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10

Timm Lampert, cited by the OP, quotes Wittgenstein (§8 of Remarks on the Foundations of Mathematics, Appendix 3): ‘True in Russell’s system’ means, as was said: proved in Russell’s system; and ‘false in Russell’s system’ means: the opposite has been proved in Russell’s system. Lampert claims Wittgenstein is assuming what needs to be proven: Whether P =...


8

It is common for beginning students of logic to read philosophical importance into the principle of explosion, but this is a mistake. The principle of explosion is merely a mathematical outcome of the way the connectives are traditionally defined in first-order logic. It doesn't reveal anything deep, however. It merely arises from the following proof: ...


6

The following truth table shows that ~(A↔B) → ~(A→B) is not a tautology: If A is False and B is True then the antecedent is True but the consequent is False making the conditional False. Because the truth table does not show a tautology, one should not be able to derive a natural deduction proof of the result. Michael Rieppel. Truth Table Generator. ...


5

Vivid designator (originally "vivid name") is Kaplan's replacement for rigid designator in the logic of beliefs and other propositional attitudes introduced in Quantifying In. The point was to eliminate the metaphysical essentialism of de re modality by relativizing rigidity to context and background of belief reports. This was a descriptivist alternative to ...


4

You can't derive ~(A→B) from ~(A↔B). Consider: A = I'm in Paris. B = I'm in France. ~(A↔B) is true, because being in Paris is not equivalent to being in France (I could be in France but not in Paris). But ~(A→B) is false, because if I'm in Paris then necessarily I'm in France. So you can't derive ~(A→B) from ~(A↔B).


4

Edward Said says: At [the core of what can be called the epistemology of imperialism] is the stubborn thesis that everyone is principally and irreducibly a member of some race or category, and that race or category cannot ever be assimilated to or accepted by others---except as itself. Thus came into being such invented essences as the Oriental or ...


3

It depends on the convention adopted by the textbook about omitting parentheses... Usually, when one connective symbol is used repeatedly, grouping is to the right: a → b → c is a → (b → c). If so, the valid argument is the second one.


3

This is not provable. Consider the following model with the domain {d,e,i,j,s}: P = {d} H = {i, j} M = {j} G = {<s,d>} R = {<s,d,e>} We get: Pd ⟷ (Hj & Mj) is true, because both sides are true. Gsd is true. ∀x∀y∃z(((Gxy & (Py ➝ Pz)) & Rxyz) ➝ Gxz) is true, because the antecedent is false for any z. Pe ⟷ ∀x(Hx ➝ Mx) is true, because ...


3

I'm hesitant to answer this question, since I believe that the OP is someone I've already given up discussing this topic with, but on the off-chance since I think this is actually a reasonably well-posed question I'll take a stab at it. I could talk about how this makes things more confusing - e.g. from the perspective of PA + "PA is inconsistent" it would ...


3

Let me add to the existing answer/comments: The key term here is truth functional. A truth functional is roughly an operation on sentences such that the truth value of the output is completely determined by the truth value of the input. E.g. "AND" is a truth functional: "X AND Y" is true iff X is true and Y is true. In my experience, although I don't know ...


3

There are no truth tables that model strict implication, for a couple of reasons. As others have pointed out in the comments, Lewis devised strict implication precisely because he felt that material implication (the kind modelled by truth tables) was not a satisfactory account of the English conditional statement 'if, then...'. Further, the conditional ...


3

(◊ ∃x Fx) ↔ (∃x ◊ Fx) can be seen as a conjunction of (◊ ∃x Fx) → (∃x ◊ Fx) (the Barcan formula in the narrower sense) and (∃x ◊ Fx) → (◊ ∃x Fx) (the converse Barcan formula). The forward direction, (◊ ∃x Fx) → (∃x ◊ Fx), says that no new objects come into existence when going from one possible world to another: If there is an ...


3

FOL is the natural logic environment to formalize mathematical theories. The basic characteristic of predicate calculus is the use of quantifiers : first-order logic is predicate calculus where quantification is restricted to individual variables (variables ranging over "objects") and quantification over predicate variables (i.e. variables ranging over "...


3

I agree it's a flawed argument. The base concept comes from Bayes' theorem, to the effect that if you randomly choose a member of a set, the characteristics of that member are most likely to be the most common characteristics found in the set as a whole. In other words, if you have a bag full of red and blue balls, and the ball you randomly pick out is red,...


3

Firstly, it would help to pin down what you mean by logical implication. It could mean syntactically that the antecedent proves the consequent, or semantically that all models of the antecedent are models of the consequent, or perhaps you have in mind some more informal notion that the conditional is necessarily true. Logical implication does not mean a ...


3

Yes, if you add a redundant premise to a valid argument, the resulting argument is also valid. If the resulting argument were invalid, that would mean that there is a case in which all the premises (say, P, Q, R) are true and the conclusion false. But then the original argument would be invalid as well, because there is a case in which its premises (Q and R)...


2

Keelan's answer is perfectly good. Just to add my two cents to your question 6: there is one (and as far as I know only one) case where you may infer, from the premise that something is F, that everything is F, and that is when there's just one object! If there's only one thing, and it has a given property, then everything (i.e., that one thing) has that ...


2

The problem with your alternative, which was recognized by Carnap himself, is that you need an infinite hierarchy of (meta)languages. Every arithmetical sentence G1 which is, for instance, irresolvable [neither provable nor refutable] in the language S1 is yet determinate [analytic or contradictory] in the language S1; in the first place there exists ...


2

Here is a shorter proof than the one provided in an answer: https://philosophy.stackexchange.com/a/42928/29944 It is similar to the first 9 lines of this proof, but proceeds with universal introduction right after that. To use the proof checker, I replaced loves(x,y) with Lxy to fit its input requirements. Once I derived line 8 that did not have the ...


2

One of the questions the OP asks is whether there is a name for the following: The assumption that because an argument contains an informal fallacy it must be false. As Dennis mentions in a comment there is the "fallacy fallacy" or "argument from fallacy" that describes this deceptive argument. Here is how Bo Bennet describes it: Concluding that the ...


2

How do I reach this goal and also get to the goal Medium(x) if it's not listed as a premise? The key point is that you want to prove that not all things are Tet and Medium. That is a job for Proof of Negation. Assume the positive, ∀x (Tet(x) ∧ Medium(x)), aiming to derive a contradiction of the premises, ie derive ∀x Tet(x), thereby deducing the ...


2

This is only a description of the justifications for the steps. How you describe the justifications in the proof checker will depend on that tool. Lines 5 and 6 used universal elimination to replace the variable x with the name a. You will likely need to reference which lines were used. Line 8 used conditional elimination which should reference the ...


2

A different way of phrasing Munchausen's trilemma (MT) might help explain what the problem is and why your attempt at a solution is inadequate. MT states that if proof is necessary for knowledge and if we accept standard rules about arguments, then nothing can be proven and knowledge is impossible. Stating that proof is possible doesn't make that problem go ...


2

The authors of forallx in section 15.7 describe explosion as: We need one last rule. It is a kind of elimination rule for ‘⊥’, and known as explosion. If we obtain a contradiction, symbolized by ‘⊥’, then we can infer whatever we like. In order to use that inference rule, we have to first derive the contradiction, that is, we have to derive a ...


2

Classical knowledge is based on how to apply certainty to arrive at other certainties. If you have doubt, classical logical approaches such as prepositional logic will fail because they have no concept of doubt built into them. There are other systems of logic which use other approaches to address doubt head on (such as using Bayesian Inference). One ...


2

I think it was Alito who is wrong here, by taking what Kagan is saying to an unreasonable extreme. I don't think Kagan slipped up, but she was appealing to some kind of (reasonable, I think) implicit principle that some properties (duties, obligations, but also things done too them) of a person get transferred over to a substitute who is tasked with ...


2

Makoto Tsukada describes a proof checking program using Prolog. Here is the abstract: A proof system for propositional and predicate logic is discussed. As a meta-language specifying the system, a logic programming language, namely, Prolog is adopted. All of proof rules, axioms, definitions, theorems and also proofs can be described as predicates of ...


2

One of the first things to consider is the goal you are trying to show: ∀x∀y∀z((Indiff(x,y)∧StrongPref(z,x))→StrongPref(z,y)) Note that it is a conditional. So we may want to assume the antecedent of that conditional to start the proof. We may start by assuming the following: Indiff(a,b)∧StrongPref(c,a) From here we want to derive StrongPref(c,...


2

It is quite common to assume ZFC as the unspoken system in which mathematical claims are to be understood. In that case, it would be unnecessary to specify "well-orderable", because the axiom of choice is (over ZF) equivalent to the statement that every set is well-orderable. The typical proof of the completeness theorem proceeds by constructing a model "by ...


2

Maybe it is useful to recall the basic definition. A deductive argument is valid : if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. In other words, for the validity of an argument is necessary that the truth of the premises implies the truth of the conclusion. A simple ...


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