14

Let me give you a historical background first. Until the end of 19th century logic was almost exclusively associated with Aristotelian logic, the syllogistic. This logic did not have quantifiers, or even propositional variables, in other words it was too weak to support even arithmetic, let alone the rest of mathematics (Chrysippus, an ancient Stoic, and ...


12

Validity in logic is a somewhat tricky notion to understand as it is different – though only subtly – from related, pre-theoretic notions. For instance, not every valid argument is ‘convincing’ or ‘useful’ in a pre-theoretic sense, nor is every ‘convincing’ or ‘useful’ argument valid. (See examples below.) To address your question more directly, suppose ...


11

You can see: Irving Anellis, The Genesis of the Truth-Table Device (2004) as well as: Irving Anellis, Peirce's Truth-functional Analysis and the Origin of the Truth Table (2012). Before Bertrand Russell (Harvard logic course: 1914) and Ludwig Wittgenstein (Russell and Wittgenstein's manuscript dated 1912; see also: Tractatus (1921), 4.31 and 4.442 for ...


11

Intuitionism is a philosophy of mathematics that was introduced by the Dutch mathematician Luitzen Egbertus Jan Brouwer (1881–1966): he developed a very personal philosophy of mathematics that founds mathematics (partially following Kant; see Kant's Philosophy of Mathematics) on a pure intuition of time. You can see his Intuitionism and formalism (1912) : ...


11

We do NOT define vacuous statements as true. A vacuously true statement is vacuously true. A "vacuously false" statement is vacuously false; although nobody ever gives this type of statement any thought. Example: Every element of the empty set is a purple flying elephant. We agree that this statement is (i) vacuous; it doesn't really say anything ...


10

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 ...


9

First, to dispel false conceptions: Informal logic is not the contrary of formal logic, at least for some established meanings of 'informal logic'. 'Formal logic' is usually reserved for the formal study of truth-preserving inference, like deduction. But there's nothing preventing a formal study of not necessarily truth-preserving inference, so-called ...


9

You've stumbled upon an old problem in philosophy, The Paradox of Inquiry, first formulated in Plato's Meno. The problem can be reformulated as follows: Either you know the answer to a question, or you don't. If you do, then there is no point searching for it. If you don't, then you will not know what to search for. The short answer is that you can ...


9

Short answer : NO. Arguments are either valid or not. Premises and conclusions are sentences, and thus they are either true or false. See Valid argument : In logic, an 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. [see also Hurley, page 44] Regarding ...


8

To put it in simple words we have to describe in a couple of words the project of Principia Mathematica, which Russell inherited from Frege: reconstructing mathematics from logic alone. For a broader context see What is the philosophical ground for distinguishing logic and mathematics? Frege himself could not complete the project because Russell discovered ...


8

My reading of Carnap's "The Elimination of Metaphysics Through Logical Analysis of Language" suggests to me that it is possible to form sentences in a language that are grammatically correct but logically meaningless. The consequence of this is that the statement, due to its logical incoherence, cannot be proven to be true or false. And likewise, if a ...


8

Here's the example from David Lewis's Counterfactuals (1973): If Boris had gone to the party, Olga would have gone. Now suppose that Boris wants to go, but not if Olga goes, because he wants to avoid her. Olga, on the other hand, wants to see Boris, and wants to go if he does. Given this supposition, the contrapositive of the above is false: If Olga ...


8

As well as the counterfactual example given by Eliran, there are an abundance of examples where the conditionals are uncertain. For example, given the poor record of the Norwegian soccer team, I might believe strongly that if Norway reach the final of the next world cup then they won't win. The contrapositive of this is that if Norway win then they won't ...


7

THe program of Universal Logic is mostly considered a joke by the professional logicians I have talked to. There seem to be two popular conceptions of logic among philosophers who think about logic. But before going into these, you have to understand first that, among present-day logicians, the philosophical concerns are most frequently secondary or ...


7

What is a variable ? A variable is a "syntactical" object: it lives in formalized languages. In the formalized language of logic and mathematics, it "works" like a pronoun. When we have a propositional function like "x is a red" or "x is odd" we have a statement like "it is red" or "it is odd". What is the menaing of the statement "it is red" ? It ...


7

Topics relating to the question are not neatly partitioned due to the involvement of complex fields and perspectives (e.g., contribution from mathematics and computer science to logic). Risking over-generalization, this is how differences among the three fields can be understood. Logic is like the tools in the shed; philosophical logic is using the tools to ...


7

It is more complicated than a merely instrumental/metaphysical division. The "logics as mere instruments" were not meaningful until well into the 20th century, after the general disillusionment in foundationalist epistemologies, and the subsequent pluralism of formal systems and interpretations. Originally, Logic meant the study of Logos, reason, its laws, ...


7

I think you are right to be impressed with the Curry-Howard correspondence. It is a detailed and extensive rule-by-rule and feature-by-feature isomorphism. This strongly suggests that proof and computability are closely related. I also agree that it is under-appreciated within the philosophy of logic and that we can and should allow it to inform our ...


7

Conditionals in English are used for a lot more than just expressing simple truth functions. Here are some general cases where the truth functional material conditional doesn't fit. Claims about causal relations. Often when we say "if A then B" we are making a causal claim. But "A causes B" is not a truth function. It is not true just in the event that A ...


7

There seem to be two parts to your question: one concerning whether all quantification is restricted, and another concerning whether free variables can carry contextual information that serves to restrict their admissible valuations. I'll address each in turn. Unrestricted Quantification It seems like your question is regarding the possibility of ...


7

Do we eventually say that we are satisfied that the premise of a conclusion can stand on its own, or do we really just not continue this exercise infinitely because we would rather do something else with our time? I think there are two ways in which this can be understood. Your question isn't one about logic, because logic doesn't say anything about the ...


7

In "Paris is the capital of France", "is" is used to mean identity. In "my pet is a cat", "is" is used to mean predication : my pet belongs to the class of cats. See Ludwig Wittgenstein's Tractatus : 3.323 In everyday language it very frequently happens that the same word has different modes of signification—and so belongs to different symbols—or that ...


7

If I understand your question correctly, you are asking in effect how do we distinguish logic from non-logic? Logical expressions give rise to valid arguments and logical truths, that is, arguments where if the premises are true it is impossible for the conclusion to be false, and truths such that there is no way for them to come out as false. But this ...


6

The answer already given by user3451767 is not in my view correct. Logical atomism characterizes Russell's work from roughly 1910-1925. POM was published in 1903. There are many differences between his views during these periods. The short answer to your question: Russell's terminology is confusing because he has two uses of the word "term" in POM: -A ...


6

The distinction between mathematics and logic was almost universally held before modern times. Aristotle's logical works (Prior and Posterior Analytics) are part of his works that later philosophers grouped as the Organon (tool). Hence, logic was seen as a tool. Ancients like Boethius and medievals like St. Thomas Aquinas, and logicians like John Poinsot, ...


6

The sentence "If Ron went to the store, Ron would be home by now." does not have a truth value. How do we further determine the validity of the argument? When you say this, it seems to me that you are conflating two different levels of logic: sentences being true and arguments being valid. So first of all, let's make our definitions of arguments being ...


6

Short answer: definitely no, that does not make logic useless. When someone makes an invalid argument, they're committing some sort of a formal fallacy. That is only to say that the conclusion does not logically follow from the premises. The invalidity of an argument does not say anything about the either the truth of the conclusion or the truth of the ...


6

Suppose you're trying to solve the following algebra problem in the real numbers: Find the solutions to x2 = 6x - 11 Your first step to approaching the problem might be to bring everything over to one side to make a quadratic equation: x2 - 6x + 11 = 0 What is the actual logical justification here? The laws of arithmetic say the following is true: ...


6

I am going to assume extra context provided by OP’s earlier related question, Does fictional discourse pose special difficulties for logic? Classical devices Even the classical set theory has a simple device for restricting the range of quantifiers without changing the domain of discourse, ∃x∈S P(x) restricts the domain to S. If the use of set theory is ...


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 ...


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