3
$\begingroup$

I saw this video: Tautologies and Contradictions; however, the example he claimed to be a tautology, "That dog is a mammal", is actually NOT a tautology, if I refer to the textbook, A Tour Through Mathematical Logic by Wolf:

(Page 11)
Definition. A statement P is called a tautology or law of propositional logic if there is a set of substatements of P such that:

(a) P is a propositional combination of those substatements, and
(b) P is true for every combination of truth values that is assigned to these substatements.

...

Example 3. The equation $2 + 2 = 4$ is not a tautology. Its only substatement is itself, [...] it's not a tautology because its form is simply "P", with no shorter substatements
...
Similarly, "For every number x, x = x" is not a tautology. Simply put, it cannot be a tautology because it includes no connectives.

I am genuinely confused. I am sure this guy in the video is well-educated in math. (He got his math PhD from University of Toronto). So can "tautology" be defined differently depending on the "version" of mathematical logic theories? Or is the guy just misleading people (which I would be very surprised about, given his credentials)?

Thanks!

$\endgroup$
6
  • 2
    $\begingroup$ No it is not a tautolgy in the context of formal logic. $\endgroup$
    – Mauro ALLEGRANZA
    Aug 7, 2022 at 14:29
  • 1
    $\begingroup$ I am genuinely confused why you are attempting to learn logic from an imprecise and misleading source. Ask most logicians and they would tell you "∀x ( x = x )" IS a tautology. It is fine if people want to define "tautology" in a different way, but not when their 'definition' is so vague as this. $\endgroup$
    – user21820
    Aug 7, 2022 at 21:31
  • 1
    $\begingroup$ @user21820 Would you please recommend me a good textbook to learn logic then? Ideally it covers Zermelo–Fraenkel set theory in detail $\endgroup$
    – Erin
    Aug 7, 2022 at 23:16
  • 2
    $\begingroup$ @Erin In propositional logic (what your book and the video are discussing), 'logical truth' and 'tautology' are synonyms. In predicate logic (i.e., first-order logic, i.e. FOL), 'logical truth' and 'validity' are synonyms, so ∀x(x=x) is always a validity. $\tag*{}$ Now, authors who never need to discuss and don't care about tautologies in the propositional-logic sense use 'tautology' interchangeably with the other two terms, and in this sense, ∀x(x=x) is "a tautology". Alternatively, one can say that ∀x(x=x) is "a first-order tautology / FOL tautology" but isn't "a tautology". $\endgroup$
    – ryang
    Aug 8, 2022 at 6:10
  • 1
    $\begingroup$ @Erin: If you have undergraduate mathematics background, Rautenberg's "Concise Introduction to Mathematical Logic" is best for logic (and much more precise than the book you quoted), and after that Jech's "Set Theory" is good for ZFC (and also precise). $\endgroup$
    – user21820
    Aug 10, 2022 at 7:45

5 Answers 5

Reset to default
7
$\begingroup$

  1. Let $d$ be that particular dog, $M(x)$ denote “$x$ is a mammal”, and ‘mammal’ be as defined by the Oxford dictionary.

    Then “that dog is a mammal” is formalised as $$M$$ in propositional logic, and as $$M(d)$$ in predicate logic. In either case, the statement is an atomic sentence, whose column in its truth table contains both True and False, so is not a tautology.

    It is not a validity (i.e., first-order tautology) either, since varying the definition of ‘mammal’ (an axiom) and, consequently, the interpretation can result in the statement becoming false.

  2. Dr. Bazett's mistake arises from his vague definition of a tautology as “a statement that is always true”.

    Sure, the given statement is (always) true in the given context, but it is not true regardless of interpretation, and certainly not always true in its truth-functional form.

  3. Note that even the stronger statement “every dog is a mammal”, which can be formalised (depending on the universe of discourse) either as $$∀x\:\Big(D(x)\implies M(x)\Big)$$ or as $$∀x\,M(x),$$ is neither valid (counterexample: the domain of discourse $\mathbb R,$ with $D(x)$ and $M(x)$ denoting “$x$ is positive” and “$x$ is even”, respectively) nor tautological; in other words, it is not logically true.

  4. On the other hand, the following are all valid arguments: \begin{align}\forall x\:\Big(D(x)\to M(x)\Big)\;\land\; D(d)\implies M(d),\\∀x\,M(x) \implies M(d),\\M(x) \;\land\; x=d\implies M(d).\end{align} To be clear: it is the implications (which aren't tautologies), not their conclusion $M(d)$ (“that dog is a mammal”), that are valid (first-order tautological).

    So, “that dog is a mammal” is true by definition then deduction, so we say that it is analytically true (as opposed to synthetically true).

    The analogous statement “that mushroom is a plant” is similarly not logically true: it too used to be analytically true, but has been false since the 1960's when fungi got redesignated as a taxonomic kingdom.

$\endgroup$
2
  • $\begingroup$ Thank you for the detailed explanation. Could you please recommend a good textbook to refer to, if I want to self-study? i think the text I mentioned in my description has some mistakes? $\endgroup$
    – Erin
    Aug 9, 2022 at 2:50
  • $\begingroup$ @Erin The text that you excerpted is perfectly accurate and precise. What mistake are you referring to? $\endgroup$
    – ryang
    Aug 9, 2022 at 4:09
2
$\begingroup$

The definition you quote is in the context of propositional logic, where statements are made using truth-valued variables. Statements like "dog X is a Spaniel" or "dog X is a mammal" are better represented in predicate logic, where statements can be made using truth-valued functions (a.k.a. predicates).

So if we're talking about predicates, it is reasonable to define a predicate as a tautology if it is true for all possible input values (keeping in mind that a predicate is a function). This would make the predicate "dog X is a mammal" a tautology, because it is true regardless of which dog you choose X to be; all dogs are mammals.

So the statement in the video is not wrong. That said, it's perhaps not the best example for explaining what a tautology is; normally we only talk about tautologies in propositional logic.

$\endgroup$
1
  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – Xander Henderson ♦
    Aug 12, 2022 at 0:14
0
$\begingroup$

What I think, is that they both mean the same thing. We can define two statements p and q such that

p says that the given creature is a dog q says that the given creature is a mammal

when we analyse p \Rightarrow q (the statement which the professor said), we find that there is only one condition in which the statement is false, i.e. , When the creature is a dog but not a mammal, which cannot be a case in real life, so rejecting that we can conclude that that statement is a tautology.

also i would've commented but the website wouldn't let me (i'm new)

$\endgroup$
0
$\begingroup$

I have not watched the video & I have no access to the book you referred to.

I am going only by the contents of this Question.

While the textbook is aimed at "Mathematicians & Logicians" using Definitions which are formal & rigorous, the video is aimed at "Newbie Non-Mathematicians & Non-Logicians" using Casual Definitions which are not formal & not rigorous.

The textbook is very strict with notation, aiming to make it a "Proper" theory.
The video is very lax with notation, aiming to introduce the Ideas & Concepts.

There is a way to treat both the video & the book in the Exact Same Way :

"That Dog is a Mammal"
"This Dog is a Mammal"
"The last Dog is a Mammal"
"The first Dog is a Mammal"

Here we have many Dogs, where "that" or "this" or some other Position is a variable to refer to various Particular Dogs.

Given Statement is true for all values of the given variable.
Thus it is a "Tautology" !

This is the outcome of "all Dogs are Mammals" which implies that "this Particular Dog is a Mammal" tautologically.

Consider :
"That Creature is a Mammal"

This is not a "Tautology", because it may or may not be true until we assign a value to the variable "that Creature" in the Statement !

$\endgroup$
0
$\begingroup$

A statement can be true or false. For example, "Aristotle is a human" is a statement. A statement's truth value is true or false, $T$ or $F$. The truth value of "Aristotle is human" is $T$.

A propositional form is an expression consisting of statements and given connectives like NOT, OR, AND. We can evaluate the truth value of a propositional form. Let's say $p,q$ are statements. The truth table (a table characterizing the truth values) of AND is:

A propositional form is a tautology if and only if it is ALWAYS TRUE, no matter what truth values the individual statements (like $p,q$) carry.

For example, the statement $p \textrm{ OR NOT}(p) $ is always true no matter if $p$ itself is true, so it is a tautology.

Now, a statement of the form $x \textrm{ is a dog} \Rightarrow x \textrm{ is a mammal}$ is a true for any $x$, but I don't know if I'd call that a tautology as it is more in the regime of predicate logic.

I can say that $2+2=4$ is not a tautology as it's just a statement. Something is only a tautology if it's true no matter what truth values of individual statements are, yet if $2+2\neq 4$, it isn't true.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .