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Note: I drew from many sources, Francis Turretin, William Whitaker, Petrus Van Mastricht, Edward Leigh, and Richard Muller. Many of these arguments are repeated throughout multiple sources. I did my best to cite those sources.
The Doctrine Stated
For the Statement of the Doctrine, one need not go further than the summary given by the Angelic Doctor in his commentary on Galatians,
Here it should be noted that “allegory” is sometimes taken for any mystical meaning: sometimes for only one of the four, which are the historical, allegorical, mystical and the anagogical, which are the four senses of Sacred Scripture, all of which differ in signification. For signification is twofold: one is through words; the other through the things signified by the words…[in Sacred Scripture] words and the very things signified by them signify something. Consequently this science can have many senses. For that signification by which the words signify something pertains to the literal or historical sense. But the signification whereby the things signified by the words further signify other things pertains to the mystical sense.
There are two ways in which something can be signified by the literal sense: either according to the usual construction, as when I say, “the man smiles”; or according to a likeness or metaphor, as when I say, “the meadow smiles.” Both of these are used in Sacred Scripture; as when we say, according to the first, that Jesus ascended, and when we say according to the second, that He sits at the right hand of God. Therefore, under the literal sense is included the parabolic or metaphorical.
However, the mystical or spiritual sense is divided into three types. First, as when the Apostle says that the Old Law is the figure of the New Law. Hence, insofar as the things of the Old Law signify things of the New Law, it is the allegorical sense. Then, according to Dionysius in the book On The Heavenly Hierarchy, the New Law is a figure of future glory; accordingly, insofar as things in the New Law and in Christ signify things which are in heaven, it is the anagogical sense. Furthermore, in the New Law the things performed by the Head are examples of things we ought to do—because “What things soever were written were written for our learning” (Rom 15:3) —accordingly insofar as the things which in the New Law were done in Christ and done in things that signify Christ are signs of things we ought to do, it is the moral sense. Examples will clarify each of these. For when I say, “Let there be light,” referring literally to corporeal light, it is the literal sense. But if it be taken to mean “Let Christ be born in the Church,” it pertains to the allegorical sense. But if one says, “Let there be light,” i.e., “Let us be conducted to glory through Christ,” it pertains to the anagogical sense. Finally, if it is said “Let there be light,” i.e., “Let us be illumined in mind and inflamed in heart through Christ,” it pertains to the moral sense. (Gal.C4.L7.n253-4)
In sum, in scripture signification may be done according to words or according to things. According to words is the genus of the literal sense. This may be done after the plain meaning of the words (proper), or after some sort of metaphor or simile (improper), thus we have the two species of the literal sense.
According to things (where the thing signified by the words is also a sign) is the genus of the spiritual sense. It can be a sign in three ways, first it is a sign of the new law (allegorical), second it is a sign of future glory (anagogical), or, third, it is a sign of what we ought to do (moral), thus we have the three species of the spiritual sense.
By way of example, if we are speaking about the unbroken lamb. The literal sense of the words (which is proper) is the unbroken lamb, yet the unbroken lamb, the thing signified by the words is itself a sign to another thing (Christ) which is the allegorical sense.
Objection 1: “Truth is only one and simple and therefore cannot admit many senses without becoming uncertain and ambiguous.” 
Response: The consequent is affirmed. The Consequence is denied, for (1) the description of “many senses” is ambiguous. Rather two-fold sense in scripture is a better descriptor. For, the signification that is the spiritual sense is the signification of the thing signified by the letter of scripture. Thus, the spiritual sense is said to be “based upon” the literal sense. Therefore, such is not uncertain. Further, (2) it is affirmed that “whatever is found in the spiritual sense is found elsewhere in the literal sense.” While the spiritual sense can itself be ambiguous, it does not therefore follow that truth is ambiguous. Such an argument would also press against the notion that “whatever is unclear in one part of scripture is clear in another part of scripture.” Thus, for ambiguity, not only a certain part, but the whole would have to be ambiguous, which is not the case. Further, (3) as the Angelic Doctor says, “It must be said that, as Augustine says in the book On Christian Doctrine, profitably is it disposed by God that truth in sacred Scripture should be manifested with some difficulty: for this is useful for destroying tedium, because unto those things which are difficult there rises a greater attention, which destroys tedium. Similarly from this the occasion of pride is destroyed, while a man can seize the truth of Scripture with difficulty. Similarly through this the truth of the faith is defended from the derision of infidels; whence the Lord says: Do not give what is holy to dogs (Matt 7:6); and Dionysius admonished Timothy that he should preserve holy things uncontaminated from the unclean. And thus it is clear that it is expedient that the truth of the faith should be handed over under diverse senses in sacred Scripture.” (QVII.Q6.A1.Rep2)
Obj. 2: “there is only one essential form of any one thing (now the sense is the form of the Scriptures).” 
Response: The Major premise is affirmed. The minor premise is distinguished. For, if by “form of the Scriptures” one means that the literal sense is the forms of the letters of scripture, it is affirmed. Yet, it is denied that the spiritual sense is the form of the letters of scripture, rather the spiritual sense is the form of those things signified by the letters of scripture (broadly speaking, the literal sense).
Obj. 3: “From the perspicuity of the Scriptures, which cannot allow various foreign and diverse senses.” 
Response: To this I respond which the Protestant principle that “whatever is found in one area of scripture to be unclear, is found in another area to be clear.” Thus, clarity is found in the whole, not in the part. In a similar way, whatever is found in the spiritual sense, is found in the literal sense in another place.
Obj. 4: To allow for multiple senses allows for equivocation.
Response: This is simply denied. As the Angelic Doctor writes, “It must be said that a variety of senses of which one does not proceed from another makes a multiplication of speech; but the spiritual sense always is founded upon the literal and proceeds from it, whence from this: that sacred Scripture is expounded literally and spiritually, there is not any multiplicity in it.” (QVII.Q6.A1.Rep1) For, equivocation would have multiple significations of the letter, on the other hand, the spiritual sense is a signification of the thing signified by the letter.
Obj. 5: “That which does not signify one thing surely signifies nothing.”
Response: The principle is affirmed. The consequence is denied. For, the words of scripture signify one thing (the literal sense). The spiritual sense is a signification of the literal sense, not a signification of the words of scripture.
Obj. 6: Bellarmine errs by defining the literal sense as “that which the words immediately present.”
Response: The assertion is subdistinguished. If by this is meant the proper literal sense, it is affirmed as a definition. If by this is meant the improper literal sense, it is denied with the Reformed.
Obj. 7: “Tropology…flows plainly and necessarily from the very words, and is…collected from the text itself…This, therefore, is not a new or various meaning, foreign to the words themselves, but absolutely one and the same with the literal sense.” 
Response: The major premise is distinguished. The hidden minor premise (that such is besides the spiritual sense) is denied. Therefore, the consequence is denied. By “flows plainly from the very words” one means that such events signify the life of the believer, it is affirmed. But, if by such is meant that the signification of the letter is the life of the believer, it is denied. This is plain, for example, the words describing Jonah’s life is, according to the letter, taken to be a historical account of the life of a certain man. But, only insofar as Jonah is a typos of the life of a believer do we get moral exhortation, i.e., that Jonah’s life, as the signification of the letter of the text is itself a signification of the life of the believer. The denial of the hidden minor premise is plain from this and from the definition of the doctrine above.
Obj. 8: “We should form a like judgment of the type or anagoge…God says, “I sware in my wrath, that they should not enter into my rest." There the rest may be understood both of the land of Canaan, and typically also of the kingdom of heaven: for the realm of Canaan was a type of the kingdom of heaven. Yet this is not a twofold sense; but, when the sign is referred to the thing signified, that which was hidden in the sign is more openly expressed. When we proceed from the sign to the thing signified, we bring no new sense, but only bring out into light what was before concealed in the sign. When we speak of the sign by itself, we express only part of the meaning; and so also when we mention only the thing signified: but when the mutual relation between the sign and the thing signified is brought out, then the whole complete sense, which is founded upon this similitude and agreement…For although this sense be spiritual, yet it is not a different one, but really literal; since the letter itself affords it to us in the way of similitude or argument….When, therefore, these are expounded literally of the things themselves, spiritually of celestial graces, we do not make two diverse senses; but, by expounding a similitude, we compare the sign with the thing signified, and so bring out the true and entire sense of the words.” 
Response: The "doctrine stated" suffices to refute this objection. For, the relationship between the literal sense and the spiritual sense is that of the sign and thing signified.
Obj. 9: St. Thomas writes, “Since the literal sense is that which the author intends, and the author of holy scripture is God, who comprehends all things together in his mind; there is nothing improper in saying that, even according to the literal sense, there are several meanings of scripture in one text ?” “Since then that is the sense of scripture, and the literal sense, which the Holy Spirit intends, however it may be gathered; certainly, if the Holy Spirit intended the tropologic, anagogic, or allegoric sense of any place, these senses are not different from the literal, as Thomas hath expressly taught us.” 
Response: The authority of St. Thomas Aquinas cannot be used to prove the consequence. For, (1) This quote is ripped out of context and the translation falsified, for, the quoting of the context makes this evident, “The author of Holy Writ is God, in whose power it is to signify His meaning, not by words only (as man also can do), but also by things themselves. So, whereas in every other science things are signified by words, this science has the property, that the things signified by the words have themselves also a signification. Therefore that first signification whereby words signify things belongs to the first sense, the historical or literal. That signification whereby things signified by words have themselves also a signification is called the spiritual sense, which is based on the literal, and presupposes it…Since the literal sense is that which the author intends, and since the author of Holy Writ is God, Who by one act comprehends all things by His intellect, it is not unfitting, as Augustine says (Confess. xii), if, even according to the literal sense, one word in Holy Writ should have several senses (in una littera Scripturae plures sint sensus). Notice “sensus” is changed for “meanings.” (2) St. Thomas’ other writings make this evident, as the section from his commentary of Galatians quoted above, and in Quodlibet Q.6, A.1, “It must be said that sacred Scripture has been handed over from heaven unto this: so that through it the truth necessary unto salvation might be manifested to us; yet the manifestation or expression of some truth can happen in two ways, by things and by words, namely inasmuch as words signify things and one thing can be a figure of another; yet the author of Scripture, namely the Holy Spirit, not only is the author of words, but also is the author of things, whence not only can he accommodate words to signify something, but also he can dispose things unto the figure of another; and according to this in sacred Scripture truth is manifested in two ways: in one way, according as things are signified through words, and in this consists the literal sense; in another way, according as things are figures of other things, and in this consists the spiritual sense. And thus many senses apply to sacred Scripture” and A.3, “It must be said that the spiritual senses of sacred Scripture are received from this: that the things running their course signify something else, which is received through a spiritual sense. Yet so to order things in their course that from them such a signification can be received, belongs to him alone who by his providence governs things, who alone is God. For just as man to signify something can present some voices or some made-up likenesses, so God unto the signification of some things presents the course itself of things subject to his providence. Yet to signify something through words or through made-up likenesses ordered only to signifying, does not make any but a literal sense, as is clear from what has been said. Whence in no human science discovered by human industry properly speaking can there be found any but the literal sense, but only in that Scripture of which, the Holy Spirit is author, whereas man is only an instrument, according to that line of the Psalmist: My tongue the reed-pen of a scribe (Ps 44:1) etc.” Further, “It must be said that the principal author of sacred Scripture is the Holy Spirit, who has understood in one word of sacred Scripture many more things than are discerned through the expositors of sacred Scripture. It is not even unfitting that a man who was an instrumental author of sacred Scripture would understand many things in one word, because the prophets, as Jerome says in his commentary on Hosea, so spoke concerning present deeds that they intended also to signify future ones; whence it is not impossible to understand many things simultaneously inasmuch as one is a figure of another.”
Obj. 10: “We must note and observe in the second place, that it is only from the literal sense that strong, valid, and efficacious arguments can be derived; which is the concession even of our adversaries themselves. It follows, therefore, that this and no other is the genuine sense of scripture. For a firm argument may always be derived from the genuine and proper sense. Since, therefore, firm inferences cannot be made from those other senses, it is evident that they are not true and genuine meanings.”
Response: The Major premise (“only from…be derived”) is distinguished. The minor premise (“For a…sense”) is distinguished. The consequence is denied. For, the utility of the spiritual sense in forming arguments is denied in a relative sense, not in an absolute sense. Relatively, it is denied insofar as we cannot know with absolute certainty what the signification of a certain sign is, for some have multiple significations, the example St. Thomas gives is “a lion on account of some likeness signifies both Christ and the devil, whence through this: that something is said about a lion in sacred Scripture, an advance to neither can be made by argumentation.” Yet, “It must be said that it is not on account of a defect of authority that an efficacious argument cannot be handed over from the spiritual sense, but it is from the very nature of likeness on which the spiritual sense is founded: for one thing can be like many, whence from that thing, when it is proposed in Scripture, it cannot be proceeded to any of those determinately, but it is the fallacy of the consequent.” This is seen from Apostolic example. For, various arguments are drawn from the Spiritual sense of scripture by Apostles, yet this does not fall under the condemnation given by the Angelic doctor, for by revelation they are certain of the precise signification of the thing signified by the words of the Old Testament.
Obj. 11: “[The Spiritual sense is denied] because it draweth away from its one true sense.” 
Response: This objection merely assumes what it is meant to prove (that there is one true sense, i.e., the literal).
Obj. 12: “[The Spiritual sense is denied] because this is the chiefest reason in explaining the Text, that the true literall sense of it may be found out.” 
Response: As was stated before, it is affirmed that the letters are explained by the literal sense, yet the things signified by the letters are better understood by the spiritual sense. Further, this assumes what it is meant to prove.
 Francis Turretin, Institutes of Elenctic Theology, ed. James T. Dennison Jr., trans. George Musgrave Giger (Phillipsburg, NJ: P&R Publishing, 1992–1997), 150., c.f. Petrus van Mastricht, Prolegomena, ed. Joel R. Beeke, trans. Todd M. Rester, vol. 1 of Theoretical-Practical Theology (Grand Rapids, MI: Reformation Heritage Books, 2018), 169., Edward Leigh, A Treatise of Divinity, consisting of Three Books (London, 1646), 174
 Prolegomena, ed. Joel R. Beeke, trans. Todd M. Rester, vol. 1 of Theoretical-Practical Theology (Grand Rapids, MI: Reformation Heritage Books, 2018), 169.
 Whitaker, Disputation on the Word of God, Disputation 5.2
 Edward Leigh, A Treatise of Divinity, consisting of Three Books (London, 1646), 174