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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
4144 questions
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How do atomic formulas with negated subjects translate into English?

I'm reading about negative normal forms. My text talks about transforming H¬x to 'negative normal form' so it reads ¬Hx. If the two sentences are interchangeable, then they mean the same thing. So, suppose that H stands for reads a lot, and that x…
Hal
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Definition set of axioma's

The definition is: "A set of formula's $\Sigma$ axiomizes $Th(M)$ if for all sentences $\phi$ the following applies: $\phi \in Th(M) \Leftrightarrow \Sigma \models \phi$" Where $Th(M) = \{ \phi | \phi$ is a sentence and $M \models \phi \}$ and $M$…
Garth Marenghi
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What is are the differences and similarities between quantifiers and assignments/mappings?

In predicate logic, you have quantifiers, a structure and a model, and something called (in Dutch) "een bedeling", which I will call "mapping" (since I have no idea what it is called in English). This mapping is a function that maps a variable to an…
Garth Marenghi
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Arbitrary meaning two different things in two different settings

The arbitrary in induction: Induction step: We assume that P(k) is true and then we need to show that P(k+1) is true as well. k is arbitrary and means "any one" Consider this example of arbitrary used in the rule of Universal…
Thomas
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Logical Equivalence Of Quantified Implications

$\forall x (P(x)\rightarrow Q(x))$ is logically equivalent to $(\exists x P(x)\rightarrow\exists y Q(y))$ The proofs I've seen use logical reasoning to prove the equivalence. Can someone supply a proof using the laws of boolean algebra?
EggHead
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Interpreting Quantified Predicate Formulas

Does $\forall x (P(x) \rightarrow Q(x))$ mean that $(P(x) \rightarrow Q(x))$ is true for all truth assignments to $x$ or does it mean that $P(x)$ and $Q(x)$ can be true or false depending on the truth assignment to $x$?
EggHead
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Proving a predicate logic statement to be valid

I've been stuck on this question for the better part of the day, and I've succumbed to asking for help. I'm not sure how to go about it honestly. I've tried to do the contrapositive to prove it, but I get stuck and end up at a dead end. The…
user41931
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Writing statements in words

I have to write out what the following statements mean and whether they are true or false: $(∃y ∈ N)(∀x ∈ N)(x < y)$ $(∀y ∈ N)(∃x ∈ N)(x < y)$ $(∀x ∈ N)(∀y ∈ N)(x < y)$ So for number 1, would I say that there exists $y \in N$ that is greater than…
Michael Turner
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Is this symbolic expression correct?

Say C: set of courses P(x,y): 'x is a prerequisite for course y' statement: 'some courses have the same prerequisites' Is this symbolic expression correct? If not, how would I write this with implication? Also how would I write this without…
merlin
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Can someone explain the solution to this statement?

Say C: set of courses P(x,y): 'x is a prerequisite for course y' statement: 'some courses have several prerequisites' symbolically: ∃ x ∈ C, ∃ y ∈ C, ∃ z ∈ C, P(y, x) ∧ P(z, x) ∧ y ≠ z I don't really understand how you get the symbolic expression…
merlin
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english to quantifiers

Let L(x,y) be the statement "x lives with y", where the domain for both x and y is all people. use the quantifiers (∀,∃) to express the following statement (1) Someone lives with exactly two people. (2) There is somebody who lives with somebody. (3)…
Daniel Zhang
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Prove this statement

Let's say we have some predicate $ P $, then how does one prove the following statement? $$\forall x \forall y ((x=y \wedge P (x))\Rightarrow P (y)) $$
Constantine
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Describe the predicate

Let's describe the predicate p(x): "x belongs to A" in these cases: $A=\{2n| n \in \Bbb N\}$ $A=\{2n+1| n \in \Bbb N\}$ $A=\{k^2| k \in \Bbb Z\}$
marinaaaa
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Should Modus Ponens and Modus Tollens Be Viewed As Implications?

Should Modus Ponens and Modus Tollens be viewed as implications? I ask because of the following. It seems that Modus Ponens can be written as $\left(P\rightarrow Q\right)\land P\rightarrow Q$ And Modus Tollens can be written as $\left(P\rightarrow…
user1148275
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represent the following sentence by predicate calculus well formed formulas

A computer system is intelligent if it can perform a task which if performed by human, requires intelligence
Mukul Sharma
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