# Who invented $\vee$ and $\wedge$, $\forall$ and $\exists$?

I can rather easily imagine that some mathematician/logician had the idea to symbolize "it E xists" by $\exists$ - a reversed E - and after that some other (imitative) mathematician/logician had the idea to symbolize "for A ll" by $\forall$ - a reversed A. Or vice versa. (Maybe it was one and the same person.)

What is hard (for me) to imagine is, how the one who invented $\forall$ could fail to consider the notations $\vee$ and $\wedge$ such that today $(\forall x \in X) P(x)$ must be spelled out $\bigwedge_{x\in X} P(x)$ instead of $\bigvee_{x\in X}P(x)$? (Or vice versa.)

Since I know that this is not a real question, let me ask it like this: Where can I find more about this observation?

• One thing is for sure: $\vee$ is for V el (= or). $\wedge$ could be for A nd. – Hans-Peter Stricker Aug 26 '11 at 17:19
• One big thing in favour of $\forall$ and $\exists$: The typesetter you went to could be counted on to have "A" and "E" in a sans-serif font and turning a single letter 180 degrees is trivial. Mathematical typesetting didn't develop in a vacuum - you had to pay more to get pages with lots of math in it typeset, so it was only with the advent of TeX that we could get virtually any symbol in our printed output. – kahen Aug 26 '11 at 17:25
• @kahen: Good answer! (Maybe it was not a logician who had the idea but his typesetter?) With $\vee$ also $\wedge$ was available at those times, and from those two signs the quantifier signs could/should have been derived. – Hans-Peter Stricker Aug 26 '11 at 17:49
• The title and body don't seem to match. Which question are you more interested in? – Qiaochu Yuan Aug 27 '11 at 2:21

See Earliest Uses of Symbols of Set Theory and Logic for this and much more.

• +1, but my question is: How could such obviously failing symbolizations so successfully survive? – Hans-Peter Stricker Aug 26 '11 at 17:57
• Maybe this is a nitpick. It may well be that $\vee$ and $\wedge$ appeared first in the papers of Russel 1908 and Heyting 1930 as claimed on that page. However, Peano, Arithmetices principia: nova methodo (1889) uses $\cap$ "et" and $\cup$ "vel" for that. See pages VI, VII and VIII. Even if that is not the exact same symbol, it seems to have the same meaning. – t.b. Aug 26 '11 at 17:58
• And here's the similar Table of Symbols from Arithmetices ![](i.stack.imgur.com/oKlyj.jpg) – Gone Aug 26 '11 at 19:36
• Although it is a very nice reference, $\bigvee$ and $\bigwedge$ are not mentioned on that page. I wasn't the downvoter, but that might be their reasoning. I agree that downvoting for a good reference to part of the answer is curious. – robjohn Aug 27 '11 at 1:18

The four types of propositions used in the classical Greek syllogisms were called A, E, I, O. Statements of type A were "All p are q". Statement of type E were "Some p are q". So of course a millennium later, mathematicians (who had a classical education) used A and E for these quantifiers, then later turned them upside down to avoid confusion with letters used for other things.

By the way: I and O were "All p are not q" = "No p are q" and "Some p are not q"="Not all p are q", but I don't remember which is I and which is O.

• There's so much to know! – Hans-Peter Stricker Aug 26 '11 at 18:45
• The only thing here comes as that "A", "E", "I", and "O" in Aristotelian logic (which isn't precisely traditional logic), aren't necessarily what we regard as quantifiers. They might come as sentence-forming functors of two name arguments. Lukasiewicz, an excellent scholar in logic and the history of logic, viewed them as sentence-forming functors of two name arguments, and thus in some way distinct from quantifiers. See Lukasiewicz's Elements of Mathematical Logic, and especially Aristotle's Syllogistic: From the Standpoint of Modern Formal Logic. – Doug Spoonwood Aug 26 '11 at 19:41
• They also had funny names as mnemoics for their inference rules, such as Barbara, Celarent, etc, where the vowels encoded the sentence forms. – starblue Aug 26 '11 at 20:28
• @Doug: Thanks for the hint on Lukasiewicz. – Hans-Peter Stricker Aug 29 '11 at 8:17
• The syllogisms were Greek, but the letter coding A E I O was given by Latin scholars. See Wikipedia: syllogism "According to Copi, p. 127: 'The letter names are presumed to come from the Latin words "AffIrmo" and "nEgO," which mean "I affirm" and "I deny," respectively; the first capitalized letter of each word is for universal, the second for particular' " – magma Jan 14 '12 at 1:11

My understanding of the quantifier symbols $\bigvee$ ("there exists") and $\bigwedge$ ("for all") was that they were supposed to be large versions of $\vee$ ("or") and $\wedge$ ("and"). Then $\bigvee_{x\in X}Fx$ would mean $Fx_1\vee Fx_2\vee Fx_3\vee\dots$ whereas $\bigwedge_{x\in X}Fx$ would mean $Fx_1\wedge Fx_2\wedge Fx_3\wedge\dots$

This is similar to the notation $\bigcup_{x\in X}S_x$ for $S_{x_1}\cup S_{x_2}\cup S_{x_3}\cup\dots$ and $\bigcap_{x\in X}S_x$ for $S_{x_1}\cap S_{x_2}\cap S_{x_3}\cap\dots$

I don't see that these symbols "fail". I can see that $\forall$ could be confused for $\bigvee$, and that would be bad since $\forall$ means the same as $\bigwedge$. However, I don't see that as being a condemnation of one over the other, and there would be no confusion if only one were being used at a time.

• "And", as well as "or" consist of functions of propositions. As long as one knows the truth values of "p" as well as "q", one can find the truth value of "p and q" as well as "p or q". "For all" and "there exists" are not functions of propositions, but rather propositional functions or predicates. They do not have truth values as their input. – Doug Spoonwood Aug 26 '11 at 23:59
• @Doug: For simplicity, suppose our universe consists of $X=\{x_1,x_2\}$. What is the difference between $\bigwedge_{x\in X}Fx$ and $Fx_1\wedge Fx_2$? What is the difference between $\bigvee_{x\in X}Fx$ and $Fx_1\vee Fx_2$? – robjohn Aug 27 '11 at 0:11
• If the universe consists of two elements, I do think you can use that interpretation. In fact, I think I've seen this to some extent in J. Lukasiewicz's "Elements of Mathematical" logic where he talks about the sentential calculus with quantifiers. But, we can consider a smaller universe of discourse. Suppose X consists of a singleton. Well, if the member of that singleton consists of a number, we can say things like "for all x, x is a number." But, we simply don't have any conjunctions there, just like how we can't meaningfully speak about the sum of just one number... – Doug Spoonwood Aug 27 '11 at 1:39
• Also, taking "for all" and "there exists" as an extended operation, as I think you've put forth, isn't quite so straight forward. Expressions like "a v b v c", especially in classical propositional logic, only work as clear under certain conditions. No (correct) definition of a wff allows "x v y v z" as a wff. Even though both "v" and "^" associate, we don't have expressions of the type "x v y v z" as clear, since "a v (b v c ^ d) v e" is not clear, since ((1v0)^0)=(1^0)=0, and (1v(0^0))=(1v0)=1, rendering a subexpression unclear here, which renders the whole expression unclear. – Doug Spoonwood Aug 27 '11 at 2:02
• @Doug: I am simply relaying what I had understood to have been the motivation for the choice of $\bigwedge$ and $\bigvee$, so an uninteresting thread of "$x\vee y\vee z$ is not clear" (when it is certainly unambiguous) or considering smaller, uninformative universes instead of larger, informative ones is pointless. – robjohn Aug 27 '11 at 9:52

I've misplaced my copy of it, but I recall S. C. Kleene in his Mathematical Logic noting that "v" came as an abbreviation of "vel". In Latin "vel" is one of the words which commonly gets translated to the English word "or", and at least people believed that the Latin word "vel" comes closer to alternation (or equivalently, inclusive disjunction) than any of the other words which commonly get translated as "or". Since Russell read Peano, and Peano's book on arithmetic got written in Latin, it does seem at least plausible that Russell first used "v" for alternation, as Bill's reference states.

That "∀" has to get interpreted by "⋀" as you correctly state in at least some cases (though not always), may seem strange at first, I agree. But, one way to think of things here comes as to have the truth set as linearly ordered. When you do this with "0" as the least truth value, and "1" as the greatest truth value, "⋀" most closely corresponds to, if not in fact is, the infimum, while "v" most closely corresponds to the supremum. Both "infimum" and "supremum", at least according to my intuition of them, involve notions just as strange if you don't look at them carefully.

• For the first paragraph: See the link I posted in a comment to Bill's answer. – t.b. Aug 26 '11 at 18:01
• @Doug: Do you appeal to the intuition that the greatest value (1) is located above (thus pointed to by $\wedge$) and the lowest value (0) is located below (pointed to by $\vee$)? At least, this would be an interesting mnemonic. ("Eselsbrücke" = donkey's bridge in german) – Hans-Peter Stricker Aug 26 '11 at 18:08
• @Theo I think we crossposted! Thanks! – Doug Spoonwood Aug 26 '11 at 18:12
• Indeed we did! It was a matter of seconds. I may have posted first but probably you started writing first... @Hans: If I read Doug correctly, he refers to commonly used notation in lattice theory. – t.b. Aug 26 '11 at 18:16
• @Hans No, I didn't have that in mind. "supremum" as meaning "least upper bound" if the words get understood properly might get described as "the least of that which is greater", while "infimum" might get described as "the greatest of that which is lesser." And Theo is correct. – Doug Spoonwood Aug 26 '11 at 18:17

That's a nice question, but you misunderstood the creation of the universal quantifier "all" and "there exists". It appears to be derived from the letter A, and I guess it is, but it didn't emerge that way. The first one to introduce quantifiers the way we know today was Gottlob Frege, who was a german mathematician, in the book Begriffsschrift. See the article about his book on wikipedia, it has the notation used for him. So, that clarifies your question about $\forall$ and $\exists$. In regard of "e" and "or", they are just Boolean connectives, introduced by George Boole in Investigation of the laws of thought. The use of "or" as $\vee$ it's just a convention, Quine explains it in his Mathematical logic, as just an abreviation of the latim word "vel". The use of "e" as $\wedge$ it's, again, a convention, many mathematicians in the analytic tradition use dots (.) instead of. Answering your last question. You're thinking about quantification in a wrong way, it's not just about "quantifying" (read Quine's ML), it's about signing, you use it just to show the variable place in a statement. Using Quine's example: suppose you want to say that every number is less than $0$, equal to $0$ or different from $0$, then you say "Whatever number you may select, it $<0$ $\vee$ it$=0$ $\vee$ it$>0$", or, "whatever number (it $<0$ $\vee$ it$=0$ $\vee$ it$>0$)". Now, for simplification, instead of using the last one, just say "$(x)$ number ($x>0 \vee x=0 \vee x<0)$". Mathematically, you just say $(x)(x \epsilon Number (x>0 \vee x=0 \vee x<0))$. So, $\bigwedge _{x\epsilon Number} P(x)$ it's just an abreviation of $(x)(x \epsilon Number (P(x)))$ but what if you wanted just to say $(x)(x=x)$ without specifying a class, like Number or $X$? You would had to introduce quantification in the notation, and it would get messy. I know my example doesn't clarify a lot, but the quantification business isn't as clear as everyone thinks, there are a lot of divergence between the professionals. Russell for example, in his Mathematical logic as based on the theory of types talks a lot about quantification, and you can see it's not a simple, it involves a lot philosophy. So, read Quine and Russell. Have a nice day. P.S.: Consider this: $(x)(x \epsilon N .\supset. P(x))$, so, suppose we want to say that using the "and" quantifier, than it becomes $(x_1,..,x_n \epsilon N)(P(x_1) \wedge .. \wedge P(x))$ which is a lot more work, and still have to use quantification. Definitions are a way of simplification. But, read Russell (Principia and the article I mencioned) and Quine, quantification used in modern mathematical logic come from their works, see if the philosophical approach correspond to what we just did.

• Perhaps consider spacing a little bit out... Adding double space at the end of the line and return to the line will return to the line. For a new paragraph, do double return to the line. – user88595 Apr 24 '14 at 0:21
• Thank you for the suggestion, I don't know how to use the text editor in SE yet. I'll take a look at it. – Ricardo Apr 24 '14 at 14:14