Existence in logic vs existence in reality

Question

Take an object which has been destroyed, we can talk about it in the past tense, how does this work logically, can we talk about objects which previously existed (in the physical sense)? For the object to be discussed it must be an object in our language, and therefore if we have a constant symbol 'y' for this object, the statement

∃x(x=y) i.e there exists an object x that is y (the object that does not exist physically)

Is the existence implied by the existential quantifier different then our 'physical' definition or do we need to adjust our definition based on the domain, i.e if we can talk about it, it is in the domain and 'exists' logically?

However, anything we say about it is limited to past tense in natural language, would all possible predicates be based on functions that map in the past tense? For example:

'The dodo was a breed of bird'

Logically let D denote The Dodo

P(D) must be 'D was a breed of bird' and not 'D is a breed of bird' as P(D) would be false if it denoted the latter.

How do we differ with the use of 'existence'?

Answer 1

There are a few distinct issues here. When we say 'exists' in logic, it is common to use this to express a kind of eternal present. In this sense, Albert Einstein exists, but he is dead, or no longer extant, or no longer alive. We have to allow that Albert Einstein exists, since otherwise a sentence like, "Albert Einstein invented relativity theory" would fail to state anything, since the name "Albert Einstein" would not denote any existing thing. Also the sentence, "There exists someone who invented relativity theory" would be false, which would be unsatisfactory. Similarly, dodos exist but they are all dead and so the species is extinct.

If you want to use logic to express temporal relations explicitly, you can use temporal logic for this purpose. Temporal logic has a similar structure to alethic modal logic, with "at all times" replacing "necessarily", "at some time" replacing "possibly", and "now" replacing "actually". The difference being that times are ordered, so temporal logic also has a primitive "earlier than" relation.

In standard logic, existence is handled using the existential quantifier. So, (∃x)(x=a) expresses the fact that a thing exists and has the name 'a'. This does not require that it exists physically; it could be an abstract object, such as a number. Standard classical first order logic assumes that the universe is non-empty, i.e. that at least one thing exists, so in fact the sentence (∃x)(x=a) is a theorem.

If you want to use logic to talk about things that don't exist, or things that only possibly exist, there are free logics that can handle that, In these, existence is treated as a predicate, and in some varieties quantification is permitted over non-existent entities.

Answer 2

From the point of view of formal logic, the existential quantifier does not commit to physical existence of any object. When a formal language that admits this quantifier is defined, like first order logic, the language definition is prior to any semantic interpretation for the language, so prior to actual existence in a physical sense. Formal logic languages can be interpreted to be about various abstract objects that have no claim to being physical. For example, a formal language can be defined then interpreted to "talk" about various algebraic structures, such as groups, which do not have any physical existence. See, e.g. Enderton.