The claim that FO 'cannot express properties in P without extensions' trivially conflates the base logic with its natural and well-motivated closure operations, which logicians since Kleene have treated as intrinsic to logical expressibility.

?Rate how convincing each reason is below to see the overall strength.

1 reason for

1 reason against

Reasons For

1 perspective

Reason for

1.Closure operations (like least fixed points) have been foundational to logic since Kleene's recursion theory, not afterthoughts.
?
Think about whether this reason is strong or weak
2.Distinguishing 'base logic' from 'closure operations' artificially fragments what logicians treat as unified expressibility.
?
Think about whether this reason is strong or weak
3.Properties definable only through natural closure operations are legitimately expressible within the extended logical system.
?
Think about whether this reason is strong or weak

Reasons Against

1 perspective

Reason against

1.First-order logic's expressive limits are precisely what motivate studying extensions; conflating them obscures important boundaries.
?
Think about whether this reason is strong or weak
2.Calling closure operations 'intrinsic' begs the question: they're added precisely because base FO cannot express those properties.
?
Think about whether this reason is strong or weak
3.Historical convention among logicians doesn't settle whether expressibility should include non-elementary closure mechanisms.
?
Think about whether this reason is strong or weak

Next step

Based on where you are in your exploration

Strongest counterpoint

Explore the most compelling reason on the other side.

Key Terms

Base logic(what FO is described as in the statement): The simplest or most fundamental version of a logical system, before you add extra tools or rules to make it more powerful.
Closure operations(described as something added to base logic): Mathematical operations that, when applied to statements in a logical system, produce new statements that stay within that same system.
Conflates(in argumentation and logic): Treats two different things as if they're the same thing, or mixes them up in a way that causes confusion.
Extensions(what FO allegedly cannot express according to the claim): In logic, the set of all things that a property actually applies to—for example, the extension of 'red things' is everything that is red.
FO (First-Order Logic)(the main subject being discussed in the statement): A formal system for reasoning that can make statements about individual things and their properties, but cannot directly talk about properties themselves the way higher-level systems can.
Logical expressibility(what closure operations are treated as intrinsic to): The ability of a logical system to formulate or represent ideas, properties, or arguments clearly and completely.
Stephen Cole Kleene(referenced as a logician who treated closure operations as important to logic): An important 20th-century logician and mathematician who made major contributions to understanding how formal logical systems work.
properties(Contrasted with substances as ontologically dependent entities.): Entities that depend for their existence on substances, being properties of individual objects.

Connections

2 topics

Truth & Knowledge1 linked Philosophy of Language1 linked

Calling closure operations 'intrinsic' begs the question: they're added precisel...Closure operations (like least fixed points) have been foundational to logic sin...

Details

Type: claim
Perspectives: 2 (1 for, 1 against)
Edits: 1 edit