Questions tagged [predicate-logic]

Questions concerning predicate calculus, i.e. the logic of quantifiers.

Some well-known formal systems covered by this term are

  • first-order logic, containing the quantifiers $\forall$ and $\exists$
  • second-order logic
  • many-sorted logic
  • infinitary logic
3819 questions
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What's the difference between predicate and propositional logic?

I'd heard of propositional logic for years, but until I came across this question, I'd never heard of predicate logic. Moreover, the fact that Introduction to Logic: Predicate Logic and Introduction to Logic: Propositional Logic (both by Howard…
Alex Basson
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Why is this true? $(\exists x)(P(x) \Rightarrow (\forall y) P(y))$

Why is this true? $(\exists x)(P(x) \Rightarrow (\forall y) P(y))$
Mats
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"If everyone in front of you is bald, then you're bald." Does this logically mean that the first person is bald?

Suppose we have a line of people that starts with person #1 and goes for a (finite or infinite) number of people behind him/her, and this property holds for every person in the line: If everyone in front of you is bald, then you are…
Færd
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**Ended Competition:** What is the shortest proof of $\exists x \forall y (D(x) \to D(y)) $?

The competition has ended 6 june 2014 22:00 GMT The winner is Bryan Well done ! When I was rereading the proof of the drinkers paradox (see Proof of Drinker paradox I realised that $\exists x \forall y (D(x) \to D(y)) $ is also a theorem. I…
29
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Can Peano arithmetic prove the consistency of "baby arithmetic"?

I am reading Peter Smith's An Introduction to Gödel's Theorems. In chapter 10, he defines "baby arithmetic" $\mathsf{BA}$ to be the zeroth-order version of Peano arithmetic ($\mathsf{PA}$) without induction. That is, $\mathsf{BA}$ is the…
WillG
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Does the unique existential quantifier commute with the existential quantifier?

Given some function involving two variables, $\mathit p(x,y)$, is the formula $$\mathit \exists!x\exists yp(x,y)$$ equivalent to $$\mathit\exists y\exists!xp(x,y)$$ I have tried writing out the formal definition for the unique existential…
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Predicate logic: How do you self-check the logical structure of your own arguments?

In propositional logic, there are truth tables. So you can check if the logical structure of your argument is, not correct per se, but if it's what you intended it to be. In predicate logic, I have seen no reference to truth tables, nor have I seen…
user2901512
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Please explain, "Asymmetric is stronger than simply not symmetric".

In some textbook I found a statement like, "Asymmetric is stronger than simply not symmetric". But as I try to perceive this statement, both appear to be same to me. For example, parentof is an asymmetric relation. If $A$ is a parentof $B$, $B$ can…
Masroor
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What's the difference between $∀x\,∃y\,L(x, y)$ and $∃y\,∀x\,L(x, y)$?

Everybody loves somebody. $∀x\,∃y\,L(x, y)$ There is somebody whom everybody loves. $∃y\,∀x\,L(x, y)$ What's the difference between these two sentences? If they are same, can I switch $\exists y$ and $\forall x$?
Ned Stack
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Two styles of semantics for a first-order language: what's to choose?

The usual classical semantics for FOL gets presented in two styles. Suppressing details irrelevant for the headline question, suppose the $L$-wff $\varphi(x)$ has only $x$ free, and let $I$ be a fixed interpretation of the non-logical vocabulary of…
Peter Smith
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Can all math results be formalized and checked by a computer?

Can all math results, that have been correctly proven so far, be formalized and checked by a computer? If so, what type of logic would need to be used there? I've heard that the first-order logic is not expressive enough: certain facts can not be…
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Any branch of math can be expressed within set theory, is the reverse true?

Set theory seems to have the property of being "universal", in the sense that any branch of math can be expressed on its language. Is there any other branch of math with this property? I am asking because set theory is a branch of math. It is…
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Why does what I've written fail to define truth?

I stumbled across a set of axioms for first order logic a bit ago. Intrigued, I decided to try to write it all down and organise what I read. After I did that, it seemed to me as though one could quite simply define 'truth' using these axioms and…
Nethesis
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Intuitive Reason that Quantifier Order Matters

Is there some understandable rationale why $\forall x\, \exists y\, P(x,y) \not \equiv \exists y\, \forall x\, P(x,y)$? I'm looking for a sentence I can explain to students, but I am failing every time I try to come up with one. Example Let $P(x,y)$…
danmcardle
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In classical predicate logics, why is it usually assumed that at least one object exists?

In classical predicate logic it is commonly assumed that the domain of objects is non-empty. This validates inferences such as $$\forall x Fx \models \exists x Fx$$ as well as, if the identity predicate is available, the logical truth of $$\exists…
Max
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