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I just came across a use of the phrase "vacuous tautology" used in a piece of writing. Now, based on my understanding, by definition a tautology has no content and is therefore always vacuous. This would make the use of the word vacuous in "vacuous tautology" redundant.

However, I have only passing familiarity with formal logic, so I wanted to know if "vacuous tautology" uncovers some lack of understanding I have of the concepts of tautology and vacuity. Are there cases in which a tautology is not vacuous? If so, what are the conditions for a statement to be called a vacuous tautology rather than just a plain old vanilla tautology?

Perhaps this question is more appropriate for the English SE, but I thought I would ask here since I am really interested in the use of tautology and vacuous from the standpoint of formal logic.

  • I take a vacuous tautology to mean a tautology based solely on negation. – user1362 Jan 14 '12 at 13:21
  • @S.T.Mannew: what is "a tautology based solely on negation"? Can you give an example of one that is and one that isn't? – Mitch Jan 15 '12 at 16:49
  • The term was also used in an answer here: philosophy.stackexchange.com/a/4361/1127 – draks ... Sep 30 '15 at 13:16
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As you seem to suspect, the phrase "vacuous tautology" is pleonastic. The modifier "vacuous" is not necessary.

However, it is probably being used to rhetorically highlight the particular vacuity of the tautology.

It is not a term of art, and has no specifically defined meaning as a sub-type of tautology.

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Some tautologies are obvious; others are not. I assume that the writer was trying to convey that the tautology was obvious. A non-obvious tautology can still be illuminating, since you may approach a problem two different ways and not realize that two different claims actually work out to be the same thing.

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    So rather than a technical usage of vacuous, it's more figurative. For example: 'P -> P' would be considered vacuous, but 'P->(Q->R) <-> (P->Q)->(P->R)' is not so obvious. – Mitch Jan 14 '12 at 4:27
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In his book "About Time" author Adam Frank explains a philosophical (opposed to the mathematical) aspect of work on the cosmological argument. Managing the troubling issue of a universe which evolved under "exactly the right conditions", scientists have used tautologies. Here are a few examples,

  1. "The universe and its laws must take a form consistent with our existence within it."
  2. "The existence of life tells us the universe has to allow life to exist."
  3. "The laws of physics must take a form that makes life a necessary feature of cosmic evolution."

As you can see statements 1 and 2 are sort of useless, while statement 3 makes a bold statement about the laws of physics. In this context I might suggest example 1 and 2 are "vacuous tautologies" while 3 is not.

  • +1; this makes me think about certain relationships in mathematics, for which some formulations may seem "trivial" or "obvious", whereas an equivalent formulation might appear much less so. – Joseph Weissman Jan 15 '12 at 0:37
  • You're not suggesting (3) is a tautology, is it? Not only is it not, it is probably not even true. (Swapping an existential quantifier for a universal one is not a tautology.) – Rex Kerr Jan 15 '12 at 11:29
  • While I can't take credit for making the allusion #3 is a tautology. And in the author's defense, he does admit the statement will not sit well with some physicists. As a neophyte, I appreciate what I perceive as a progression from tautology: (true) human rules of physics must take a form consistent with humans ruled by (true) physics --versus-- human rules of physics must take a form consistent with humans existing in an evolving physics. – xtian Jan 17 '12 at 2:17
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One possible meaning of "vacuous tautology" is for those tautologies where there is no line in the truth table where all of the premises are true. This would not include all tautologies. So maybe that was the intent.

However, the phrase "vacuous tautology" should raise questions. Since the source of the phrase is missing all we have to go on is to try to make sense of the phrase as it stands using other references to get a clearer definition.

The OP offers the following for consideration:

Now, based on my understanding, by definition a tautology has no content and is therefore always vacuous.

A tautology, however, is not without content. It has a semantic content and means that the well-formed formula is true for all possible valuations of the atomic sentences composing it. This is how forall x (page 70) describes it for truth-functional logic (TFL) also known as propositional logic:

...we explained necessary truth and necessary falsity. Both notions have surrogates in TFL. We will start with a surrogate for necessary truth.

A is a TAUTOLOGY if it is true on every valuation.

We can determine whether a sentence is a tautology just by using truth tables. If the sentence is true on every line of a complete truth table, then it is true on every valuation, so it is a tautology.

The idea of a tautology being true on every valuation is semantic content. So it does have some content, some semantics, but nothing else.


Quine discusses the semantics of English sentences (syllogisms) in more detail and whether the words in those sentences are essential or vacuous. (page 2):

A word may be said to occur essentially in a statement if replacement of the word by another is capable of turning the statement into a falsehood. When this is not the case, the word may be said to occur vacuously. Thus the words 'Socrates' and 'man' occur essentially in the statement 'Socrates is a man', since the statements 'Bucephalus is a man' and 'Socrates is a horse' are false; on the other hand 'Socrates' and 'mortal' occur vacuously in ['Socrates is mortal or Socrates is not mortal'] and 'Socrates', 'man', and 'mortal' occur vacuously in ['If every man is mortal and Socrates is a man then Socrates is mortal']. The logical truths, then, are describable as those truths in which only the basic particles alluded to earlier occur essentially.

Note that this description applies to English sentences and syllogisms not to truth-functional logic where most of this content would be lost when the English sentence is symbolized as some letter, say, "P".


Graham Priest describes a tautology "q and ~q implies p" (page 18):

There is no row in which both of the premises are true and the conclusion is false. Indeed there is no row in which both of the premises are true. The conclusion doesn't really matter at all! Sometimes, logicians describe this situation by saying that the inference is vacuously valid, just because the premises could never be true together.

This might be the criteria for a vacuous tautology. It would be a tautology for which no row of the truth table has all of the premises true. Non-vacuous tautologies would have at least one row where the premises are all true, such as the tautology, "p implies p". Here is a truth table for the tautology Priest considered:

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Tautologies are often considered together with the ability to construct a truth table. This restricts them to truth-functional logic, but the concept of vacuous has additional value in first order logic where there are quantifiers over terms representing members of a domain. The domain is assumed to be non-empty unless one allows "free logic", but a predicate in first order logic may be empty. These empty predicates can be called vacuously true if there are no elements of the domain having that predicate.

Here is how forall x (page 165) describes this situation:

...we emphasized that a name must pick out exactly one object in the domain. However, a predicate need not apply to anything in the domain. A predicate that applies to nothing in the domain is called an EMPTY PREDICATE.

Consider the domain of animals. Let 'Rx' be 'x is a refrigerator' and let 'Mx' be 'x is a monkey'. How should we treat '∀x(Rx → Mx)'? They continue (page 166):

This symbolizes ‘every refrigerator is a monkey’. This sentence is true, given our symbolization key, which is counterintuitive, since we (presumably) do not want to say that there are a whole bunch of refrigerator monkeys. It is important to remember, though, that ‘∀x(Rx → Mx)’ is true if any member of the domain that is a refrigerator is a monkey. Since the domain is animals, there are no refrigerators in the domain. Again, then, the sentence is vacuously true.


Keeping the above in mind, let's summarize why the phrase "vacuous tautology" should cause the OP to be suspicious.

The idea of a term being vacuous usually applies to words in English sentences (syllogisms) or to sentences of first order logic involving empty predicates. Tautologies which are well-formed formulas of truth-functional logic are at least intuitively separate from both of these contexts. So referring to something being a vacuous tautology requires more explanation.

However, if a tautology is formed when there is no situation, or row of a truth table, where all the premises are true, then that might represent a vacuous tautology in a way that is not redundant.


Reference

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/ Wikipedia, "Fitch notation" https://en.wikipedia.org/wiki/Fitch_notation

Nolt, John, "Free Logic", The Stanford Encyclopedia of Philosophy (Fall 2018 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/fall2018/entries/logic-free/.

Priest, G. (2010). Logic: A brief insight.

Quine, W. V. O. (1951). Mathematical Logic... Revised edition. Harvard University Press.

Stanford Truth Table Generator, http://web.stanford.edu/class/cs103/tools/truth-table-tool/

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