The statement 'Water is H₂O' is necessary a posteriori. In two-dimensional semantic terms, this means:
AIts primary intension is necessary (the same across all worlds considered as actual) while its secondary intension is contingent
BIts secondary intension is necessary (true in every world of evaluation), while its primary intension is contingent (could pick out something other than H₂O in an epistemically possible world)
CBoth its primary and secondary intensions are necessary, making it knowable a priori after all
DThe statement is actually contingent because we could have discovered water was XYZ
The secondary intension holds actual reference fixed: since water is in fact H₂O, in every possible world we evaluate, water (= H₂O) is H₂O — necessary. But the primary intension asks what 'water' would pick out if a given world were actual. In an epistemically possible world where the watery stuff turns out to be XYZ, the primary intension picks out XYZ — contingent. The a posteriori character comes from the primary dimension: we had to investigate the actual world to discover which watery substance we were referring to.
Question 2 Multiple Choice
In Kaplan's framework, the 'character' of the indexical 'I' is best described as:
AA rigid designator: it picks out the same individual in every possible world
BA rule that takes a context of utterance and returns the speaker as the semantic content
CThe specific individual picked out, which varies depending on which world is being evaluated
DAn empty placeholder with no semantic value until a speaker fills it in
In Kaplan's two-level framework, character is a function from contexts to contents. The character of 'I' is the linguistic rule: 'refers to the speaker of this utterance.' This rule is stable across all contexts. But the content — the actual person picked out — varies with the context of utterance. Once content is fixed, it is then evaluated for truth across possible worlds. Two-dimensional semantics generalizes this character/content split to all expressions, not just indexicals.
Question 3 True / False
Two-dimensional semantics implies that most necessary truths should be knowable a priori.
TTrue
FFalse
Answer: False
The whole point of two-dimensional semantics is to explain necessary a posteriori truths — statements that are metaphysically necessary (true in all worlds at the secondary intension level) yet not knowable without empirical investigation (contingent at the primary intension level). 'Water is H₂O' and 'Hesperus is Phosphorus' are paradigm cases. Two-dimensionalism separates metaphysical modality (which intension is necessary) from epistemic modality (which intension is knowable a priori), showing they can come apart.
Question 4 True / False
The primary intension of 'water' picks out H₂O in nearly every epistemically possible world considered as actual.
TTrue
FFalse
Answer: False
The primary intension of 'water' picks out whatever the watery stuff turns out to be in whichever world is considered actual. It is a description-like function: 'the watery substance in the actual world.' In an epistemically possible world where the watery stuff is XYZ rather than H₂O, the primary intension picks out XYZ. This is precisely how two-dimensionalism explains why 'Water is H₂O' was a discovery: the primary intension is contingent, leaving open which substance is water until investigation determines it.
Question 5 Short Answer
Explain in your own words why 'Water is H₂O' counts as necessary but not a priori in two-dimensional semantics.
Think about your answer, then reveal below.
Model answer: It is necessary because the secondary intension holds actual reference fixed: since water in our world is H₂O, across every possible world of evaluation, the thing that is actually water (H₂O) is H₂O — true everywhere. It is not a priori because the primary intension is contingent: there are epistemically possible worlds where the watery stuff is XYZ, and we cannot rule this out from the armchair. We had to do chemistry to discover which world we inhabit. The two dimensions come apart: secondary-intension necessity coexists with primary-intension contingency, producing a truth that is both metaphysically necessary and discovered empirically.
This is the two-dimensionalist resolution of Kripke's puzzle about necessary a posteriori truths. The apparent paradox disappears once you distinguish the dimension along which reference is fixed (the primary intension, tracking epistemic possibilities) from the dimension along which truth is evaluated (the secondary intension, tracking metaphysical possibilities).