If Abrams arrived, Browne arrived.
Had Abrams arrived, Browne would have barked.

Complex sentences consisting of a main clause modified by a subordinate clause include conditional expressions where the modifying clause provides the semantics of the antecedent and the main clause is the consequent. One of the two primary syntactic variants for this construction introduces the dependent clause with the subordinating conjunction if, while in the other variant, the dependent clause exhibits the subject-auxiliary inversion used in yes-no questions, but with a constrained inventory of permissible auxiliary verbs in subjunctive mood. The only auxiliaries found in this construction are had, were, should, and might. In both variants, the dependent clause can appear either before or after the main clause, and the main clause can be optionally modified by an overt then adverb. In the variant introduced by if, the head verb of the dependent clause can be in either indicative or subjunctive mood.

• Subjunctive mood in both clauses: If the dog were angry, it would bark.

• Indicative mood in both clauses: If the dog is angry, it barks.

• if-clause order variation: The dog would bark if it were angry.

• Inverted clause order: The dog would bark, were it angry.

The two-place relation takes the top handle of the main clause as its first argument, and that of the dependent clause as its second, in both instances mediated by a qeq relation to permit quantifiers to scope inside or outside.

if_x_then[ARG1 h1, ARG2 h2]
h3:[ARG0 e1]
h4:[ARG0 e2]

{ h1 =q h3, h2 =q h4 }

• In the inverted clause variant, negation of the auxiliary verb is not possible: *Hadn't the dog been angry, it would not have barked.

• The subjunctive form of the syntactic head verb in the dependent clause (in both variants) is the morphological past-tense form, though the semantics of the verb is not constrained to be in the past.

• Should non-clausal variants of phrases introduced by if be coerced to the same clausal semantics? Consider If chased, the dog will bark. or If the victim, the dog will bark. The grammar currently does this coercion, and it seems justified for the predicative phrases like chased, since we also see If being chased, the dog will bark. where the complement of if is clearly a proposition, but it's less clear with the NP-complement case. Even in the predicative variant, the question remains about whether to try to bind the external argument of the verbal predication; it is tempting, but consider If chased, I'm pretty sure the dog will bark. where there is no ready mechanism to provide the index to unify with the XARG of chase. The same issue arises for the NP complement variant, even if we choose to hallucinate an identity relation where the overt NP is identified as the ARG2; binding that relation's ARG1 to an index in the main clause is not straightforward, as in If the victim, it seems like the dog always barks.

• Should the semantics of when clauses be made the same or similar? Consider When the dog is angry, it barks. alongside If the dog is angry, it barks..

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