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Deductive Logic IV: Quantification and Predicate Logic^[1]

I. Affirmative Singular Predicates

The formal proof methods covered in Chapter V help us construct valid arguments regarding a combination of simple and compound statements. However, if you recall the syllogisms covered in Chapter 3 on Aristotelian syllogisms, then you may see that our formal proofs have a large gap. Chapter V’s system doesn’t help us express some basic valid arguments.

Consider “Syllogism Example #1”:

(P1): All humans are mortal.

(P2): Socrates is a human.

∴ Therefore, Socrates is mortal.

Well, the argument is clearly valid. A Venn diagram would show us as much. But our current method of formal proofs will not. Why? Because here we have three distinct propositions. If we were to try and write them out for a formal proof, we’d be stuck with:

1. A

2. H /∴ M

The tools we’ve encountered so far simply do not give us a way grab onto the salient parts that make the argument valid; namely, the features that let categorical propositions (universal affirmative, universal negative, particular affirmative, and particular negative) express what they do about subjects and predicates (recall that the subject refers to what a statement is about, and predicate refers to whatever descriptive attribution is pinned on to the subject).

Affirmative Singular Propositions

We want to translate Aristotelian Syllogisms into SL so that we can apply our formal proof techniques to a wider pool of arguments. For this, we need Predicate Logic. Predicate logic is logic that includes a formal system for talking about subject/predicate relationships. Our first step in gaining this skill will be to learn a system for writing out phrases like “Socrates is a human.” Here, “Socrates” refers to an individual subject, while “is human” stands for the predicate term, i.e., the term that gives more illustrative information about the subject. We call statements like this a singular proposition, i.e., a proposition that describes a singular subject. The next sections will show how to construct singular propositions in a notation system with our SL language.

Our new system of Predicate Logic will use lower-case letters to refer to the subjects of propositions. So, to start, we could us “s” for Socrates. Typically, the letters a through w are used to stand for whatever subjects are being discussed and are called individual constants. Finally, we will abbreviate the predicate term to a single capital letter, called a predicate symbol. We will also put the predicate before the subject.

And so for example, we can use “Hs” to stand for the singular proposition, “Socrates is human.”

To recap:

Hs = Socrates is human.

Where “Socrates” = s

and “is human” = H.

“s” is the individual constant standing for the subject

“H” is the predicate symbol standing for the predicate

The combination of the predicate symbol and the individual constant express the singular proposition.

In a manner analogous to general forms, we might want to replace “Socrates” or “s” with some empty placeholder to simply express the predicate “is human” that could be applied to any particular, singular subject. We will use “x” for that placeholder, also known as an individual variable. Henceforth, we will be able to express ideas such as:

Hx = x is human

The entire phrase “Hx” is an example of what is called a propositional function. It has a predicate symbol, and individual variable, and when that variable is filled in, with, say, an “s” for Socrates or a “g” for Gandalf, we create a singular proposition. Turning a propositional function into a proposition by filling in the individual variable is called instantiation. We replace the variable with an instance of its being the case. The predicate “is human” is a simple predicate. Simple predicates are propositional functions that could be said of various things, in this example such as Socrates or Yoda, wherein some instances of filling the placeholder yield true propositions while other instances yield false propositions.

The predicate “is human” is just one of a million possible illustrative, simple predicates we might find ourselves representing in our new system. One might say Socrates is wise by using the predicate symbol W for wise, yielding Ws. Or, one might say of some undisclosed x that x is wise; Wx.

Just as we used x as an individual variable, we might also ditch our illustrative predicate for a placeholder that could stand for any simple predicate. A convention is to use Greek symbols, such as “Φ”. And so, Φs means “Socrates is ____.” Moreover, Φx means “_____ is _____.” Blank is blank might be an odd thing to say, but it is helpful to pause upon. As we proceed to learning about categorical syllogisms, Φx helps us get ready to deal with predication in the abstract and not worry about what is being predicated to whatever subject.

Let’s return to our example, Syllogism Example #1:

(P1): All humans are mortal.

(P2): Socrates is a human.

∴ Therefore, Socrates is mortal.

At this point, we have the ability to translate P2 into “Hs”. However, Aristotelian syllogisms require using the terms “all,” “some,” “none” or “some are not.” Our example just uses a universal affirmative as its major premise, but ideally we want to handle any sort of syllogism of the type we covered in chapter 3. To do this, we need to add quantification into our system. We will do this by introducing what are called universal and existential quantifiers.

II. Existential and Universal Quantifiers

The four types of categorical propositions are universal affirmative (all), universal negative (non), particular affirmative (some), and particular negative (some are not). To move toward expressing these categorical propositions, we will introduce existential and universal quantifiers.

Take the sentences “Everything is mortal,” and “All things are mortal.” Unlike our singular proposition, “Socrates is mortal” which only talked about a single subject, “all things are mortal” talks about everything; therefore, we call this a general proposition. Another way to state “all things are mortal” is to say, “given any x, Mx.” We can reduce the “given any x” to just “(x)” for short. Thus,

(x)Mx = All things are mortal.

Here, by adding (x) to Mx we’ve turned the propositional function into a general proposition through what is called “quantification.” Because the (x) here means we are talking about any given thing, and thus all things, we call the (x) symbol the “universal quantifier.”

The universal quantifier lets us get to the notion of your first type of categorical proposition, “universal affirmative” (A) type statements. The idea of the “universal affirmative” can be expressed by the general quantification, (x)Φx.

(x)Φx = universal affirmative quantifier – all things are Φ

In order to say, “no thing is mortal” and express a universal negative, we can negate the predicate function in the affirmative version, thus changing (x)Mx into the negative (x)~Mx. The more abstract idea of expressing any “universal negative” can be achieved by dropping our illustrative predicate for a predicate variable, yielding:

(x)~Φx = universal negative quantifier – no things are Φ

Both (x)Φx and (x)~Φx talk about any given thing, whatsoever. They are both talking about “All x’s.” The only difference is that one says “all x are Φ” while the other says “all x are not Φ.” And this is like saying “no x are Φ.”

To talk about the particular affirmative and particular negative categorical propositions, we need a way to say “some x” rather than just “for any given x.” To do this we will introduce a special symbol, “∃.” When we have “∃x” we have what is called the “existential quantifier.” Putting an existential quantifier before some illustrative predicate, like mortal, gives us a way to say, “There is some x that is mortal” or “some x’s are mortal.” In predicate logic, we have (∃x)Mx. Swapping our M for a placeholder gives us our third categorical type:

(∃x)Φx = particular affirmative quantifier

To say, “some x’s are not ____” and express a particular negative, we just add a negation before the predicate function:

(∃x)~Φx = particular negative quantifier

Here are the 4 types of general quantifiers all set together for your reference:

(x)Φx = universal affirmative quantifier

(x)~Φx = universal negative quantifier

(∃x)Φx = particular affirmative quantifier

(∃x)~Φx = particular negative quantifier

EXERCISES

Translate the following statements into quantified predicate notation. To do this, first state your abbreviations for the predicate terms.

Sample Problem.                  Sample Solution

Nothing makes sense          S = makes sense

                                               (x)~(Sx)

1. Everything counts in large amounts.

2. Some things happen for a reason.

3. Some things do not last.

4. Nothing is permanent.

III. Subject-Predicate Propositions

We need more than just the four types of general propositions. We need to express full general propositions involving specific predicates and specific subjects, like “All humans are mortal” to express syllogisms such as:

(P1): All humans are mortal.

(P2): Socrates is a human.

∴ Therefore, Socrates is mortal.

We can do this by initially reframing our subject phrases into predicates themselves. For example, take H to stand for being human, as a predicate. Thus Hx = x is a human. We can rephrase “all humans are mortal” into, “for any given x, if x is a human then x is a mortal.” Using the notation we’ve just learned, we get:

(x)(Hx ⊃ Mx) = all humans are mortal.

Schematically presented, universal affirmative (A) propositions such as “all ____ are ____” would look like:

(x)(Φx ⊃ Ψx) = universal affirmative subject-predicate proposition (A)

Notice we’ve added a second predicate variable, Ψ, to cover the other term. Since (x)Φx just means “all x’s are __[insert predicate here]__”,  we can insert something more complex, and more specific, like “(Φx ⊃ Ψx)” into the blank space. Remember the discussion of universal affirmative statements and existential import as discussed in the chapter on Aristotelian Syllogisms. For a long time in history, our study of logic was complicated by assuming that A statements meant that their subject existed. However, following Boole, it was decided that our inferences went much more smoothly were we to drop this assumption. To say, (x)(Φx ⊃ Ψx) is not to affirm that any Φ’d x’s exist. Rather, our A statement just says that for any given thing, if it were to be a Φ’d thing, then it is Ψ’d. For example, look at a specific, instantiated universal affirmative claim about the now extinct ivory-billed woodpecker. Suppose we say, “All ivory-bills are birds,” with “Ix” as the predicate function for ivory-bill-edness and “Bx” being for birds. Then we’d have (x)(Ix ⊃ Bx). And this is true, even though, sadly, no ivory-billed woodpeckers still exist.

Moving on, to cover a universal negative version, we can begin by recognizing that claims such as no “humans are mortal” can be rephrased as something like “for any given x, if x is a human, then x is not mortal” into our notation. This rephrasing helps disclose what the symbolic notation for the general proposition will look like:

(x)(Φx ⊃ ~Ψx) = universal negative subject-predicate proposition (E)

The above shows that if Φ holds for x, then Ψ does not. Applying an E statement to human immortality, we’d have (x)(Hx ⊃ ~Mx); if something is human, it’s not mortal, which is equivalent to saying, no humans are mortal. Not a true statement, but at least we’ve gained the ability to write it out in a well-formed manner in our notation.

Once again, it is important to note that universal propositions (A) and (E) statements do not assert that anything exists. Moreover, they are not directly opposite statements. Rather, they are contraries. Both cannot be true; it can’t be that if any given thing were, it’d both be and not be. However, both can be false at the same time. If some things were not while some were, then either universal claim would be false. “All fish are blue,” and “No fish are blue” are both false in a pool with one red and one blue fish.

When we want to move away from saying, “All x’s are [or are not],” we can start using our existential quantifier – the ∃ symbol. For particular affirmatives and negations, we want to talk about at least one thing, but not everything, and affirm that such a thing exists. And so, to say, “Some humans are mortal,” we might first rephrase the statement as “There is at least one x such that x is both human and mortal.” In our notation, the general proposition, about any random set of predicates, would take the form:

∃x(Φx • Ψx) = particular affirmative subject-predicate proposition (I)

The particular affirmative uses the ∃x existential quantifier. The existential quantifier affirms that something exists. The above just says there is something for which both predicates hold.

Finally, if we want to say, some humans are not mortal, we can write out, ∃x(Hx • ~Mx). Dropping human and mortal predicates functions for general placeholders, we’d get the general proposition,

(∃x)(Φx • ~Ψx) = particular negative subject-predicate proposition (O)

This statement simply says some x’s are such that they are Φ and not Ψ.

For your convenience, here are all four general propositions again:

(x)(Φx ⊃ Ψx) = universal affirmative subject-predicate proposition (A)

(x)(Φx ⊃ ~Ψx) = universal negative subject-predicate proposition (E)

(∃x)(Φx • Ψx) = particular affirmative subject-predicate proposition (I)

(∃x)(Φx • ~Ψx) = particular negative subject-predicate proposition (O)

These general forms are useful because they let us talk about any categorical proposition, not just ones about humans and mortality. We can substitute illustrative predicates for the placeholders as context demands.

By using quantifiers (either (x) or (∃x)) and two predicate functions (Φx and Ψx) linked with either a conditional or a conjunction, plus an optional negation sign, we have gained a new way to write out categorical propositions that move us one step closer to being able to construct formal proofs of Aristotelian Syllogisms.

Because we are able to translate categorical propositions and predication into our formal language, we can now rewrite Syllogism Example #1.

(P1): All humans are mortal.

(P2): Socrates is a human.

∴Therefore, Socrates is mortal.

Becomes

(P1): (x)(Hx ⊃ Mx)

(P2): Hs

∴ Ms

Our next step toward formal proofs will be to add some more inference rules that justify making inferences between lines. The next section will pick up this task in a manner similar to how our existing inference rules were originally explained in Chapter 5.

EXERCISES

Translate the following A, E, I, or O into quantified predicate notation. To do this, first state your abbreviations for the predicate terms.

Sample Problem.                  Sample Solution

All birds are nice.                 B = bird

                                               N = nice

                                              (x)(Bx ⊃ Nx)

1. Some fish are green.

2. Mules are stubborn.

3. No elephants forget.

4. Some birds do not fly.

IV. Instantiation, Generalization, and the Quantifier Negation Rule

All of our general quantifiers are about any random x. Because they are just using variables, such as the “x” they literally are “generally speaking.” (x)(Mx ⊃ Lx) is a general statement that says all mortals love logic. (∃x)(Mx • Lx) is a general statement saying some human out there loves logic.

We sometimes want to move from general statements to speaking about particular instances, such as Socrates. In this case, as you saw in the previous section, we assign a constant, such as “s,” to mean Socrates and fill in the x’s. If all mortals love logic, and we know Socrates is a mortal, we should be able to say he’s mortal and loves logic, right? Right. To say, if Socrates is mortal, then he loves logic, we’d then write, (Ms ⊃ Ls). Notice, too, how the “(x)” at the beginning disappeared because we are no longer talking about any given “x.” Moving from general statements with individual variables to a specific statement with an individual constant is called “instantiation.” (Think finding an “instance” of the thing).

Sometimes we want to move in the other direction. Suppose we know Socrates is mortal and loves logic (Ms • Ls) and we want to perform a “generalization,” and say there is some general person who is mortal and loves logic. First, we’d replace our individual constant, s, with a variable, x. We also want to re-introduce a quantifier, ∃x, to say, “there exists an x such that . . .”. Thus, we end up expressing this generalization as, (∃x)(Mx • Lx). Again, this just means something exists that is mortal and loves logic.

To summarize, when we move from a variable to a constant, that is called “instantiation.” When we move from a constant to a variable, that is “generalization.” Generalization and Instantiation are important for manipulating quantified predicates, and we will be exploring rules for their use in the following sections. But first, it is important to detour for a minute and recognize an important fact about how we can negate our general quantifiers.

Suppose I want to say the opposite of (x)(Mx ⊃ Lx). Well, to say not all mortals love logic, I could just write out ~(x)(Mx ⊃ Lx). However, because our negation is assigned to the quantifier, but not the propositional functions, we don’t have a clean way to instantiate the statement. The negation is attached to the quantifier (x), and if we drop the quantifier we risk losing the negation and misrepresenting our statement. To get around this issue, we need to think of a different way to negate our quantifier. We need to ensure the negation sign only applies to simple predicates, such as Mx or Lx. We call quantified statements that are expressed this way as being normal-form formulas. And so, we need to look for ways to express negations in normal form. Well, the negation of “Everything is Φ” is “There is something that is not Φ.” And so, we can say,

~(x)Φx ≡^t (∃x)~Φx.

Similarly, there is not a thing that is not Φ, i.e., “~(∃x)~Φx” is the same as everything is Φ. i.e.,

~(∃x)~Φx ≡^t (x)Φx

The negation equivalences for “nothing is not Φ,” i.e., ~(x)~Φ, and “there is not a thing that is Φ” can be equated into normal-form formulas as,

~(x)~Φx ≡^t (∃x)Φx

And,

~(∃x)Φx ≡^t (x)~Φx

By moving the negation from the quantifier to the propositional function, we will be able to smoothly instantiate and generalize our quantified statements. Therefore, again, it is important to recognize which general quantified statements are negations of each other.

Here, therefore, is a table of quantifications and their negations:

Table 6.IV.1, Quantifier Negations
(x)Φx = universal affirmative proposition	Can have its negation expressed as	(∃x)~Φx = particular negative proposition
(x)~Φx = universal negative quantifier	Can have its negation expressed as	(∃x)Φx = particular affirmative propostion
(∃x)Φx = particular affirmative proposition	Can have its negation expressed as	(x)~Φx = universal negative proposition
(∃x)~Φx = particular negative proposition	Can have its negation expressed as	(x)Φx = universal affirmative proposition

Our four traditional subject-predicate categorical propositional forms, the A, E, I, and O statements, can also be negated without needing to add a negation to the quantifier symbol itself. We already saw the contraries of these categorical propositions in the “Square of Opposition.” Translating this square into quantified predicate notation, we can yield a table for which categorical propositions can negate each other.

Table 6.IV.2, Categorical Proposition Negations
(x)(Φx ⊃ Ψx) = universal affirmative A	Can have its negation expressed as	(∃x)(Φx • ~Ψx) = particular negative O
(x)(Φx ⊃ ~Ψx) = universal negative E	Can have its negation expressed as	(∃x)(Φx • Ψx) = particular affirmative I
(∃x)(Φx • Ψx) = particular affirmative I	Can have its negation expressed as	(x)(Φx ⊃ ~ Ψx) = universal negative E
(∃x)(Φx • ~Ψx) = particular negative O	Can have its negation expressed as	(x)(Φx ⊃ Ψx) = universal affirmative A

The logical relationship between the quantifier negation pairs and the categorical proposition negations can be established using the rules of inference covered in chapter 5. For example, let’s negate the particular negative, O. We know,

~(∃x)Φx ≡^t (x)~Φx

And so, filling in the illustrative predicates will compound predicates we have our negated-O and its normal-form formular equivalent,

~(∃x)(Φx • ~Ψx) ≡^t (x)~(Φx • ~Ψx)

Notice how we have consistently replaced the illustrative predicate Φx with the compound (Φx • ~Ψx). Since Φx means x is ____ we are just filling in the blank with (Φx • ~Ψx). Now, take the right side of the biconditional equivalence. If we apply De Morgan’s to it, we get,

(x)~(Φx • ~Ψx) ≡^t (x)(~Φx ∨ ~~Ψx)

And we can apply the rule of double negation to the right side to that to get,

(x)(~Φx ∨ ~~Ψx) ≡^t (x)(~Φx ∨ Ψx)

And we can use Material Implication to say,

(x)(~Φx ∨ Ψx) ≡^t (x)(Φx ⊃Ψx)

Hence, condensing this chain of biconditional equivalences, we get,

~(∃x)(Φx • ~Ψx) ≡^t (x)(Φx ⊃Ψx)

(x)(Φx ⊃ Ψx) is thus logically equivalent to the original, non-normal-form formula of ~(∃x)(Φx • ~Ψx). Moreover, (x)(Φx ⊃ Ψx) is our expression of the universal affirmative A statement. Hence, we have proved that A statements are the negation of O statements.

There might be times when a formal proof involves negating a quantified statement. In this case, we may want to translate it into its normal-form formula for further use. To do so, we can appeal to a new replacement rule:

Quantifier Negation Rule Q.N.

One can substitute a negated quantified statement with its equivalent normal-form formula.

Always Valid!

Here’s a table of negated quantified statements and their equivalent normal-form formulas:

Table 6.IV.3, Quantifier Negation Rule
negated quantified statement		normal-form formula
~(x)(Φx ⊃ Ψx)	≡T	(∃x)(Φx • ~Ψx)
~(x)(Φx ⊃ ~Ψx)	≡T	(∃x)(Φx • Ψx)
~(∃x)(Φx • Ψx)	≡T	(x)(Φx ⊃ ~ Ψx)
~(∃x)(Φx • ~Ψx)	≡T	(x)(Φx ⊃ Ψx)

The quantifier negation rule can be used in a formal proof just like any of the other replacement rules so that we may move back and forth between different expressions of the same A, I, E, or O statements.

V. Formal Proofs of Validity for Aristotelian Syllogisms

Sections II, III, and IV have given us the notational tools for translating Aristotelian Syllogisms into our formal language. And, we just picked up a new quantifier-specific rule, Q.N. Now, we just need to add four more quantifier-specific inference rules and we will be ready to learn about formal proofs of validity for Aristotelian Syllogisms.

To begin, let’s re-articulate just what the issue is, and why we have to add these new rules in the first place. To start, normal propositional logic gives us 19 rules for drawing valid inferences and replacements when we are dealing with simple propositions or compounds of propositions [i.e. (A ⊃ B) or (A ∨ B) or (A • B), etc.]. Our quantifiers give us noncompound statements, too, but these are like “All humans are mortal.” Translating this proposition into something like a mere H hides the quantified information inside that gives us the ability to draw inferences. While (x)(Hx ⊃ Mx) tells us what we need to know about human mortality, if we take the entire proposition as a whole none of the previously introduced inference rules help us conclude Ms from the two premises (x)(Hx ⊃ Mx) and Hs. Why? Although we are close to being able to apply modus ponens, Hs does not actually appear in the first premise; only Hx does. Instead, we need to get something like Hs ⊃ Ms from the first premise. If we had that, then Ms would follow from Hs.

Generally speaking, for proofs in predicate logic, what we need is a set of rules that helps us move between general quantified statements like (x)(Φx) and (∃x)(Φx) singular propositions, such as Φs. The following four quantification rules of inference allow just that.

Universal Instantiation

Our first quantification inference rule is called ‘Universal Instantiation’ and lets us go from a general universal claim to a singular proposition. This rule states, “any substitution instance of a propositional function can validly be inferred from its universal quantification” (Copi et al, 2020: p 469). To explain, let’s think about what a universal quantification says: “for any given x, Φx.” Well, if Φx is really true for any given x, then we are justified in giving an instantiation of the x. For example, if any given thing whatsoever is mortal, and Socrates is a potentially given thing, then from (x)Mx we are welcome to conclude Ms, i.e., that Socrates is mortal. And this sort of inference holds valid for any use of (x)Φx and any individual constant we want to give it. Rather than the individual variable “x” we will use “v” to stand as some individual constant, which could really be “s” or “a” or whatever. Schematically we can say:

With Universal Instantiation, we take the phrase “any given x” up on its offer and say, okay, how about “v”, let’s give it that.

In the above example and those that follow, we will stick to single function statements in our formal display. Such as, U.I. allows Φv from (x)(Φx). However, any combination of propositional functions can go after the quantifier. For example, U.I. also allows (Φv ⊃Ψv) from (x)(Φx ⊃ Ψx), or any other replacement provided the propositional functions, and logical operators remain consistent following the application of the rule.

Universal Instantiation just so happens to be all that we need to write out a formal proof of our example Aristotelian syllogism about Socrates’s human mortality. Here now is the full proof as an example of Universal Instantiation in action.

Proof of Syllogism Example # 1
1.	(x)(Hx ⊃ Mx)
2.	Hs	/∴ Ms
3.	Hs ⊃ Ms	1, U. I.
4.	Ms	3, 2, M.P.

Here you see the application of Universal Instantiation on line 3. Notice also that we are treating the singular propositions Hs and Ms just like we treat other propositions, such as A or B, as we proceed through the remaining inferences. In this case, we applied modus ponens to achieve our conclusion.

Universal Generalization

Our next inference rule works like Universal Instantiation but in the other direction. Here, we go from a special sort of particular back to the universal. Suppose we pluck an arbitrary individual out of thin air, call it “y,” and not knowing anything else about it, find out through other inferences that Φy holds true to it. Because we know nothing about y, it is valid to acknowledge that y could just as well have been “any given thing.” Well, if any given thing could be Φ, then everything is Φ. This brings us to our second quantifier inference rule; universal generalization.

The following example (adapted from Copi, p. 470) will help further illustrate and explain just what Universal Generalization is and how it works. Suppose we want to build a proof for the following argument:

(P1): All humans are mortal.

(P2): All Aztecs are human.

∴ All Aztecs are mortal.

First, we can write this out with our new notation system, where Hx functions for Human, Mx mortal, and Ax predicates Aztec to its subject. Thus, we have:

Syllogism Example 2:
1. 2.	(x)(Hx ⊃Mx) (x)(Ax ⊃ Hx)	/∴ (x)(Ax ⊃Mx)

Applying Universal Instantiation and our existing inference rules to this argument gets us close to the desired conclusion, but not close enough. Here’s as far as we get with Universal Instantiation and hypothetical syllogism:

Partial Proof of Syllogism Example 2:
1. 2. 3. 4. 5.	(x)(Hx ⊃Mx) (x)(Ax ⊃ Hx) Hy ⊃ My Ay ⊃ Hy Ay ⊃ My	/∴ (x)(Ax ⊃ Mx) 1, U.I. 2, U.I. 3,4, H.S.

So far, we’ve seen that Universal Instantiation lets us plug any particular instance into the universal affirmatives provided in lines 1 and 2. But, suppose for a particular instance we use unknown individual “y.” In Syllogism Example #2, the rule Hypothetical syllogism lets us get line 5 from lines 3 and 4. And so, we affirm that if unknown y is Aztec, then y is human. However, our conclusion is not about “unknown individual y.” Rather, it is about all Aztec people. Needless to say, because “y” is so vague, it could represent any given person. Nothing has been said about y that means it couldn’t just as well be anyone. All we’ve done was apply the premised claims that for any given thing, if that thing is human then it’s mortal, and if that thing is Aztec then it’s human. Because any substitution from U.I. is true, then all substitutions must be true. Hence, we can move from y back to (x)Φx.

Our new inference rule, Universal Generalization follows the above line of reasoning. Here is the general schema:

There is an important rule we must follow. Jumping from “y” to the universal quantifier only works when “y” is any given individual. Hence, we will only use “y” when it’s been introduced as the assumed “any given random thing.” For our purpose, this would be when y is introduced through U.I. in the first place or as an assumption in a conditional proof. I mean, we cannot reasonably infer that just because Socrates is Greek, then everyone is Greek. Our individual constant must represent an “unknown thing.”

Syllogism Example #2, shows how we can properly use U.G. to finish a proof.

Complete Proof of Syllogism Example 2:
1. 2. 3. 4. 5. 6.	(x)(Hx ⊃Mx) (x)(Ax ⊃ Hx) Hy ⊃ My Ay ⊃ Hy Ay ⊃ My (x)(Ax ⊃ Mx)	/∴ (x)(Ax ⊃ Mx) 1, U.I. 2, U.I. 3,4, H.S. 5, U.G.

Existential Generalization

The preceding rules cover generalization and instantiation for universal statements. We also need two inference rules that let us move back and forth between singular propositions, e.g., Φv’s, and quantified functions, ∃x’s.

Recall that ∃x(Φx) is true if at least one individual can have Φ predicated to it. And so, if we have an instance of Φv, (where v is any individual constant), we’ve satisfied the condition for ∃x(Φx). We can articulate this valid inference via the Existential Generalization rule.

Existential Generalization is, therefore simply the rule that if we have at least one specific example of something, we can say that there exists an example of that thing.

Existential Instantiation

It may help to introduce our final quantification rule with a less abstract example. Suppose you have ordered a pizza, which you momentarily leave on the counter. When you return, you find the pizza gone. Being a keen logician, you say, “we know there exists something which stole our pizza. Let’s call that “something ‘v’.” Maybe not a conclusive step, but so far this is a deductively valid way to proceed.

The rule of Existential Instantiation formalizes the reasoning that occurs in examples like the above. If we know there exists something with a certain predicate, i.e., (∃x)(Φx), then we are warranted to give some arbitrary name to “that thing which exists” in order to talk about it further.

The important caveat here is that the name we give it must not occur anywhere prior in our argument. Return to the pizza example. No matter how guiltily your dog, Shadow, looks at you, without further evidence, you are not entitled to just say, “we know there exists something which stole our pizza. Let’s call that something . . . Shadow. Bad dog, Shadow.” The name we use needs to be neutral in our context, lest we accidentally contaminate our argument with any invalid inferences. In other words, the name cannot be used earlier in the argument.

Thus, our formal rule is,

With this rule in place, let’s review our pizza problem to see exactly how much trouble we get into if we break the, “only use new constants” rule.

Let’s use, Px to predicate “ate the pizza” to something and Sx to predicate “being Shadow-dog” to something. And so, our givens at the start of the argument are that something ate the pizza, and something [something else potentially] is Shadow:

1. 1. 1. 1. 1. 1. 1. 1. (x)Px
                     2. (x)Sx           /∴ (x)(Px • Sx). [Invalid!]

Now, our invalid argument tries to reach its unjust conclusion by using the same constant twice in repeated applications of Existential Instantiation. The unjustness occurs when we assume or let the idea slip into our argument that the same thing which is Shadow is what ate the Pizza. Here’s the formal argument with the ledger column and an added “notes” column to help clarify the steps and missteps:

Invalid Example #1

(∃x)Px (∃x)Sx.           /∴ (∃x)(Px • Sx)    Invalid Conclusion! Pa                   1, E.I.                     Okay step Sa                   2, E.I.                    NO! “a” is already in use! Pa • Sa           3,4, Conj.              Okay step, but tainted by error on line 4 (∃x)(Px • Sx)  5, E.G. v                Invalid, due to error on line 4

 

 

 

 

 

 

 

 

Because the second use of E.I. on line 4 uses a constant that previously occurs, rather than a new one, the argument then invalidly proceeds to identify “a” with the pizza eater. Had the argument followed the rules, it couldn’t have reached the conclusion.

Here is an example of a valid syllogistic proof using two particular statements and a correct application of the Existential Instantiation rule:

Syllogism Example #3:

P1: No mammals are cold-blooded

P2: Some egg-layers are mammals

∴ Some egg-layers are not cold-blooded

Let M, C, and E predicate mammal, cold-blooded, and egg-laying. Here is a formal proof of our argument in predicate logical notation (with explanatory notes):

Syllogism Example #3
(x)(Mx ⊃ ~Cx) (x∃)(Ex • Mx)      /∴ (x∃)(Ex • ~Cx) Ea • Ma                2, E.I. Ma • Ea                3, Com. Ma                        4, Simp. Ma ⊃ ~Ca             1, U.I. ~Ca                       5,6, M.P. Ea                         3, Simp Ea • ~Ca              7,8, Conj. (∃x)(Ex • ~Cx)        9, E.G.	We get to randomly assign “a” We can re-use “a” when using U.I. . .

The above syllogism is valid, and thanks to the Platypus, it’s sound as well.^[2]

VI. Asyllogistic Inferences

The method of assigning predicates to things and using our logical operators of ⊃, ∨, •, ≡, and ~ can help us formalize arguments beyond Aristotelian syllogisms. Inferences that utilize predicates but move beyond Aristotelian syllogism are asyllogistic inferences. With our new system, we can talk about how different predicated objects related to each other, beyond just how objects might or might not fit into categories.

Here is an example of using predicate logic beyond Aristotelian syllogisms. I used to work in an outdoor shop that repaired gear. Behind the cash register, there was a sign that said:

A job can be fast, cheap, or good, but not all three.

If a job is fast and good, it won’t be cheap

If a job is good and cheap, it won’t be fast.

If a job cheap and fast, it won’t be good.

We can formalize the above axioms into asyllogistic predicates and make asyllogistic inferences. For example, if we use Fx, Cx, Gx, and Jx for Fast(x), Cheap(x), Good(x), and Job(x), we can rewrite our example as follows:

Translation of “the Sign”
(x){Jx ⊃ ([(Fx ∨ Cx) ∨ Gx] • ~[(Fx • Cx) • Gx])} (x){Jx ⊃ [(Fx • Gx) ⊃ ~C]} (x){Jx ⊃ [(Gx • Cx) ⊃ ~Fx]} (x){Jx ⊃ [(Cx • Fx) ⊃ ~Gx]}	A job can be F, C, or G, but not all three. If a job is fast and good, it won’t be cheap. If a job is good and cheap, it won’t be fast. If a job is cheap and fast, it won’t be good.

Had a customer come in and asked for a good job, I could used the signs axioms and predicate logic to formally demonstrated the validity of telling them that a good job would either be a) not cheap or b) not fast!

Asyllogistic Formal Proof #1

(x){Jx ⊃ ([(Fx ∨ Cx) ∨ Gx] • ~[(Fx • Cx) • Gx])} (x){Jx ⊃ [(Fx • Gx) ⊃ ~C]} (x){Jx ⊃ [(Gx • Cx) ⊃ ~Fx]} (x){Jx ⊃ [(Cx • Fx) ⊃ ~Gx]}                                  /∴ (x)[(Jx • Gx) ⊃ (~Cx v ~Fx)] | Jy • Gy                                                               /∴ (~Cy ∨ ~Fy) (A.C.P.) | Jy                                                                       5, Simp | Gy • Jy                                                               5, Com. | Gy                                                                      7, Simp. | Jy ⊃ [(Cy • Fy) ⊃ ~Gy]                                       4, U.I. | (Cy • Fy) ⊃ ~Gy                                                 6, 9, M.P. | ~~Gy                                                                  8, D.N. | ~(Cy • Fy)                                                          10,11, M.P. | ~Cy ∨ ~Fy                                                          12, De.M. (Jy • Gy) ⊃ (~Cy ∨ ~Fy)                                       5 – 13, C.P. (x)[(Jx •Gx) ⊃ (~Cx ∨ ~Fx)]                                 14, U.G.

In the above proof, you will notice a conditional proof begins on line 5, where we suppose “y” is both a job and is good. Conditional proofs are totally fine to use within predicate logic, as long as we mind the rules. From this particular conditional proof and our prior premises, we arrive at the conclusion that it will either be not cheap or not fast. At line 14, we then export the conditional as a validly derived claim.

The next step is important to pay attention to, as it involves the complicated U.G. Since our instantiated job “y” was any given job and introduced into the main proof at line 14 from a mere assumption – with no other information assumed about “y” (or predicated onto “y”) we can apply Universal Generalization – after the conditional proof is done – to conclude that for anything, if it’s a job that is good, then it either won’t be cheap or won’t be fast.

We can only use U.G. for constants introduced in a conditional proof after the statement has been exported because using U.G. beforehand would result in creating an invalid universal claim about just a part of the overall conditional. For example, applying U.G. to line 8 would have given us the nonsense claim (x)(Gx), i.e., all things are good from the mere assumption that “y” was a good job.

Predicate logic can also be used with indirect proofs, as shown in the example, below:

Indirect Proof with Predicate Logic Example
1. (x)(Mx ⊃ ~Cx) 2. (∃x)(Ex • Mx) 3.  | (x)(Ex ⊃ Cx) 4.  | Ey • My 5.  | My ⊃ ~Cy 6.  | My • Ey 7.   | My 8.  | ~Cy 9.  | Ey 10. | Ey ⊃ Cy 11.  | Cy 12. | Cy • ~Cy 13. ~(x)(Ex ⊃ Cx) 14. (∃x)(Ex • ~Cx)	/∴ (x)(Ex • ~Cx) /∴ (A.I.P.) 2, E.I. 1, U.I. 4, Com. 6, Simp. 5, 7, M.P. 4, Simp. 3, U.I. 9,10, M.P. 8, 11, Conj. 3-12, I.P. 13, Q.N.	Here we start and indirect proof New use of “y” following E.I. rules Any symbol, include y can be used with U.I.         Any symbol can be used with U.I.   Yes, it’s a contradiction, but that’s the point. Export the negated I.P. assumption Apply Q.N. to return to the normal-form formula.

Asyllogistic inferences use all the rules of inference that formal proofs utilize. The additional skill that is required in their application to arguments lies beyond formal logic. That skill is of accurate interpretation of propositions into formal notation so that the intended meaning of the propositions is preserved.

For example, consider the following example – “oysters and clams are delicious” (from Copi, 2020: 482). If we take this to mean oysters are delicious and clams are delicious, we need to be careful in what our symbols represent. We can take it as a conjunct of propositions, one about oysters, and one about clams,

(x)[(Ox ⊃ Dx)] • (x)[(Cx ⊃ Dx)]

Or, we can take it as non-compound claim that if a thing is either an oyster or a claim, then it’d be delicious,

(x)[(Ox ∨ Cx) ⊃ Dx]

But we don’t want to just say,

(x)[(Ox • Cx) ⊃ Dx]

Because this statement entails that anything that is delicious is somehow both an oyster and clam at the same time. Seems fishy to me. Moving effectively from thought to language is an art regardless of whether or not the language is English, Spanish, Latin, Swahili, Hawaiian, Arabic, Mandarin, or a formal logical notation system.

EXERCISES

Complete the ledger column for the following arguments.

1.

1. (x)(Ax ⊃ Bx)

2. (x)(Bx ⊃ Cx)     /∴ (x)(Ax ⊃ Cx)

3. Aa ⊃ Ba

4. Ba ⊃ Ca

5. Aa ⊃ Ca

6. (x)(Ax ⊃ Cx)

2.

1. (x)(Fx ⊃ Gx)

2. (x)(Gx ⊃ ~Hx)       /∴ ~(∃x)(Fx • Hx)

3. Fa ⊃ Ga

4. Ga ⊃ ~Ha

5. Fa ⊃ ~Ha

6. (x)(Fx ⊃ ~Hx)

7. ~(∃x)(Fx • Hx)

3.

1. (∃x)(Sx • Gx)

2. (x)(Gx ⊃ Mx)       /∴ (∃x)(Sx • Mx)

3. Sa • Ga

4. Ga • Sa

5. Ga

6. Ga ⊃ Ma

7. Ma

8. Sa

9. Sa • Ma

10. (∃x)(Sx • Mx)

4.

1. (x)(Rx ∨ Dx)

2. (∃x)(~Rx)         /∴ (x)(Dx)

3. ~Ra

4. Ra ∨ Da

5. Da

6. (∃x)(Dx)

5.

1. (x)(Hx ⊃ Px) • (x)(Sx ⊃ Wx)    / ∴ (x)[(Hx v Sx) ⊃ (Px v Wx)]

2. (Ha ⊃ Pa) • (Sa ⊃ Wa)

3. | Ha v Sa

4. | Pa v Wa

5. (Ha v Sa) ⊃ (Pa v Wa)

6. (x)[(Hx v Sx) ⊃ (Px v Wx)]

Complete formal proofs for the following arguments.

1.

1. (x)(Mx ⊃ ~Px)

2. (∃x)(Fx • Px)     /∴ (Ex)(Fx ⊃ ~Mx)

2.

1. (∃x)[Gx ⊃ (Hx ∨ Jx)]

2. (x)(Hx ⊃ ~Jx)             /∴ (x)[Gx ⊃ ~(Hx • Jx)]

Translate the following arguments into predicate logic and then construct formal proofs of their validity.

1.

(P1) All skunks are mammals.

(P2) No invertebrates are mammals.

No invertebrates are skunks.

2.

(P1) Any given teapot is neither here nor there.

(P2) Some folks are there.

Some folks are not teapots.

3.

Some hobbits are Bullroarers, and all Bullroarers are tall. And so, some hobbits are tall.

VII. Proving Invalidity with Predicate Logic

It only takes one bad oyster to disprove the claim that all oysters and clams are tasty. We have seen that proving validity using quantifiers and predicate logic works along the lines of the formal proofs we explored above and in Chapter 5. It is also possible to assign truth values to the premises and the conclusion and use the Truth Table technique developed in Chapter 4.

One of the strengths of deductive logic is that we can use its stripped-down ‘formal-relationship-only’ approach to investigate the validity and invalidity of any given argument’s logical analogy. If two arguments have the exact same logical structure and we can prove one to be invalid, then since invalidity depends on structure alone, we’d know the other is also invalid. This can be very useful when the original argument in question is clouded by vested interest, bias, complications, or whatnot.

Refutation by logical analogy can be achieved in a special way when dealing with syllogisms and quantified propositions; the use of the counter-example. Copi et al put the idea very well,

An argument involving quantifiers is valid if, and only if, it is valid no matter how many individuals there are, provided there is at least one. So, an argument involving quantifiers is proved invalid if there is a possible universe or model containing at least one individual such that the argument’s premises are true and its conclusion false of that model (2020, p 477).

In other words, just as with Venn diagrams, if we can find conceptual space for an individual counterexample to an argument, we debunk the entire deductive argument. Consider the invalid syllogism from earlier in the text:

No M are P.

Some S are not M.

∴ Some S are P.

A Venn diagram would show where a counter-example can be found. Constructing a Venn Diagram shows that it is possible to preclude anything from being in the M/P overlap zone (per premise 1), have an object in the S circle that is outside the M circle (per premise 2), and yet still not be forced to fill in an object in the S/P overlap (per conclusion). Again, for an argument to be valid, the premises must necessitate the conclusion – force it on us. With true premises and a false conclusion, this argument is invalid.

In our new quantified notation, we can see the same thing:

(P1.) (x)(Mx ⊃ ~Px) – Universal Negative (E): no M are P

(P2.) (∃x)(Sx • ~Mx) – Particular Negative (O): Some S are not M

∴ (∃x)(Sx • Px) – Particular Affirmative (I): Some S are P

General propositions, like universal affirmative (x)Φx or (x)~Φx universal negatives, are only true if they hold of all the individuals a world with any possible number of individuals. This means they are true in scenarios where there is only one thing, true in scenarios with 2 things, 3 things, etc.

And so, in a world with just one thing, called “a”

(x)(Φx) ≡^t Φa ≡^t (∃x)(Φx)

The first part states the universal quantifier. The second gives the predicate to the only thing that might exist. The third part says the Φ’d thing does exist.

In a world with two things, a and b, we know the following two principles must be true:

(x)(Φx) ≡^t [Φa • Φb]

and (∃x)(Φx) ≡^t [Φa ∨ Φb]

The first part, here, deals with universal statements and says everything, including a and b are Φ. The second deals with existential statements and says (∃x)(Φx) is logically equivalent to saying that at least one of the two things that exist is a Φ’d thing, but we don’t know which one.

In a world with three things, we have

(x)(Φx) ≡^t [Φa • Φb • Φc]

and (∃x)(Φx) ≡^t [Φa ∨ Φb ∨ Φc]

This says the same as above but with a third object in our possible world. If everything is Φ, then everything listed out is Φ. Dealing with the next line, if we know at least one thing is Φ, then we know at least one thing out of the entire list of things is Φ, but we don’t know which one.

To cut the matter short and write out our rule for any given number of things, we’ll use “. . . n” to represent the continuation from c to some arbitrary amount “n”. Hence:

(x)(Φx) ≡^t [Φa • Φb • Φc • . . . • Φn]

and (∃x)(Φx) ≡^t [Φa ∨ Φb ∨ Φc ∨ . . . ∨ Φn]

When we want to prove invalidity of a bad syllogistic inference, we can start by assuming a world where just one thing exists and see if we can assign true values to the premises but a false one to the conclusion. If we cannot, we can try to use a two-world model and see if we can have true premises and a false conclusion. The upshot is that when we are dealing with quantified statements, we logically allowed to look for things that would serve as counterexamples to our conclusion.

We find counter-examples by trying to assign true values to instances of the premises and a false value to the conclusion here (while being consistent in our truth-value attributions, of course), thereby showing that the argument is not valid. Rather than by using a Venn Diagram, we can assume a world where only one thing exists and assign a constant to the “x” in the argument in both the premises and the conclusion. If the argument holds true for that, we can see if there is space for a second individual which falls outside the conclusion, or a third, etc.

In the following argument, we are told that one individual exists, but we can invalidate the conclusion by appealing to a second individual that stands as a counterexample.

Now, can we make the premises true but the conclusion false? Yes, we can! Just because we are told that there is an individual that is S and ~M doesn’t mean that there might also be and second individual thing that is S and M. And we know that if a thing is M then it is ~P. So, our S thing would be M and thus ~P. But this would give us (∃x)(Sx • ~Px), which is the same as ~(x)(Sx ⊃ Px). And so, we have a situation where the premises are true, but the conclusion is false. Hence, the argument does not force the conclusion and is therefore invalid.

Sometimes it’s not easy to pull a counter-example from thin air. A much more mechanical way to proceed is to systematically search for options where the conclusion is false, but the premises are true by using a truth table. We initially try to demonstrate this by utilizing a truth table where our premises are true, but the conclusion is false in an imagined world where only one individual exists. For this example, we will walk through building up the complete truth table one step at a time. First, let’s lay out the premises and conclusions as if M, P, and S only described “a”.

Truth-Table Proof of Invalidity for Invalid Example #2

Ma

Pa

Sa

Ma

⊃

~

Pa

Sa

•

~

Ma

Sa

⊃

Pa

Next, let’s assign a false value to our conditional-form conclusion statement:

Ma

Pa

Sa

Ma

⊃

~

Pa

Sa

•

~

Ma

Sa

⊃

Pa

F

 

From here, we use the truth conditions for the conditional to force the values for Sa and Pa. Since a conditional is only false if the antecedent is T while the consequent is false, we have:

Ma

Pa

Sa

Ma

⊃

~

Pa

Sa

•

~

Ma

Sa

⊃

Pa

F

T

T

F

F

Now, we use those assigned values to backfill the other instance of Sa and Pa where they occur.

Ma

Pa

Sa

Ma

⊃

~

Pa

Sa

•

~

Ma

Sa

⊃

Pa

F

T

F

T

T

F

F

Next, we start filling in the premises in an attempt to make them all true. Let’s start with the first premise. If Pa is F, the ~Pa is T. Hmmm. A conditional with a true consequent, here ~Pa, is true regardless of the antecedent. This leaves Ma undetermined. Let’s try the second premise, then. Here, Ma must be false for ~Ma to be true, which is the requirement of the conjunct. And so, we have:

Ma

Pa

Sa

Ma

⊃

~

Pa

Sa

•

~

Ma

Sa

⊃

Pa

F

T

T

F

T

T

T

F

T

F

F

Now we have the value for Ma, so we can fill out the rest of the table:

Truth-Table Proof of Invalidity for Invalid Example #2

Ma

Pa

Sa

Ma

⊃

~

Pa

Sa

•

~

Ma

Sa

⊃

Pa

F

F

T

F

T

T

F

T

T

T

F

T

F

F

And voila! We’ve given a consistent description of a possible scenario where the premises are true, but the conclusion is false.

This method works for worlds with two individuals as well. But the truth table gets quite big, fast. Recall that for two individuals:

(x)(Φx) ≡^t [Φa • Φb]

and (∃x)(Φx) ≡^t [Φa v Φb]

And so, this tells us we need to take whatever statement is being quantified, such as (Ma ⊃ ~Pa), and adjust it accordingly depending on the quantifier (x) or (∃x). For example compare “Single Item Instantiation” with “Two-Item Instantiation” of our example argument as given in the table below:

Invalid Example #2 Instantiated in Single Individual and Two-Individual Worlds
Single Item Instantiation	Two-Item Instantiation
P1. Ma ⊃ ~Pa P2. Sa • ~Pa ∴ Sa ⊃ Pa	P1. (Ma ⊃ ~Pa) • (Mb ⊃ ~ Pb) P2. (Sa • ~Pa) ∨ (Sb • ~Pb) ∴ (Sa ⊃ Pa) • (Sb ⊃ Pb)

The truth table ends up having many columns (to save space on the page, we’ve cut out the columns that list the variables). In our new two-item world it is possible that either a or b might end up being the instantiation of our counterexample. In the following fleshed out truth-table, it just so happens that the item that is S and ~P is the “b” item:

Truth Table Proof of Invalidity for Invalid Example #2 in a Two-Item World

(Ma

⊃

~

Pa)

•

(Mb

⊃

~

Pb)

(Sa

•

~

Ma)

∨

(Sb

•

~

Mb)

(Sa

⊃

Pa)

•

(Sb

⊃

Pb)

T

T

T

F

T

F

T

T

F

T

F

F

T

T

T

T

T

F

T

T

T

F

T

F

F

Luckily, because it only takes one possible, consistent model of a counterexample to prove invalidity, our truth-table only needs one row.

EXERCISES

Use the Truth-Table Method to show the following arguments are invalid:

1.

(P1) (∃x)(Fx • Kx)

∴ (x)(Fx ⊃ Kx)

2.

(P1) (x)(Hx ⊃ Mx)

(P2) (∃x)(Hx ∨ Px)

∴ (x)(Mx ⊃ Px)