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How academics would go about explaining in everyday English, so without any philosophical or mathematical jargon, what is meant by the expression ∃xHx, if H would stand here for “is a human being”.

On the face of it, ∃xHx seems to mean that there is at least one thing such that it is a human being. This then seems to mean that there exists at least one human being... This is apparently true, but is this really what ∃xHx means?

Does ∃xHx mean for example that there is at least one thing such that it is true that this thing is a human being?

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    More simply: "There is a Human being". Feb 24 at 20:29
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    For some x, x is a human being. Feb 24 at 20:58
  • @Speakpigeon What is the differencce between your two sentences “there is at least one thing such that it is a human being” and “there is at least one thing such that it is true that this thing is a human being”? - IMO: If a property holds, then it is true that the property holds. Without jargon both sentences say: “There exists a human being”.
    – Jo Wehler
    Feb 25 at 4:08
  • Keep in mind, this alone does guarantee there truly are any humans beings, in the same sense ∃xP(x) where P(x) means "x is a prime number" does not guarantee there truly are prime numbers/numbers, i.e. it does not mean there there are numbers/humans--the fictionalist account.
    – J Kusin
    Feb 25 at 4:25
  • ∃xHx literally means "there is a thing which is a human being". T(∃xHx) literally means "it is true that there is a thing which is a human being", where T is the truth predicate. By Tarski's Convention T, ∃xHx entails T(∃xHx), but that is not what it means.
    – Conifold
    Feb 25 at 5:32

3 Answers 3

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As always when considering quantified expressions in logic, the domain of discourse is essential. If the domain is ‘everything’, then ∃xHx is true, but if the domain is the Natural Numbers, then it is false. That is, Peano Arithmetic does not prove “there is a human.”

This is an important point, since for example, it may well be that the physical universe is finite, even though axioms like the Axiom of Infinity of ZF set theory still carry meaning in the domain they satisfy. Indeed, there are infinitely many sets in the universe of sets, and the Axiom of Infinity helps us to formalize the notion of infinity in math. Just as the existence of humans does not require that PA proves the existence of humans, axioms of math and set theory do not need to be physically realizable in order to analyze them with a model-theoretic notion of truth.

That is all to say that what is true in formal logic is determined by models in a like fashion that what is true in reality is determined by facts of the matter. It is our job to connect “truth in a model” with our understanding of the world.

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  • So ∃xHx is neither true nor false and doesn't mean anything on its own. Feb 28 at 10:49
  • @Speakpigeon it is clearly true, but just not in every domain that predicate logic can express or theory that it can formalize.
    – PW_246
    Feb 28 at 10:56
  • "it is clearly true" Sure, but not in your theory since there is no domain specified. Feb 28 at 17:16
  • @Speakpigeon I’m not working in a domain. Model-theoretic truth is different from truth simpliciter. The proposition represented by ∃xHx is true, but real-world truth doesn’t have to hold in every hypothetical scenario we cook up. That’s the whole point of counterfactual reasoning.
    – PW_246
    Feb 28 at 18:00
  • "real-world truth doesn’t have to hold in every hypothetical scenario we cook up" Sure, but surely you can formalise the fact that it is true in the real world? 2. "counterfactual reasoning" Nah, nothing to do with counterfactual reasoning. Feb 29 at 10:43
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For others, this is a question of existential quantification. But you know that. Then I'll do my best to interpret your question. To review:

"∃xHx" is a fragment of an artifical language. It is formal semantics, and as such it can be defined to mean anything desired according to the natural language metatheory that governs the artificial language. Symbols are constructed as a tool of language. This particular form, the existential declaration, is an abstraction related to linguistic binding, such as the pronoun, a simple form of endophora.

Instead of he or she, or er or sie or es, if you prefer, this form of reference is an abstraction of predication. The rules of the predication for this abstraction are a convention devised by logicians. "∃xHx" is generally taken to mean one or more of some thing exist, and that's in distinction to another binder ""∃!xHx" in which is used to indicate exactly one thing to which the predicate applies exists. If "Hx" is "is human" (the copula and predicate adjective form the full predicate), then one or more beings are conferred the property "human".

Does ∃xHx mean for example that there is at least one thing such that it is true that this thing is a human being?

This is an interesting question. What you seem to be asking here is, does the fragment carry with it assertoric force? It is obviously truth-apt, that it can be true, but does it assert by virtue of its articulation a claim that it is true? 'There exists a unicorn' is truth-apt. But does it assert by virtue of its construction that in the world, there exists a unicorn?

According to WP's article Deflationary Theory of Truth:

a deflationary theory of truth (also semantic deflationism1 or simply deflationism) is one of a family of theories that all have in common the claim that assertions of predicate truth of a statement do not attribute a property called "truth" to such a statement.

Therefore, according to Frege, and many who follow, "it is true that this thing is a human being" doesn't carry any more assertoric force than "this thing is a human being". But what about the existence? Is it same to say a "a thing is human" and "a thing is human exists". That depends on the context of the propositions. Binding by definition applies to a domain of discourse. So, when one talks about something existing in a domain of discourse, that is relative to the context which the variable and the domain are related.

Obviously, common sense dictates that there are human beings in the real world, so if "∃xHx" is used to make an assertion about Earth presently, it is indeed a true statement, as it makes a claim about physical reality.

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Dijkstra and Scholten have written a book on predicate calculus that redoes traditional logic using '≡'.

Traditional logic more often uses ⇔ instead of ≡; but traditional logic at the proof level uses the consequence relation for proofs ⊨. The benefit of the Dijkstra approach is that logic becomes equational like algebra. The predecessor is Boole. ie one can drive an equational proof from left to right or from right to left. For ⊨ this option is not available.

D&S call ≡ 'equivalence'

A shorter paper: Gries and Schneider
A beginner textbook: Gries and Schneider

A basic axiom for this logic is
A ≡ True ≡ A

Grouped as (A ≡ A) ≡ True
it indicates that for any A
A ≡ A is the same as B ≡ B
And we can define this constant to True

Grouped like
(A ≡ True) ≡ A
it shows the idempotence of '≡ True'

Normally one would use it to simplify, ie. we would reduce 'A ≡ True' to A

But one can drive it backwards, ie replace A by A ≡ True

This of course can be carried on to an 'infinite series'!

    A
≡ (True ≡ A)
≡ (True ≡ (True ≡ A))
≡ (True ≡ (True ≡ (True ≡ A))) ...

tl;dr

English does not have a very good equivalent to equivalence but if we admit "is" as ≡ , what D&S logic shows is that "... is true" can generally be added or dropped on any truth term.


Added later

This answer addresses the question implied in the last paragraph. viz.

It is true that his thing is a human being

is equivalent to

This thing is a human being

The answer does this more generally

      This thing is a human being
It is true that this thing is a human being
It is true that it is true that this thing is a human being
≡ etc

But from your comments it seems your question is about something else. It could be about:

  • Bounded quantification:
    • which itself a can of worms
    • especially when it's most implicit.
  • It also indirectly touches on the difference in approach between classical and intuitionist approaches to math/logic, especially in the existential quantification case

Until you say a little more on what it is you are asking about, one cant do much more.. One could try a more elaborate answer but before that you need to clarify what it is you are asking

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    Your answer may be interesting but I fail to see how it even begins to address my question. Feb 25 at 10:21
  • "what it is you are asking" How academics would go about explaining in everyday English, so without any philosophical or mathematical jargon, what is meant by the expression ∃xHx, if H would stand here for “is a human being”? I don't see how you answer this question. Feb 27 at 16:42
  • @Speakpigeon You cant claim to stay within the realm of everyday English when your question is fundamentally — every paragraph and title — about the formal logic symbolism ∃xHx
    – Rushi
    Feb 27 at 16:49
  • "You cant claim" I didn't claim anything. I asked a question and your answer doesn't address it. If you think it is not possible to answer my question, then don't pretend you do. Feb 28 at 10:27
  • @Speakpigeon Your question asks about a bridge between two worlds — the formal world of symbolic logic and the informal world of meanings. Also adding ...without any philosophical of mathematical jargon... in everyday English. contradicts that. How does one explain predicate calculus without starting with predicate calculus??
    – Rushi
    Feb 28 at 10:34

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