Mathematics Stack Exchange
Mathematics Stack Exchange

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
0
votes
2 answers

The meaning of a standalone predicate statement as an assumption or step in a formal proof

In Example 9 of section 17 of 'First Order Mathematical Logic' by Angelo Margaris, it is stated that: In this example from number theory, we show what can happen if the restriction on gen ($v$ is not free in $\Delta$) is violated. $x < 2$…
user695931
  • 517
0
votes
0 answers

convert from english to predicate formulas

Propositions, Predicates and Quantifiers [16 Points] Consider the predicates defined below, where the domain is the ISU community. $L(x, y)$ means "$x$ and $y$ went to lunch together today" $S(x)$ means "$x$ is a student" $F(x)$ means "$x$ is…
Barkha Mathur
  • 31
0
votes
0 answers

Predicate logic: no need to introduce a new predicate letter?

Consider the following argument. 1) Arthur is taller than Brian 2) Brian is taller than Chris Conclusion: 3) Chris is shorter than Arthur. Here is how I would formalise this reasoning using predicate logic: 1) Tab 2) Tbc Conclusion: 3) Tac T= taller…
user405159
  • 353
  • 2
  • 12
0
votes
2 answers

Predicate logic Vs propositional logic: a specific example

Consider the following argument. 1) People who write novels are more sensitive than people who play soccer. 2) Alf writes novels. 3) Brian plays soccer. Conclusion: 4) Alf is more sensitive than Brian. Here is how you formalise this argument using…
user405159
  • 353
  • 2
  • 12
0
votes
1 answer

Basic Logic/ Predicate Proof help

I'm having difficulty writing with these proofs. If you could write these statements as correct predicate logic statements and explain briefly how I would prove each one, I'd really appreciate it. 1) For any real number, there is a number larger…
3.14Pie
  • 93
  • 1
  • 2
  • 7
0
votes
1 answer

Logical Equivalence Proof

Let A(x) and B(x) be predicates over the same nonempty domain. State True/False for each statement. If True give non formal/formal proof in English. If False give a counter example. 1) ∃x, (∼ A(x)) equivalent to ∃x, (A(x) → B(x)) 2) ∀x, (A(x) →…
Sss87
  • 11
  • 1
0
votes
1 answer

Re-index the range of a universal quantification

I have been wondering if the following equivalence holds: $(\forall i \, | \, 0 \leq i \leq n : f.i \leq f.n) \, \land \, f.n \leq f.(n+1) \equiv (\forall i \, | \, 0 \leq i \leq n + 1 : f.i \leq f.(n+1))$ So, I try to prove the $\Rightarrow$…
Adam
  • 97
0
votes
1 answer

prove or disprove a quantified proposition

I have to prove or disprove the following: $\forall x \exists y P(x)\Rightarrow \forall y \exists x P(y)$ There are other similar questions. I couldn't find a counterexample to it. I think it is true but don't know how could I show it.
madeel
  • 287
0
votes
1 answer

What condition of $L(x)$ and $L(y)$ implies $T(x)=T(y)$?

I have the following boolean expression $e_1\land e_2\Leftrightarrow 1$: $$ \begin{align} \large e_1&:=(\exists t\ge T(y):(t=T(x)\implies L(x)))\\ \large e_2&:=(\exists t\ge T(x):(t=T(y)\implies L(y))) \end{align} $$ Where $t\in\mathbb{R},\enspace…
Garmekain
  • 3,124
  • 13
  • 26
0
votes
1 answer

Predicate logic: definition of a closed formula

I'm learning predicate logic and I'm reading some lecture notes from my professor that say: From theorem XXX (that is, "In a model M, if two assignments σ and σ' are such that σ(x) = σ'(x) for every variable x free in a formula α, then σ satisfies α…
user405159
  • 353
  • 2
  • 12
0
votes
1 answer

why is this formula not valid for n=0 and i=0?

$i$ and $n$ are integers. $\exists (i)\Big(\big((n\geq 0) \wedge (0\leq i)\wedge (i\leq n)\big)\implies (3\leq n)\Big) $ It should be not valid for n=0 and i=0.
Ramanuj
  • 21
0
votes
2 answers

Predicate Logic - Simple explanation needed

I have been working through some tutorials (of which I have the answers to) trying to understand predicate construction. Anything I've found online has been quite well laid out and easy to understand and I understand the role of the quantifiers but…
B--
  • 15
0
votes
0 answers

Working out if a predicate formula is true or false given a domain {0, 1}

I can't figure out how to interpret this. Is my understanding of the statement correct? "There exists an x such that for all y, if p(x) is true then x = y" i. P(0) = true, if y = 1 then x != y, formula is not true. p(1) = true, if y = 0 then, x !=…
Shiny_and_Chrome
  • 363
0
votes
1 answer

Different uses of the keyword 'any' while writing predicate logic?

I came across two statements which highlight the my problem with the keyword 'any' in mathematical english. The statement "Anyone who eats any pumpkin is a nutrition fanatic." translates to $\forall x ((\exists y (PUMPKIN(y) \land EAT(x,y)))…
user3508140
  • 539
0
votes
1 answer

Prove that $\forall x \forall y (A(x,y) \rightarrow \neg A(y,x)) \vdash \forall x \forall y (A(x,y) \rightarrow \neg x = y)$

In predicate logic how to prove that: $$\forall x \forall y (A(x,y) \rightarrow \neg A(y,x)) \vdash \forall x \forall y (A(x,y) \rightarrow \neg x = y)$$ First it should be proven that: $$x=y \vdash A(x,y)\leftrightarrow A(y,x) $$ (I did prove the…
Tony
  • 1,044
1 2 3
…
22 23