When it is stated that "logic is by extension not intention," it means that logical analysis primarily deals with the extensional aspect of terms or concepts rather than their intended meaning or defining properties. Logic is concerned with the relationships between different categories, the truth values of statements, and the validity of arguments based on how terms are related extensionally. It focuses on the logical form and structure of reasoning, rather than the specific content or intended meaning of the terms involved.

In essence, logic abstracts away from the specific details of the intended meanings of terms and instead focuses on their extensional relationships and implications, where the role of the interpreter is given at least equitable importance as a logical theory synthesiser and ultimate say over the interpretation of theories of logic.

Ehrenfeucht Fraïssé Games exemplifies this where the synthesis of a theory of finite-model theory, and the intended standard interpretation, means naught in context of an EF Game where the duplicator may, by extension, find another interpretation over the set of played theorems in an EF game.

In an Ehrenfeucht-Fraïssé game, two players, commonly referred to as the spoiler and the duplicator, compete against each other. The spoiler selects an element from one structure (as defined by a set of theorems of a theory under finite-model theory), while the duplicator aims to find an element in anther structure that behaves in a similar way. The game progresses by alternating moves between the players. In the context of Ehrenfeucht-Fraïssé games, if the duplicator has a winning strategy, it means that they can find a structure in the second model that is isomorphic to the structure chosen by the spoiler. This isomorphic structure, however, may be defined by a different theory or a different set of axioms. The purpose of Ehrenfeucht-Fraïssé games is precisely to compare structures based on their elementary properties, disregarding the specific theories or interpretations associated with them. The duplicator's strategy aims to demonstrate that the two structures are elementarily equivalent, meaning they satisfy at least the same first-order logic sentences. This equivalence can be achieved by finding an isomorphism between the structures, even if they are defined by different theories.

This means that for every formula in the first-order language, if it is true in one structure, it must also be true in the other structure, and vice versa. However, it is possible that there are additional formulas that one structure satisfies but the other does not. By stating that two structures satisfy at least the same first-order logic sentences, we allow for the possibility of additional sentences being true in one structure but not in the other, while still ensuring that the essential logical properties are preserved. This formulation acknowledges that the structures may have further differences beyond the shared sentences.

The game can be seen as a formal way to compare the extensional properties of structures. The duplicator tries to demonstrate that the two structures are indistinguishable in terms of their behaviour or the truths of certain formulas. If the duplicator has a winning strategy, it means that the two structures are elementarily equivalent, indicating that they satisfy the same first-order logic sentences.

The idea behind Ehrenfeucht-Fraïssé games is that they focus on the extensional aspects of structures rather than their intended interpretations or the specific content of their elements. The game captures the notion that if the duplicator can consistently mimic the behaviour of one structure using the elements of another, then from a logical perspective, the two structures cannot be distinguished.

Therefore, in the context of Ehrenfeucht-Fraïssé games, the notion of logic being by extension, not intention, is exemplified. The games allow us to analyse structures based on their extensional properties, disregarding the specific intended interpretations or the intended standard interpretation. The duplicator's ability to find another interpretation over the set of played theorems demonstrates that the focus is on the extensional relationships and implications rather than the intended meaning or defining properties.

The very existence of EF Games and that a duplicator can have a winning strategy cements that logic is by extension rather than intention.