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I realize this is an older OP but since some of the postings contain misinformation, perhaps this reply is relevant. The second statement DOES NOT mean there is at least one one comedian who is funny. The translation for 'There is at least one comedian that is funny' is ∃x[C(x) & F(x)]...

This is a day late and a dollar short since your question is from November but ... I think the axiom scheme should be ∀vP→P(t/v) so we are given the following as premises: ∀vP ∀vP→P(t/v) notice that this has the form 1. A 2. A→B Modus ponens is the rule that says that given 1 and 2 we may...

A set of well-formed formulas can have a model and not be complete. The set propositional tautologies has a model but is incomplete with respect to the deductive system that has only one axiom scheme p > (q > p) and modus ponens as its only rule.

Implication and Equivalence Quine has called this an unfortunate choice of terminology dating back at least to Russell of calling the statement connective '\supset' or '→' "implication". This invites confusion with the notion of "logical implication" which is the relationship between...

2+2 I would question what exactly is meant by saying that "set theory isn't entirely consistent".

I think so. I was assuming that there was some way of deriving q→p from {p} without really saying what it was because systems can differ. I think what you did was show how you could get there in the system you are using.

Did you mean to include that Ʃ was maximally consistent? If not then let Ʃ be the unit set that contains only the sentence letter p. If it is granted that we can then derive q→p, we have a situation where (1) Either q→p is derivable from Ʃ or ~(q→p) is derivable from Ʃ (because q→p is...

If you are familiar with how to determine a formula is logically true, then you can use the fact that formulas are logically equivalent just in case their biconditional is logically true. If there is an interpretation that makes the biconditional of (1) and (2) false, then they are not logically...

The fact that the same word "implies" is sometimes used for the statement connective often written with an arrow (→) and sometimes used for "logical implication", i.e., the relationship one sentence has to another when it is not possible for the first to be true and the second false adds to the...

I am not saying that the use of the different quantifiers makes a difference, it does not. I am saying that Existential Elimination does not allow you to assume that the x referred to in the first existential claim is one and the same x as the x referred to in the second existential claim. It...

This argument is fallacious. You cannot infer that one and the same x is referred to by the two existential quantifers. If you could, it would be easy to prove that something is a Dodge and something is a Toyota implies that some one thing is both a Dodge and a Toyota. Maybe it would help to see...

Count the number of distinct sentence letters, say this number is n. Then the total number of rows will be 2n. For your first sentence letter divide 2n in half. Say the result is m. So make m T's and m F's under the first letter. Then take m and divide it in half coming up with, say, p, and make...

Hello, I just downloaded the Java Development Kit (J2EE) and during the install I was asked to configure something called Glassfish. I don't understand what Glassfish is for. Is it only for Web development? What does it do for me?

Let 'S' be 'x smokes two packs of cigarettes over an extended period' Let 'L' be 'x has lung problems' Let 'a' be 'Jones' Then the argument (\forallx)( Sx \rightarrow Lx) Sa Therefore, La is deductively valid. So if it turned out that \negLa, I would say we must give up one of the...

Hi RabbitWho, I am okay with your amendment. We could even go further and require that the conclusion be "Jones has a 95% chance of contracting lung disease". I am appealing to our pre-analytic notion of what a valid inductive argument is since I know of no precise definition of inductive...

Two observations. 1. All deductive reasoning does not proceed from the general to the specific, at least not in any straightforward way. Consider the following valid, deductive argument that appears to go from the specific to the general. Only Alice and Bill are on the hill. Alice has blue...

I gave an English rendering of the non-negated sentence earlier. This translates to: (\forallx)(\existsy)(Rxy \wedge \negLxy) Negating this we get \neg(\forallx)(\existsy)(Rxy \wedge \negLxy) Driving the negation successively inward we get: (\existsx)\neg(\existsy)(Rxy \wedge...

Without actually doing the translation, here is an English rendering: For every x, there is a y, such that y is related to x and x does not like y.

Your first example is typically translated into sentential logic the same way as If I score 100 on every exam, then I will pass logic. This first sentence is false if I score 100 on every exam and do not pass logic. Your second example is typically translated the same way as If I pass...

Hello, I am trying to learn Powershell and for my first exercise I started reading a file that we use at work. The file is pipe delimited and contains a header row. The field names contain underscores. Here is the header row (wrapped to fit this window). H|Seq_Nbr|Data_Key |Plan_Type...

I wouldn't have been able to reply without Stephen's assistance because I am not in proximity to my copy of Halmos. I find his example a little contorted as well. But as long as you understand the point that set membership is distinct from identity (Halmos says 'equality'), I wouldn't sweat the...

That sir, was excellent criticism. Thank you so much for taking the time to do that. Let me try again. Successor is just like the standard successor except that S(a) = a ¬S(0) = a a + a = a Multiplication is just like standard multiplication except that a \cdot 0 = 0 for any n > 0, a...

Ah, my bad. So the axiomatization I am using is the system N from Chris Leary's book A Friendly Introduction to Mathematical Logic and he produces a proof that the formula does not follow from those axioms. I got a little over-zealous there. Sorry

I would challenge you to prove \forall x, \neg (x<x) from the axioms of number theory.

Right. So let me amend. The function S is just like that of standard number theory except that S(a) = a \negS(0) = a The relation < is just like that of standard number theory except that a < a 0 < a for any n > 0, S\cdots_{}S(0) < a where S\cdots_{}S are n occurrences of S...

I am trying to construct a non-standard model < A,0,S,+,*,E,< > that has as its domain the natural numbers plus the letter a such that the model 1. Makes all of the axioms of number theory true (Say, Mendelson's S) 2. Makes a < a. So in this model the domain has already been specified. We...

Personally, I'm going to tough it out without Flash on my iPad. I think it's just a matter of time before Flash goes the way of the albatross http://m.ibtimes.com/adobe-flash-dead-250077.html [Broken]

It will depend on the particular deductive apparatus. You might be using a set of axioms together with a set of rules of inference, or you might be using a system of natural deduction, etc. Let's suppose it's a natural deduction system. Assume your antecedent. Often you will have a rule of...

Thank you that was it!

Hello nobahar First, in ∃x(Fx ∧ ∀y(Fy → y=x) & Bx) what does ∧ mean if it is different than &? It looks like two different notations for conjunction so I'll assume the interpretation ∃x(Fx & ∀y(Fy → y=x) & Bx). Ala Russell's 1905 paper "On Denoting", let's take F to be "is the king of France"...

Let L be a first order language. Let A be any set of sentences of L. We extend L0 (=L) to L1 by adding denumerably many constants c1, …,cn,… to L. We enumerate the existential formulas of L. We add Henkin axioms to L by taking each formula in the enumeration and making it the antecedant of a...