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I’m having a hard time determining if this sentence would be a tautology or an open sentence.

x is a multiple of 7 or x is not a multiple of 7.

I’m not sure if this would be a tautology because of the free variable x. For example, the sentence “x is a multiple of 7.” would be an open sentence and therefore not have a truth value. “x is not a multiple of 7” would also be an open sentence and therefore not have a truth table. Therefore I don’t see the sentence shown as a tautology, I see it as an open sentence. If the statement was:

$(\forall x \in \mathbb{R})$ (x is a multiple of 7 or x is not a multiple of 7)

I would see this as a tautology because x was is bounded by the quantifier $\forall$. Can someone explain if I’m wrong? Thank you!

Dr. J
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2 Answers2

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Informally, an open sentence is a formula depending on some variables whose truth value might, but does not necessarily have to, depend on these variables.

For instance, if $A(x)$ is the formula $x=0$, then $A(x)$ is an open sentence whose truth value clearly depends on $x$. However, we might also consider the formula $A(x)$ given by $0=0$. In that case, $A(x)$ is always true, independent of the actual value of $x$. In that case, $A(x)$ is an open sentence, but still a tautology.

For any formula $A$, the formula $B$ given by $A\vee\neg A$ is a tautology. Similarly, for a formula $A(x)$ depending on some variable $x$, also the formula $B(x)$ given by $A(x)\vee\neg A(x)$ is a tautology.

Zuy
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  • I have two questions.

    1. I thought a statement or open sentence that is atomic can’t be a tautology. So how is A(x) given by 0=0 a tautology?

    2. If A(x) or ~A(x) is a tautology, does it have a truth value?

    Thank you for your help!

     – Dr. J Aug 04 '23 at 20:45
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Everybody agrees that A ∨ ¬A is a tautology, but in the context of first-order logic only some authors also call ∀x (Ax ∨ ¬Ax) a tautology. For clarity, we can always say that the latter is a first-order validity but not a propositional-logic tautology.

x is a multiple of 7 or x is not a multiple of 7

I don’t see this sentence shown as a tautology, I see it as an open sentence.

This open formula (i.e., not a sentence, due to the free variable) can be formalised as P or not P, so it is a propositional-logic tautology, so it is also a validity.

  1. I thought an atomic statement or atomic open sentence can’t be a tautology. So how is 0=0 a tautology?

Being atomic, it indeed does not make sense for the validities x=x (an open formula) and 0=0 (a sentence) to be propositional-logic tautologies.

  1. Does A(x) or not A(x) have a truth value?

∀x∈R (x is a multiple of 7 or x is not a multiple of 7)

I see this sentence as a tautology.

The open formula A(x) or not A(x) is valid, i.e., logically true. So, the sentence ∀x∈R (x is a multiple of 7 or x is not a multiple of 7) is also valid; but it is not a propositional-logic tautology, since in propositional logic it is atomic.

ryang
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  • I think I’m understanding but can you explain what you mean by “valid sentence” and “first-order tautology”? Thank you! – Dr. J Aug 05 '23 at 01:58
  • @Dr.J Roughly speaking, a validity (I don't generally use the phrase "first-order tautology" except to disambiguate) is a formula that is true no matter how you assign meaning to it, that is, that is true in the abstract, by the sheer force of logic. For example, I can tell that your second example is logically valid without knowing what "multiple of 7" means. Determining a formula's validity is a culture-free process. (PL) tautologies is a strict subset of logical validities. – ryang Aug 05 '23 at 02:08
  • so you mentioned that x=x is a validity. From my understanding of your explanation x=x is an open formula but it is a validity because no matter what x is this is true. But how would 1=1 be a validity? My understanding is that statements can take on a truth value of either true or false. 1=1 is a statement that evaluates to true, so what makes it a validity? What am I missing here? – Dr. J Aug 05 '23 at 02:11
  • I guess what I’m trying to ask is this. I’m thinking you’re using validity and first-order tautology interchangeably. So what I’m thinking is that 1=1 wouldn’t be a first-order tautology because it isn’t true based on the “form” of the statement. The statement evaluates to true because in fact 1 does equal 1. If I wrote 2=1 this would be false because 2 does not equal 1. – Dr. J Aug 05 '23 at 02:19
  • "But how would 1=1 be a validity? 1=1 is a statement that evaluates to true, so what makes it a validity?" Haven't you just answered your own question? To be clear: the given sentence is true even when the 1 in it does not stand for the usual arithmetical one; it is logically true. $\quad$ P.S. The link in the link contains a related discussion. – ryang Aug 05 '23 at 02:21
  • @Dr.J 2=1—unlike 1=1—is certainly not a validity. (Very tangential note: in a single-object universe, 2 and 1 stand for the same object, in which case the sentence 2=1 is actually true.) – ryang Aug 05 '23 at 02:34
  • I see. So 1 = 1 is a validity because we are thinking of 1 as some type of object. What I mean by this is 1 is some object (it doesn’t have to be the number 1) so since the object is on both sides of the equal sign the statement is true no matter what. And 2 = 1 is not logically false in a single-object universe because 2 and 1 would be the same object. So in the single object universe 2 = 1 would be logically true right? Is this just an abstract way of thinking in logic? – Dr. J Aug 05 '23 at 02:35
  • @Dr.J You're very close in your understanding: everything absolutely correct except your last sentence. In a single object universe, 2 = 1 is true. I would not say that 2 = 1 is logically true nor say that it is logically false, to avoid giving the impression that it is valid (the former) or that it is unsatisfiable (the latter). – ryang Aug 05 '23 at 02:39
  • Why would it not be logically true? I’m having a hard time understanding why if 2 and 1 are the same object. Would it be because 2 and 1 are not the same object in every universe? – Dr. J Aug 05 '23 at 02:45
  • So I think I get it. So I know why 1 = 1 would be valid. 2 = 1 wouldn’t be valid because if we think of 2 and 1 as the numbers we are used to seeing this would be a false statement. It would not be invalid(i.e. false no matter what the meaning of the statement is) because we can think of the single object universe. But the statement “1 = 1 $\wedge$ 1 $\neq$ 1” is invalid. Am I understanding this correct? – Dr. J Aug 05 '23 at 03:06
  • @Dr.J 1. An invalid sentence can be true (it just isn't true regardless of interpretation). So, your last comment is almost correct: just replace every instance of "invalid" with "unsatisfiable"; fuller explanation here. $\quad$ 2. "Logically true" strongly suggests—if not outright means—that a formula is a validity. Refer to my first comment. $\quad$ (SUMMARY: For avoidance of confusion, I'd refrain from saying "tautology" when I merely mean validity, and I'd refrain from saying "logically true" when I merely mean true.) – ryang Aug 05 '23 at 03:14