "Logic is easy if it is fun" and "if
logic is fun then it is easy" both would be expressed as F->E so that we see that
being easy is the result and being fun is the sufficient condition
that, if met, gives us that result. In the same way "Only if
Alice studies hard will she pass" and "Alice will pass only if she
studies hard" are both P->S,
with study a necessary condition (and, yes, if we negate the antecedent
and use a wedge to get ~P v S--"Alice
will not pass unless she studies hard"--we have another way of
expressing that condition).We also saw that we can atomic statements that express something we say is true about an individual when we use a capital letter for the predicate and a small letter for the name. We can have Sa for "Alice studies hard" and Pa for "Alice passes" and we could have the same truth table as we did before.
| Pa |
Sa |
Sa
& Pa |
Sa
v ~Pa |
Sa
-> Pa |
Pa
-> Sa |
Pa
<-> Sa |
| 1 |
1 |
1 |
1 |
1 |
1 |
1 |
| 1 |
0 |
0 |
0 |
1 |
0 |
0 |
| 0 |
1 |
0 |
1 |
0 |
1 |
0 |
| 0 |
0 |
0 |
1 |
1 |
1 |
1 |
We could mix and match these two types of notation, but ordinarily we don't. The reason that we do have these predicate expressions is that for much of our reasoning we are interested in statements such as "everyone studies" or "everyone who studies will pass." We use what we call quantifiers to express the idea that we are talking about everyone in a group or just some (at least one individual) in a group. We can then reason from generalizations such as "Everyone who studies will pass" (x)(Sx->Px) to a particular statement about Alice (Sa -> Pa) and if we know it's true that Alice does study we can then reach the conclusion that she indeed will pass. How we do this will be a major part of the course still ahead.
For what you need to know now, make sure you understand how to symbolize using both universal and existential quantifiers. The following examples are models for what you need to do on the first review test.
Everyone
studies. (x)Sx
No one studies. (x)~Sx
Not everyone studies. ~(x)Sx
Some individuals pass. (Ex)Px
Some individuals do not pass. (Ex)~Px
Everyone who studies will pass. (x)(Sx -> Px) we use an arrow to express what we now think of as a conditional relationship
No one lazy will pass. (x)(Lx -> ~Px)
Anyone not lazy will pass. (x)(~Lx -> Px)
Only someone who studies will pass. (x)(Px -> Sx) think about the importance of having the correct order here so that we see study as a necessary condition
Some who study do pass. (Ex)(Sx & Px) we use an ampersand to state that both things are true about person x
There are individuals who study but do not pass. (Ex)(Sx & ~Px)
No one studies. (x)~Sx
Not everyone studies. ~(x)Sx
Some individuals pass. (Ex)Px
Some individuals do not pass. (Ex)~Px
Everyone who studies will pass. (x)(Sx -> Px) we use an arrow to express what we now think of as a conditional relationship
No one lazy will pass. (x)(Lx -> ~Px)
Anyone not lazy will pass. (x)(~Lx -> Px)
Only someone who studies will pass. (x)(Px -> Sx) think about the importance of having the correct order here so that we see study as a necessary condition
Some who study do pass. (Ex)(Sx & Px) we use an ampersand to state that both things are true about person x
There are individuals who study but do not pass. (Ex)(Sx & ~Px)
Keep in mind that x (or the other variables y and z) is not actually a name in the same way as the a in Sa. What these letters do is allow us to point to a group and indicate we are talking about potentially anyone at all or just about some individual(s) not identified. A universally quantified expression will be be true when there are no exceptions, and an existentially quantified expression is true when there is at least one instance where it applies. We could, of course, set up a truth table if we have only a few individuals in a group to talk about, but, as we will see later, there are other ways to test whether expressions are equivalent or whether one expression has to follow from one or more others.
A final bit of symbolization, which we will make use of much later, involves setting up relationships among individuals by using more than one name.
Fab could symbolize "Al is Bob's father" and ~Fba could symbolize "Bob is not Al's father."
~(x)(Ey)Fxy could symbolize "Not everyone is someone's father." Later we will see how we turn this into the expression (Ex)(y)~Fxy, telling us that there is someone who is no one's father.
By now you should be ready to do three things: (1) symbolize various English sentences using the suggested letters, (2) decide the truth value of any given expression, and (3) use truth tables to decide whether or not two expressions are equivalent.
