23

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 ...


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 =...


9

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".


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 is a ...


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

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 ...


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

Model theory, as it is today understood, is a formal way to study how bits of language manage to represent the world. The fundamental idea of model theory is that you have a structure that assigns interpretations to bits of language in such a way that the structure makes each sentence in the language either true or false. You know this. The metalanguage thus ...


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

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

All philosophy uses logic, but what you've asked suggests that what you really want to know is who uses symbolic logic in drawing out their arguments. For the obvious reasons, the branch of philosophy that you'll see applying symbolic logic with the highest frequency is the philosophy of logic itself: Russell's Principia Mathematica has enough to make your ...


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 ...


5

The two different symbols on the page you link to are indeed different. The first is the turnstile symbol Ⱶ which may be read as 'proves', while the arrow → is material implication. These are very different. Material implication is a symbol in the object language defined by the truth table that you give, i.e. T/F/T/T. Turnstile is a symbol in the ...


5

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) ...


5

Premise : 0) (T⇒E) ∧ (A⇒L). Use Material Implication on the premise to get : 1) (~T∨E) ∧ (~A∨L) Use Simplification to get respectively : 2) (~T∨E) and : 3) (~A∨L) Using Addition on 2) get : 4) (~T∨E) ∨ L Using Association : 5) ~T ∨ (E∨L) Using Addition on 3) get : 6) (~A∨L) ∨ E Using Association : 7) ~A ∨ (L∨E) and using Commutation : 8)...


5

another approach giving the minimum number of steps (though not a formal proof): 1. (P & Q) v ~(P & Q) law of excluded middle 2. (P & Q) v (~P v ~Q) DeM 1


5

Because of the double negation (no + unless) I would first try to paraphrase the sentence before trying to translate it into predicate logic. All the following have the same meaning: No dolphin sings unless it jumps All dolphins don't sing unless they jump All dolphins don't sing if they don't jump All dolphins jump if they sing The last ...


5

First, let's observe (in response to a comment) that we can get infinitely nonequivalent sentences; this isn't what you want, but it demonstrates that the situation is nontrivial. Namely, we can talk about cardinality, via the following statements which are easily seen to be appropriately expressible: N(n): There exist x_1,...,x_n which are all distinct. ...


5

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 ...


5

The rule is called material implication in classical logic. Here's Wikipedia's description: In propositional logic, material implication is a valid rule of replacement that allows for a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that P implies Q is logically equivalent to not-P or Q and that ...


5

They are "irrelevant" in the sense that standard first-order logic does not give us a way to systematically treat them, so in translation these more fine-grained distinctions are usually dropped in order to keep the symbolization simple. For example, in natural language, from Bill has never eaten a hamburger we should be able to infer Bill didn'...


4

If you take A -> B to be a material conditional, such that A -> B iff ~A v B is a prior definition rule MC, then a very simple proof might go something like this: A->A (established in the subproof below:) A (suppose for a conditional) A (by reiteration) A->A (by conditional proof ~A v A (by MC) But the actual details of ...


Only top voted, non community-wiki answers of a minimum length are eligible