Author: Timothy Eshing

Category: Logic and Reasoning

Word count: 1000

*Editor’s note: for many readers, this essay would be more profitably read after reading Formal Logic: Symbolizing Arguments in Sentential Logic by Thomas Metcalf.*

There are many varieties of formal logic of varying complexity. Here we survey one that’s somewhat more complex than *sentential* or *propositional* logic, but still fairly elementary: *quantificational *or *predicate* logic.^{[1]}

### 1. Atomic Formulas

We call symbolized sentences “formulas”. And we call some formulas “atomic” because they can’t be split into simpler formulas. We call all other formulas “molecular” or “complex”. In sentential logic, atomic formulas consist of single *sentence letters*.^{[2]}

In quantificational logic, by contrast, atomic formulas have a bit more structure. Now they consist of a *subject* or subjects and a *predicate*. Subjects are typically symbolized using lowercase letters, and predicates using uppercase letters. Three examples:

English | Sentential Logic | Quantificational Logic^{[3]} |

“ocrates is Suman.”h |
H | H(s) |

“ate is Cortal.”m |
M | M(c) |

“be Aoves leth.”B |
L | L(a,b) |

Our atomic formulas now contain more structure, but they can still behave just like they did in sentential logic, because all of the sentential logical constants are retained in quantificational logic:

“It is not the case thatocrates is Suman” or ““hocrates is Suman”.not h |
¬H | ¬H(s) |

“ate is Cortal mandbe Aoves leth.”B |
M ∧ L | M(c) ∧ L(a,b) |

“ate is Cortal morbe Aoves leth.”B |
M ∨ L | M(c) ∨ L(a,b) |

“Ifocrates is Suman, hthenate is Cortal.”m |
H → M | H(s) → M(c) |

“ocrates is Suman hif and only ifbe Aoves leth.”B |
H ↔ L | H(s) ↔ L(a,b) |

### 2. Variables and Quantifiers

Quantificational logic introduces two new kinds of symbols: variables and quantifiers.

*Variables *are typically symbolized with *x*, *y*, etc. Just like in algebra, variables in quantificational logic stand in for unknown or unspecified items. In algebra, variables stand in for numbers. In quantificational logic, they stand in for the subjects of our formulas.^{[4]} Three examples:

English^{[5]} |
Quantified Logic |

“ is xuman.”h |
H(x) |

“ is yortal.”m |
M(y) |

“xoves l.”y |
L(x,y) |

In these examples, the variables are “free”. But formulas with free variables are not *sentences* in quantificational logic.^{[6]} Variables have to be “bound”.

To do that, in quantificational logic we add two more symbols to the roster from sentential logic: the *existential* and *universal* *quantifiers*:

Name | Symbol^{[7]} |
English |

Existential Quantifier | ∃x |
“ …” or “For some x, …”.There is/exists an x that |

Universal Quantifier | ∀x |
“ …” or “For all x, …”.Every x^{[8]} |

Quantifiers *quantify over* or “bind” the variables in our formulas. Eight examples:

Formula | English |

∃x[H(x)] |
“For some x, is xuman” or “h is There is/exists an x thatuman”.h |

∀y[H(y)] |
“For all y, is yuman” or “h is Every yuman”.h |

¬∀z[M(z)] |
“It is not the case that is for all z,zortal” or “m is a Not every zortal”.m |

∃z[¬M(z)] |
“For some z, is zortal” or “not m is There is/exists a z thatortal”.imm^{[9]} |

∃y[H(y) ∧ M(y)] |
“ is For some y,yuman h is and yortal” or “m is There is/exists a y thatuman handortal” or simply “Something is a human mortal”.m |

∀x[H(x) → M(x)] |
“For all x, is if xuman, hthen is xortal” or “mEveryuman is hortal”.m |

∃y∀z[L(y,z)] |
“For some y, for all z,yoves l” or “zThere is/exists a y thatoves l” or simply “Something loves everything”.every z |

∀y∃z[L(y,z)] |
“For all y, there is/exists a z thatyoves” or “lEvery yoves l” or simply “Everything loves something”.some z^{[10]} |

Notice how we can bind both atomic (simple) and molecular (complex) formulas. Also notice how we can employ multiple quantifiers, and how their order matters.

### 3. Proofs

To prove things in quantificational logic, we must specify a *domain* containing all the individuals that can be subjects in our formulas.

Sometimes our domain includes absolutely everything there is. In that case, “Something is human”, ∃*x*[H(*x*)], is true, because humans such as you and me exist. But “Everything is human”, ∀*x*[H(*x*)], is false, because other things exist, too, like potatoes. Yet here is something seemingly true of absolutely everything: ∀*x*[P(*x*) ∨ ¬P(*x*)], “Everything is a potato or not a potato”!

Our domain doesn’t have to include absolutely everything though. Suppose our domain only contains three classmates: ** A**be,

**eth, and**

__B__**ate. They’re all**

__C__**tudents: S(a), S(b), and S(c). Now we can**

__s__*universally generalize*over our domain and infer that

*everyone*is a student: ∀

*x*[S(

*x*)].

We can also go in reverse by *instantiating* (giving an instance of) that universal generalization. If we know that everyone is a student, ∀*x*[S(*x*)], then we can infer that some specific individual in our domain, like Abe, is a student: S(a).

Suppose not all the members of our domain are the same gender, but at least one of them, ** B**eth, is

**emale: F(b). Now we can**

__f__*existentially generalize*from that fact and infer that

*someone*is female: ∃

*x*[F(

*x*)].

We can go in reverse here too, but we have to be careful. If we know that someone is female, ∃*x*[F(*x*)], but don’t know who that is, then we can instantiate by giving that someone a new name, one we haven’t used already, like “d”. Then we can infer that whomever “d” refers to is female: F(d). It just so happens that “d” could refer to the same person as another letter we’ve already used, like “b” for Beth.

Here’s a formal proof (whose domain includes absolutely everything) using two such inferences. It demonstrates that from “every human is mortal” and “Socrates is human” we can infer “something is mortal”:

Number | Formula | English | Rule^{[11]} |

1. | ∀x[H(x) → M(x)] |
“Every human is mortal.” | Assumption |

2. | H(s) | “Socrates is human.” | Assumption |

3. | H(s) → M(s) | “If Socrates is human, then Socrates is mortal.” | From 1 |

4. | M(s) | “Socrates is mortal.” | From 2 and 3 |

5. | ∴∃x[M(x)] |
“Therefore, something is mortal.” | From 4 |

### 4. Beyond Quantificational Logic

Once we know the rules for quantificational logic, it’s not difficult to symbolize even more complex sentences and arguments. For example, we might want to quantify over predicates and not just subjects, or formalize generalizations that are probably or necessarily true or false; but those are topics for other entries.

### Notes

^{[1]} Sentential logic is also known as “propositional” or *zeroth-order* logic. Quantificational (sometimes “quantified”) logic is also known as “predicate” or *first-order *logic.

Note: the labels “predicate” and “first-order” and even “quantified” are far more common in the literature than “quantificational”. But “quantified” isn’t quite apt, because the logic itself is not quantified. And “predicate” isn’t quite apt, because what’s distinctive about the logic is not just the subject-predicate structure of its sentences, but also the use of quantifiers ranging over subject variables. “First-order” is apt, because herein we only quantify over subjects, and not predicates (as we do in second-order logic); but that’s a more mysterious label than “quantificational”, so the latter is the preferred term in this essay.

^{[2]} See section 1 of Formal Logic: Symbolizing Arguments in Sentential Logic by Thomas Metcalf.

^{[3]} Notice how compared to English the order of subject and predicate is reversed: the predicate comes first, next comes the subject(s) to which it applies.

^{[4]} But not predicates. However, in *second-order *logic we can also quantify over predicates.

^{[5]} Technically, we should write “⌜*x*⌝ is human”, “⌜*y*⌝ is mortal”, etc., where the variables *x* and *y* are placed in *quasi-quotes*. This is because we’re not saying that the letter *x* itself is human, or that the letter *y* itself is mortal, but rather that something as yet undefined we’ll put in place of *x* or *y *(like Abe or Beth) is human or mortal. For more information, see note 6 of Formal Logic: Symbolizing Arguments in Sentential Logic by Thomas Metcalf. As in that essay, herein quasi-quotation marks are left out to avoid confusion.

^{[6]} To be more precise: such formulas are *well-formed* expressions, but they are not *semantically complete*, which means we cannot evaluate them for truth or falsity. “Sentence” is a technical term defined as a well-formed formula with no free variables. In quantificational logic, only sentences can have truth-values. Formulas with free variables are often called “sentence functions” (or something similar) to indicate that they are not themselves sentences, but mere building blocks of sentences: they can be turned into sentences by either (i) replacing all their free variables with one or more individual constants representing one or more definite subjects, or (ii) binding their variables with at least one quantifier as described in the main text above.

^{[7]} Alternative symbol for the universal quantifier: “(*x*)”. Sometimes people place the symbols above in parentheses too: “(∃*x*)” and “(∀*x*)”, but doing so is just for readability; it doesn’t change how the logic works.

^{[8]} Note: the universal quantifier does not imply that anything exists. For example, we can rightly say that “All unicorns are horned”, translating that into quantified logic as “∀*x*[U(*x*) → H(*x*)]”, without committing ourselves to the existence of any unicorns. So “All unicorns are horned” can be true, even when “There exists a unicorn”, “∃*x*[U(*x*)]”, is false.

^{[9]} Notice that this example means the same as the previous one: we can swap “¬∀*x*” with “∃*x*¬”, as well as “∀*x*¬” with “¬∃*x*”. This is because the quantifiers are *duals*, which means we can define one in terms of the other plus the negation operator: “∀*x*” is equivalent to “¬∃*x*¬”, and “∃*x*” is equivalent to “¬∀*x*¬”. For example, “Everything is a potato” means the same as “nothing is not a potato”, and “something is a potato” means the same as “not everything is not a potato”.

^{[10]} To be clear: “Everything loves something, but not necessarily the same thing”. To formalize “Everything loves one and the same thing” we’d write “∃*y*∀*z*[L(*z*,*y*)]”, that is, “There is something that everything loves”. Compare this to the previous example!

^{[11]} In most proof systems, we don’t just say where the line came from; we also say why, by citing a specific rule for each line. To save time and avoid confusion, in this presentation we’ve just said where each line comes from, without saying how. Line 3 follows from line 1 by *universal instantiation *and line 5 follows from line 4 by *existential generalization*. Both of these rules were discussed in the main text above. The other two rules we discussed, *universal generalization* and *existential instantiation*, are more complicated to use, so we’ve avoided them in this example. Line 4 follows from lines 2 and 3 by a different rule, called *modus ponens*, which works like this: from the truth of both “P” and “If P, then Q”, we can infer that “Q” is true, too; but this and other rules of inference are topics for other entries.

### For Further Reading

*The following are all excellent introductory textbooks covering quantificational logic:*

Allen, Colin and Michael Hand: *Logic Primer* (2nd ed., Bradford, 2011).

Barker-Plummer, Dave, Jon Barwise, and John Etchemendy: *Language, Proof, and Logic* (2nd. ed., CSLI Publications, 2011).

Copi, Irving M., Carl Cohen, and Kenneth McMahon: *Introduction to Logic* (14th ed., Routledge, 2016).

Gensler, Harry J.: *Introduction to Logic* (3rd. ed., Routledge, 2017).

Hodges, Wilfred: *Logic: An Introduction to Elementary Logic* (Penguin, 2001).

Howard-Snyder, Frances, Daniel Howard-Snyder, and Ryan Wasserman: *The Power of Logic* (6th ed., McGraw-Hill, 2020).

Hurley, Patrick J. and Lori Watson: *A Concise Introduction to Logic* (13th ed., Cengage Learning, 2018).

Magnus, P. D. et al.: *forall x: Calgary: An Introduction to Formal Logic* (issued under a CC BY 4.0 license, 2011).

### Related Essays

Formal Logic: Symbolizing Arguments in Sentential Logic by Thomas Metcalf

Critical Thinking: What is it to be a Critical Thinker? by Carolina Flores

### About the Author

Timothy Eshing is a philosophy graduate student at Saint Louis University.

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