QUANTIFIERS AND LOGICAL INFERENCE
Adapted from Patrick J. Hurley, A Concise Introduction to Logic (Belmont: Thomson Wadsworth, 2008).
Predicate Logic
To introduce quantificational logic, let me address different ways we can symbolize statements in logic. Previously, we used one letter to symbolize a statement (e.g. P=Bananas are yellow). This kind of rendering is based on “propositional logic”. It takes the whole proposition as the basic logical unit. But there is another way to symbolize statements that differentiates the subject and predicate, which is pertinent to “quantificational logic”. Here is an example:
E.g. Luna is a cat. = Cl (C= is a cat; l=Luna)
Predicates, constants, variables
In the statement about Luna in the previous slide, the small case letter ‘l’ represents an individual or constant. It only talks about one specific individual in the world (namely, my pet Luna). This individual ‘l’ is predicated by the capitalized “C” and symbolizes “is a cat”. This predication gives the subject a fixed attribution, and the subject term can be replaced by other individuals, like “Tom”. If I had another cat, Tom, the sentence ”Tom is a cat” would be symbolized ‘Ct’.
Variables
But sometimes, we have statements whose subject term is not an individual, but a category. Like what if we want to predicate something about all cats in general? For example, that all cats are animals. This statement is about more than just my pet Luna, but about all cats in the world. In the forgoing slides, we will see that we can symbolize such statements with quantifiers, but to do so, they require symbols that do not point to just one individual, but a multiplicity. What we need are variables, and we typically reserve the letters ‘x,’ ‘y,’ and ‘z’ for variables. ‘All’, ‘No’, ‘Some’, or ‘Some are not’ refer to the quantity and quality of things we want to refer to.
Quantifiers in predicate logic
Let’s take another example of a quantificational statement: “All cats are whiskered.” In symbolizing this statement, the letter ‘c’ to symbolize ’cat’ would be inadequate because it would imply that there is one individual cat being referred to. My intention is to predicate all cats in the world, not just a single, individual cat. Also, how about the statement “Some cats are black.’? In that statement, I am neither referring to one cat, nor am I referring to all cats, but only some.
Symbolizing quantifiers in predicate logic
Predicate logic is versatile because it allows me to symbolize all these statements with their nuances.
Luna is whiskered = Wl
All cats are whiskered. = (∀x)(Cx → Wx)
Some cats are black. = (∃x)(Cx Λ Bx)
I will talk more about the symbolic translations in a moment.
Quantifiers
There are four kinds of quantificational statements in quantificational or predicate logic. They can be characterized by their quantity or quality. Quantity refers to the number of subject terms in the world that the statement makes reference to, while quality refers to whether the statement is affirmative or negative. ‘All’ and ‘No’ statements refer to all of the pertinent subject terms in the world and are therefore universal in quantity. On the other hand, ’Some’ or ‘Some…are not…’ refer to only some (or at least one such subject term in the world) and are particular in quantity. ‘All’ and ‘Some’ have the affirmative quality, while ‘No’ and ‘Some…are not…’ are negative.
Quantifiers
To sum up the previous slide, this is how we would characterize each of the four kinds of quantificational statements:
Statement: Quantity Quality
All S are P. Universal Affirmative
No S are P. Universal Negative
Some S are P. Particular Affirmative
Some S are not P. Particular Negative
Universal Statements
∀x is the symbol for the universal quantifier and can be rendered affirmative or negative in these two ways:
All cats are animals. = (∀x)(Cx → Ax)
(Translation: For every x, if x is a cat, then x is an animal.)
No cats are dogs. = (∀x)(Cx → ~ Dx)
(Translation: For every x, if x is a cat, then x is not a dog.)
Universal Statements
The previous slide contains statements about x such that for EVERY x, if x is a cat, then it is an animal. It is not just about one individual, but about the whole category of cats, dogs and animals.
See the next slide for Existential Statements.
Existential Statements
∃x symbolizes the quantifier in existential statements.
Some apples are red. = (∃x)(Ax Λ Rx)
(Translation: There exists an x such that x is an apple and x is red. Another way to say it: There exists at least one x such that x is an apple and x is red.)
Some apples are not red. = (∃x)(Ax Λ ~Rx)
(Translation: There exists an x such that x is an apple and x is not red. Another way to say it: There exists at least one x such that x is an apple and x is not red.)
Examples of universal and existential statements.
Statement Symbolic translation
companions.
on time.
to the wedding.
Exercise
Translate the following statements into symbolic form. Avoid negation signs preceding quantifiers. The predicate letters are given in parentheses.
1. Elaine is a chemist. (C)
2. Nancy is not a sales clerk. (S)
3. Intel designs a faster chip only if Micron does. (D)
4. Some grapes are sour. (G, S)
5. Every penguin loves ice. (P, L)
6. There is trouble in River City. (T, R)
7. Tigers exist. (T)
Key to Exercises
1. Ce
2. ~Sn
3. Di → Dm
4. (∃x)(Gx Λ Sx)
5. (∀x)(Px →Lx)
6. (∃x)(Tx Λ Rx)
7. (∃x)Tx
Logical Inference
In demonstrating the rules of inference in the following slides, I will be reverting back to propositional logic in order to keep things simple. The rules of logical inferences reflect the every day way we use inferences in language, logic, and math. Formal logic provides specific names for these common moves. Once you make them explicit, it is easier to identify and use them. Your textbook mentions six such inferences, and I will give them these labels that are typical in philosophical logic: modus ponens, modus tollens, disjunctive syllogism, conjunction, simplification, and addition.
Modus Ponens
One of the most common inferences we make in logic is the modus ponens. In this kind of inference, a conditional relation is given. Then, the antecedent or first proposition is asserted. Given the two statements, the consequent of the conditional or second proposition inferentially follows.
E.g.
If it rains, then the ground gets wet.
It is raining.
Therefore, the ground gets wet.
Modus Ponens
This inference can be symbolized this way:
R → W
R
W
Any proposition W can be inferred given the other two statements. Note that if W was the second line, R would not inferentially follow. In other words, if the ground is wet, it does not mean we can assert that it had rained.
Modus Tollens
This is an inference that also relates to a conditional statement, but it asserts something about the antecedent. If you have a conditional, and assert the negation of the consequent, you can infer the negation of the antecedent.
E.g.
If it’s a cat, then it’s an animal.
It’s not an animal.
Therefore, it’s not a cat.
Modus Tollens
This inference can be symbolized this way:
C → A
Note that this move would not be logically necessary if the second line was the negation of the antecedent and the third line the negation of the consequent. In other words, if we assert that ‘it’s not a cat’, that does not mean we can infer that ‘it’s not an animal.’
Disjunctive Syllogism
In the disjunctive syllogism, if you have a disjunction or propositions connected by either/or, and then if you negate one of the propositions of the disjunction, you may infer the non-negated proposition of the disjunction.
E.g.
Either the banana is yellow or the banana is green.
The banana is not yellow.
Therefore, the banana is green.
Disjunctive Syllogism
This inference can be symbolized this way:
Y V G
~Y
G
If indeed bananas are yellow or green, and we learn that this banana is not yellow, then it must be green. Sometimes though, we assert a false dichotomy which would involve providing two options that are equally false. That would be a fallacy and would not conform to this rule. Either the unicorn put the gold there or the leprechaun did.
Conjunction
In this type of inference, if you have two true statements, then you may also conjoin them to make a single true statement.
E.g.
Chalk is white. Snow is white. Chalk and snow are white.
Conjunction
This inference may be symbolized this way:
C
S
C Λ S
If one of the propositions is false, however, the entire conjunction would be false. This inference only works with two true statements.
Simplification
In this inference, if you have two statements that are conjoined into a complex statement, you may logically infer one component of the complex statement.
E.g.
Chalk and snow are white.
Chalk is white.
Simplification
This inference may be symbolized this way:
C Λ S
C
If you remember the truth tables, both propositions in a conjunction must be true for the conjunction to be true. So if the original statement is true, so must its component statements be true.
Addition
In this inference, if you have a true statement, you may properly add a disjunction providing any statement whatsoever without falsifying the original statement.
E.g.
Chalk is white.
Chalk is white or a unicorn appeared in the sky.
Addition
The previous inference can be symbolized this way:
C
C V U
We probably agree that ‘A unicorn appeared in the sky’ is false. But if you remember the truth tables for the disjunction, only one of the statements need to be true for the disjunction to be true. Any statement can be added by a disjunction to a true statement without affecting the truth value.
Exercise
Fill in the missing premise and give the inference that justifies the conclusion.
2. ______
3. K ____
2. ______
3. S ____
2. ______
3. ~K ____
Exercise – Fill in the blanks.
2. A v E
3. ______ ____
4. ~A Λ E ____
2. T → G
3. (T v U) → H
4. ___________ ____
5. H ____
2. (M Λ E) → D
3. E
4. __________ ____
5. D ____