When is a quantified sentence true?
In short, ∀xP(x) is true when EVERYTHING is P. Similarly, ∃xP(x) is true when SOMETHING is P. But let's be more precise, by introducing two notions: SATISFACTION and the DOMAIN.
An object (call it *c*) satisfies a wff like Loud(x) or OnFire(y) just in case, when we substitute the name *c* for the variables x and y, we get a true sentence. So we say that the object c SATISFIES a wff like Loud(x) if this replacement of the variable with the constant gives us a true sentence.
(Clarification: this lecture is about SATISFACTION. For the related notion "Satisfaction", see M. Jagger's seminal (1965): [ Ссылка ])
And the DOMAIN is just all the (existing) things we're talking about, so everything.
On these lights, ∀xP(x) is true just in case every object in the domain satisfies P(x). And ∃xP(x) is true just in case at least one object in the domain satisfies P(x).
This is what we mean when we talk about the truth of quantified sentences: so we CAN get some satisfaction (at least if our domain isn't empty).
LEGAL GUNK RELATING TO MICK JAGGER CUTOUT .PNG
The Mick Jagger .png image that appears at the beginning and end of the lecture when I say the word "satisfaction", as well as in the thumbnail, is a cutout (i.e. modification) of an image taken by Ralph_PH (link: [ Ссылка ]).
It is used under the Creative Commons Attribution 2.0 Generic license (link: [ Ссылка ]).
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