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30 votes
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How do you differentiate between At least one X and Exactly one X in predicate logic?

You sometimes find the notation ∃!x as an abbreviation for "exactly one x". With the standard symbol inventory, "exactly one" can be defined in terms of "at least one and ...
Natalie Clarius's user avatar
11 votes

Why is the identity predicate needed?

The standard approach in predicate logic is to use names to identify things, and predicates to identify properties. Samuel Clemens is a name, not a property. It is possible to do as you suggest and ...
Bumble's user avatar
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8 votes
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Is it possible to construct infinitely many non-equivalent formulas in predicate logic?

Below 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 ...
Noah Schweber's user avatar
6 votes

Why is first-order logic interesting to philosophers?

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 ...
Bumble's user avatar
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6 votes

Is ¬(a = b) the same as (a ≠ b) in logic

In classical mathematics these are the same, and indeed more generally the "strike-through" notation is just shorthand for logical negation. In constructive mathematics, however, my understanding is ...
Noah Schweber's user avatar
6 votes

What does "unqualified notion of truth" mean in this passage?

You understood that correctly A qualified truth is simply a truth that is conditioned, ie. the truth of it depends on other conditions. In this case, a theorem is a qualified truth in the sense of it ...
Philip Klöcking's user avatar
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5 votes
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Proof Using Model Universe

Yes-ish: it takes some work to formalize it, but it can be done. Specifically, the proof of the relevant model checking theorem gives a general method for proving, for an appropriate sentence p, a ...
Noah Schweber's user avatar
5 votes
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Definite Descriptions VS 'Exactly' Statements

To expand a little on Conifold's comment... (∃x){Ax ∧ Hx ∧ (∀y)[Ay → y=x]} This states, there is at least one thing that is the author of Evangeline and is named Henry Wadsworth, and there is at most ...
Bumble's user avatar
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4 votes

Truth Value of Definite Descriptions

The purpose of Russell’s Theory of Descriptions is precisely to give meaning (i.e. truth value) to a statement concerning a non-existent entity. The basic assumtpion is that names of individuals must ...
Mauro ALLEGRANZA's user avatar
4 votes
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What are the advantages of Aristotle's term logic over predicate logic?

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 ...
Bumble's user avatar
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4 votes

Is ¬(a = b) the same as (a ≠ b) in logic

Although x ≠ y stands for ¬(x = y) to be more reader-friendly in the standard presentations of predicate logic, there is a syntactic distinction between them. ¬(x = y) is the negation of an atomic ...
Tankut Beygu's user avatar
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4 votes
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Philosophy book written using logic statements

Prolog doesn't check the consistency of programs. Doing so would be equivalent to solving the (provably unsolvable) halting problem. In fact, in Prolog, there is no way to assert a negative ...
David Gudeman's user avatar
4 votes

General sentence operators

There isn’t a general procedure to determine the restrictions required for combining certain operators. There isn’t even agreement on how the standard connectives of propositional logic should ...
PW_246's user avatar
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4 votes

General sentence operators

To my knowledge, the phrase occurrent in the literature such as covers this topic is "combining logics": The subject of combinations of logics is still a young topic in contemporary logic. ...
Kristian Berry's user avatar
3 votes

Why is first-order logic interesting to philosophers?

Let me add to the existing (very good) answers. First of all, there's an implicit assumption in your question that philosophical interest comes from strength. This is unjustified, especially given the ...
Noah Schweber's user avatar
3 votes

How does one tell if logical expressions are equivalent?

The first observation is that moving the ∀b quantifier in the second formula to the front is unproblematic since we can safely move a quantifier from the right-hand side of an implication to the ...
Natalie Clarius's user avatar
3 votes

Truth Value of Definite Descriptions

What might seem odd to you is that Russell treats the description operator in a syncategorematic way. That is, the operator itself is not associated with an explicitly defined operation, but formulas ...
sequitur's user avatar
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3 votes
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Is al-Farabi right that predicates must add information and existence is not a predicate?

Rescher has an informative discussion of al-Farabi's views on existence as a predicate in A Ninth-Century Arabic Logician on: Is Existence a Predicate?, which includes their comparison to the more ...
Conifold's user avatar
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3 votes
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What exactly is the relationship between first-order logic and the axioms of ZFC? Which one is more fundamental?

From Benacerraf's identification problem (which undermines or at least underdetermines the reduction of natural numbers to sets) to the charge that second-order logic is "set theory in sheep's ...
Kristian Berry's user avatar
3 votes
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What does "unqualified notion of truth" mean in this passage?

The entire point here is that truth is always relative to an intended model. Any logic textbook worth reading would precisely define truth via the satisfaction relation ⊨. It would define when exactly ...
user21820's user avatar
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2 votes
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SELF-REFERENCE AND THE LANGUAGES OF ARITHMETIC

See e.g. Gödel’s Incompleteness Theorems : Diagonalization: it is a general result of FOL that : Let A(x) be an arbitrary formula of the language of F with only one free variable. Then a sentence D ...
Mauro ALLEGRANZA's user avatar
2 votes
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How does one tell if logical expressions are equivalent?

Expressions are equivalent if they can be related with an if and only if connection. One way to check if they are equivalent is to use a tree proof generator. Putting these expressions into such a ...
Frank Hubeny's user avatar
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2 votes
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How does identity work in modal predicate logic?

A general summary of this topic can be found on the SEP. "Some predicates are necessary to objects and others are not" is exactly right: at least sometimes, we use predicate modal logic in order ...
Noah Schweber's user avatar
2 votes

Why is first-order logic interesting to philosophers?

There's a point of view that topics of inquiry move out of the realm of philosophy and into the realm of science when they become codified, standardized, well-understood and reliable. In contrast, ...
Chris Sunami's user avatar
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2 votes

Why is first-order logic interesting to philosophers?

Short Answer FOL is a simple model of human reasoning, and much like simple models in general, it is a pedagogical aid in introducing students to the formal aspects of logic without being unwieldy and ...
J D's user avatar
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2 votes

Why is first-order logic interesting to philosophers?

There are certain limitations to FOL, particularly the Lowenheim-Skolem theorem which is why we have to use HOL for models which are uncountably infinite because using an countably infinite number of ...
Anirban Mandal's user avatar
2 votes
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Can 'All S is P' ever be false if S is empty?

Since you keep saying "boolean interpretation", I guess what you mean here is the contrast made in some introductory texts (e.g. Copi or similar) between the interpretations of Boole vs ...
against very long user names's user avatar
2 votes

Can 'All S is P' ever be false if S is empty?

In the standard semantics for the universal quantifier (which is not restricted to classical logic) the falsity of a universally quantified statement of the form 'All Fs are Ps' indeed implies that ...
sequitur's user avatar
  • 292
2 votes

How to prove ∀x∃y(Fx→Gy) ⊢ ∃y∀x(Fx→Gy) using natural deduction?

Well, first understand what it's saying and why it would intuitively be true. ∀x∃y(Fx→Gy) means that: If Fx is true, then there must exist a y such that Gy is true. It could be a different y for ...
causative's user avatar
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2 votes
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ontological commitments to properties

Quine was generally rather nominal than realistic towards the relative abstract within the context of a predicate as described here. A predicate is a sentence that contains a finite number of ...
Double Knot's user avatar
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