24

Here is part of the question: My only idea is v must be introduced, but how would I use subproofs to show one of A/\C or B/\D is never false if A v B? It might be best to think of using disjunction elimination initially although disjunction introduction may be needed later. The OP notes the following: Obviously since A → C and B → D then if A v B one ...


16

Explosion is a property of logical consequence relations, and thus of logics, that is not trivial: Some logics have it, some don't. So there is simply no sense asking Why can't we simply get rid of the Principle of Explosion? If the logic you start with, say classical propostional logic, is explosive, then you cannot get rid of explosiveness and at the ...


11

The the set {¬,∨,∧,⇒,⇔} is usually used because it includes the "most natural" ones. If we start with a minimal set, like {¬,⇒}, the other ones are usually introduced as abbreviations. There is no "deep" reason : mainly tradition, and a "reasonable" trade-off between savings (minimality) and readibility (to express p ∧ q as ¬(p ⇒ ¬q) is not so "natural".


10

It is true that : (Q≡⊤) ⊨ (P∧Q≡P) we can check it with a truth-table. But the other "direction" does not hold : when P and Q are both false, we have that (P∧Q≡P) is true but (Q≡⊤) is not.


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 =...


10

Gensler's star test is a simplified method for determining the validity of a syllogism proposed in 1973. According to the test, one stars (asterisks) the first (capital) letter after "All", and all letters after "not" or "No". The syllogism is valid if and only if every capital letter is starred exactly once and there is exactly ...


9

There seem to be several overlapping concerns in your issue with proof by contradiction. You have an objection to the truth table for material implication: I've never been satisfied with the argument that two false propositions create a true implication however. What good justifications are there for agreeing with this? You seem to misunderstand proof ...


8

From Jan von Plato, Elements of Logical Reasoning (Cambridge UP, 2013), pag 10 : The two sentences "if A, then B" and "B if A" seem to express the same thing. Natural language seems to have a host of ways of expressing a conditional sentence that is written A → B in the logical notation. Consider the following list : From A, B follows; A ...


8

EDIT: here I've looked at sentences, as opposed to mere formulas, since the resulting situation is more interesting. If we allow formulas with free variables, then as Toothpick Anemone essentially observes below the situation is quite trivial. Perhaps surprisingly - and contra Veedrac's comment above - we can indeed get infinitely many inequivalent ...


8

Rather than give you the correct technical answer (edit: okay I ended up going into some technical details, oops), which has been provided, I think I'll try and diagnose where you're intuitions have led you astray. It seems to me that you think that if a formal system can derive (P & ¬P), then we have to accept it and thus accept the resulting ...


8

You can use proof by contradiction: p1: A v B p2: A -> C p3: B -> D assume ~(C v D) ~C & ~D (from 1, De Morgan's law) ~C (from 2, conjunction elimination) ~D (from 2, conjunction elimination) ~A (from 3, p2, modus tollens) B (from 5, p1, disjunctive syllogism) D (from 6, p3, modus ponens) D & ~D (4, 7) Since D & ~D is a contradiction, our ...


8

It's a terrible and outdated system of notation where dots (.) and colons (:) function both as conjunctions and as parentheses. When a . or : occurs between expressions, it denotes conjunction. For example, p ⸱ p ⸧ q means p ∧ (p ⸧ q). When a . or : occurs next to a connective such as ⸧, it has the function of parentheses. There are exact rules how to ...


8

Answer According to Wikipedia's article on logic symbols, := is used for definition. The truth of a proposition can be determined through empirical or rational means, but sometimes it is assigned axiomatically: p = q means there is an equivalence of values p := q means that p is equivalent to q by definition, assignment, or declaration, such as when a ...


7

There is no strict universally accepted convention. In mathematical logic, ≡ is logical equivalence, as you said, and it is a connective between proposition (in propositional calculus) or formulae (in predicate calculus). Example : (p → q) ≡ (¬p ∨ q). Usually, = stands for equality and it is a binary relation between "objects", like numbers in ...


7

Option b looks correct to me. Your descriptions for a and b seem to be the wrong way round. Option c says that anything that is both a dog and a cat and well-trained is a good pet. Option d is weird - it says that if anything is a cat then well-trained dogs are good pets.


7

You can't, because it isn't valid. Think about it with numbers, consider: a=1 b=2 c=1 It's true that a≠b & b≠c, yet a=c.


7

I guess it refers to this passage later in the article/review you were quoting from: the debate between Heidegger and Carnap -- Shirley's next topic -- precisely turns on whether Heidegger's account is compatible with a different aspect of mathematical logic: the use of existential quantification in first-order predicate calculus. As Shirley presents it, ...


6

The EI rule formalizes the fact that if we know that ∃xP(x), we are licensed to give to "that P" a name. But we have to avoid that the said name is not already "in use" because, if so, it may denote an object that has some properties incompatible with its "being P". This intuitive restriction is formalized with the proviso : the term (variable or constant) ...


6

One should keep in mind that the meaning of "logic" changed over the last century, and is now more confined to formal logic, although it is broader than deductive or mathematical logic in the narrow sense. The interface between the formal and the informal, formalization, formal semantics, is also included. But this is not what Kant meant by "transcendental ...


6

It is contentious even to suppose that logic is concerned with being 'self-evident' at all. The old-fashioned idea that logic represents the immutable laws of thought that hold everywhere for all rational beings has fallen by the wayside. Logic has to do with accounting for how it is that some propositions follow from others, or how some combinations of ...


6

Yes, the existential quantifier expresses existence. If you assert that Some pegasus are flying then you do assert that pegasuses exist, at least by the classical logical treatment of the existential quantifier, and I would claim also by the intuitive understanding of the sentence. If there are some pegasuses which are flying, then well, there are some ...


6

It means, "is equal by definition to" or "is defined to be equal to".


5

It depends of context. But, usually := is to define a objects from anothers. Usually one define variables from values, x:=7, and maps, f(x):=x+1. Many programming languages use = to do that, but Pascal language uses := to assign values. = is to compare two objects and ensure they have the same "identity". For instance, some programming languages use "==" to ...


5

(A) : For the left-to-tight direction we have : 1. P v Q _ | 2. ¬Q --- assumed [1] | 3. P --- assumed [2] for v-Elim | 4. P --- from 3 | 5. Q --- assumed [3] for v-Elim | 6. ⊥ --- from 2 and 5 by →-Elim (recall that : ¬Q is abbrev for Q → ⊥) | 7. P --- from 6 by ⊥-Elim | 8. P --- from 3-4 an 5-7 with 1 by v-Elim, discharging [2] and [...


5

HINT Here is a proof with Natural Deduction : 1) ¬(¬¬¬P ∨ P) --- assumed 2) ¬P ∨ P --- Law of Excluded Middle (is a tautology) 3) P --- assumed [a] 4) ¬¬¬P ∨ P --- from 3) by ∨-introduction 5) ¬P --- assumed [b] 6) ¬¬¬P --- from 5) by Double Negation : φ ⊢ ¬¬φ 7) ¬¬¬P ∨ P --- from 6) by ∨-introduction Now, from 3)-4) and 5)-7) we have derived ¬¬¬...


5

The difference between Excluded Middle and Bivalence: Excluded Middle says every proposition of the form P v ~P is true Bivalence says every proposition is true or it is false and that's all she wrote (forget all the technical jousting)


5

You can try to prove it by cases (∨–Elim). The general form is: If you have: ⊢ (A ∨ B), A ⊢ C, and B ⊢ C Then you can conclude: (A ∨ B) ⊢ C This means that if you've proved (A ∨ B) and you have proved (i) C from assumption A, and (ii) C from assumption B, then you have proved C from assumption (A ∨ B). ...


5

gnasher729 raised an important point that deserves some expansion: "In formal logic, implication x ⇒ y and equivalence x ⇔ y are very obviously useful - they directly express the possibly most important concepts of formal logic." The main point that I want to bring up is this: Naive set theory is not the only important set theory, and Classical logic is not ...


5

The "basic" mathematical logic axioms for identity are : x = x (reflexivity) x = y → y = x (symmetry) x= y ∧ y = z → x = z (transitivity). Thus, from a = b we can derive b = a by simmetry, and from a=b and a=c, we derive c=a from the second one by symmetry and then, from c=a and a=b, we derive c=b by transitivity, followed by b=c by symmetry ...


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