Kreisel's squeezing argument supports the view that our pre-theoretic notion of logical validity is determinate even when formal metatheories disagree, so Cohen-style independence results do not destabilize the semantics of second-order logic.

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Destabilize(as describing what independence results might do to our understanding of logic): To undermine or call into question the reliability or fixed nature of something.
Formal metatheories(as competing frameworks for understanding logic): Mathematical systems that study other logical systems themselves—basically, formal rules about how logic works.
Independence results (Cohen-style)(as potentially problematic for the stability of logical meaning): Mathematical discoveries (pioneered by Paul Cohen) showing that certain statements cannot be proven true or false within a given logical system.
Kreisel's squeezing argument(as a defense of our basic understanding of logical validity): A logical argument by philosopher Georg Kreisel that tries to show our intuitive understanding of something (like what makes reasoning valid) is actually precise and definite, even if we can't fully prove it yet.

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