33

Edit in response to your comment: Okay, long answer: What is the meaning of "the"? (A previous version of the question had the statement "The goblins are pink"; this is an elaboration on that formulation.) First of all, as noted in the comments, the "the" makes things a bit problematic; it is not obvious that "the" is ...


24

As Conifold comments, a real-life intuitionist would not shy away from assuming LEM ... when appropriate. Intuitionism merely permits the failure of LEM, it doesn't assert that it always occurs. For example, consider equality: in intuitionistic mathematics, equality is decidable (= subject to LEM) in the context of the natural numbers but usually not in the ...


20

Because there was a calculus for one-place predicates, Aristotle's syllogistic, roughly equivalent to monadic predicate calculus. Aristotle does discuss "relatives" in Categories, which refer to multi-place relations, or rather to objects entering them. What will later be called oblique syllogisms involving relatives is mentioned in passing in ...


17

There are two ways in which these statements can be non-contradictory: Option A: Non-mutually exclusive It is possible for a goblin to be both pink and yellow, therefore it is possible for a goblin to be both pink AND yellow simultaneously. Option B: Vacuous truth (which is what it seems you are angling for) From wikipedia: In mathematics and logic, a ...


9

Propositions in intuitionistic logic are probably best understood as statements about provability. P ʌ Q means that you can prove P and prove Q, ¬P means that from P you can derive a contradiction, ∃x.P(x) means that you can exhibit a particular x and a proof of P(x) for that x, and so on. There is a law of noncontradiction because there can't be a proof of ...


8

The thought experiment is known as Chrysippus’s Dog and goes back to the named ancient Stoic. It was discussed by many modern philosophers, including Dennett, see Chrysippus’s Dog as a Case Study in Non-Linguistic Cognition by Rescorla for a survey. Here is the description of Chrysippus's dog given by Sextus Empiricus: "[Chrysippus] declares that the ...


7

An n-ary relation gives rise to parameterized unary predicates if one fixes n-1 arguments. Wilfrid Hodges argues that this is what logicians did before the nineteenth century. (There may be other works of his that better explain this.) More concretely, they would re-write the relevant statements by using natural language reasoning so that all relations are ...


6

You seem to confuse belief (which is subjective) and the actual truth value of a proposition. The LEM only applies to the latter, not to the former. If you wish to stay inside a mathematical framework, one might view probabilities as being degrees of belief. This is the subjective probability interpretation, or the Bayesian view. In your example, we would ...


6

It would be worthwhile distinguishing between a conditional sentence in the object language and a conditional in the metalanguage. Some deductive arguments have a conditional in the object language, e.g. those of the form modus ponens or modus tollens. Some arguments do not, e.g. those of the form conjunction elimination, disjunction elimination, etc. But ...


6

You give the example: Amy said you didn't go to school yesterday. She lied about it though! So you did go to school? What makes you say that? The issue here is that typically you would not say that Amy lied unless you knew that what she said was false, and the implicature would be that you know what she said was false because you know you ...


6

Firstly, the fact that the ancestor relation cannot be defined in FOL is not itself a philosophical difficulty. It relates mainly to the issue of consistency and completeness and their omega counterparts over infinite domains. It hardly means that FOL is extremely limited. Your question could reasonably be split up into separate components. Why are ...


5

There are ways we could apply normal propositional logic that might seem insane. We might utter a contradiction (any contradiction), look our interlocutor squarely in the eye, and then confidently state, "Therefore, a cedar tree draped with polkadot cloth strips ought to be the first democratically elected leader of [insert country name]," or any ...


4

The logical opposite of "I believe X is true" is not "i believe X is false" but "I do not believe X is true". While "X true" and "X false" are contradictory, and can't be both part of your beliefs without contradiction, not believe that X is either true or false is valid, and equivalent to "I don't know ...


4

This has to do with how we translate statements from natural language into formal logic. There are many different possible ways to do so, and some of those yield different results. Statements like these are typically translated into a Tarskian second-order-logic where "All Goblins are yellow" would first be converted to "For all things, if ...


4

From a riddle perspective I imagine both statements are simultaneously possible if you consider the definition of yellow to be cowardly. All goblins are cowardly and pink.


4

Classical logic can't be incompatible with an acausal universe, because it never talks about causality. It is merely rules of inference. Truth tables say nothing about something causing something else (although, of course, we can still prove the validity of a syllogism that mentions causality, or any other noun). In particular, to a logician "if p then ...


4

Gareth Evans is arguing that Aristotelean logic is closer to natural language usage and as such introduces fewer unfamiliar logical devices and has fewer counterintuitive features. This is true, but the vast majority of logicians consider this to be a price worth paying to have a much more powerful and expressive logic. Natural languages such as English have ...


4

There is a rule of first-order predicate logic (FOPL) called existential generalization that allows you to go from "Fred is a mechanic" to "something is a mechanic". But in standard logic, there are two issues with your example. Firstly, you cannot name things that do not exist, so "Fred is a mechanic" fails to be a proposition ...


3

Hint for the first question: An argument scheme being valid means that all instances of sentences of this form are valid; if the form is invalid, then not all instances are valid. According to this definition, could it be the case that there exist valid instances of an invalid form? Hint for the second question: An argument is valid iff in all structures, ...


3

Well, in what way do you think the truth of the premises does not guarantee the truth of the conclusion? In which situations is the promise "If the premises are true, the conclusion will be true" broken? The definition of validity is: For all interpretations it holds that if all premises are true under that interpretation, then the conclusion is ...


3

This is really a question about the definition of "rational person." Rationality can be subdivided into a number of different types or categories of reasoning, which include: Deductive reasoning (drawing logical inferences from rigorous application of the rules of classical logic). Inductive reasoning (inferring that a general statement is "...


3

Potentially, you could be asking one of two different questions. One is how did our ability to do logic get started? i.e. a question about origins. The other is how do we consider logic to be grounded or justified? i.e. a question about the epistemology of logic. In the case of the former question, we really just don't know. We know from studies of animals ...


2

You state (slightly paraphrased): the truth of the premises doesn't guarentee the truth of the conclusion. But the stance of classical logic (re: this, see below) is that in fact the truth of the premises does guarantee the truth of the conclusion. There is no conceivable situation where the premises are true but the conclusion is false, since there is no ...


2

A set of axioms cannot, without further argumentation, carry any ontological weight. Hence, objects definable based on a set of axioms cannot, without further argumentation, be considered existing. However, sometimes there exist further argumentations. In the case of mathematics, the axioms (e.g. by Zermelo-Fraenkel) can (arguably) carry the ontological ...


2

You may define a statement of a theory as "true" if it holds in every model, and as "provable" if there is a logical deduction from the axioms of the theory. Then Goedel's first incompleteness theorem tells us, that any theory, which is as least as powerful as number theory, contains true statements which are not provable. Hence "...


2

Just to add the very trivial point that at the very least the Intuitionist Mathematician must also accept Definition in their sources of legitimate truths. These are taken to be in some sense “analytic” truths, declaring them not so much to be matters of substance than semantics, but our choices of definiens greatly influences the resulting mathematical ...


2

A misleading feature I observe in many textbooks on logic is that they restrict translations to and from natural language to a few strict patterns, upshot of which is a habit of interpretations bereft of the huge power of informal semantics, which, however, mathematicians freely enjoy (I have always thought that it would be very nice if the English language ...


2

It is misleading to speak of an "equivalence thesis". Mathematicians typically use material implication (MI) as a conditional because it is useful to do so. In simple contexts it obeys the implicational rules that we expect a conditional to obey. It is possible to show that if you require a truth-functional dyadic connective for bivalent classical ...


2

Recall that (see also your previous post) Graham Priest is a dialetheist, i.e. a proponent of the view that there are true contradictions. Priest's discussion of the Liar starts rejecting the view that there are (truth-value) gaps and supporting the view of gluts. If we analyze the "classical Liar" (sentence (2)) in terms of truth values, we ...


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