The Teaching of the Roman Catholic Church:
The Holy Spirit proceeds from the Father and from the Son as from a Single Principle through a Single Spiration. (De fide.)
It draws it's argument from these Proofs from Holy Scripture:
a) The Holy Spirit, according to the teaching of Holy Writ, is not merely the Spirit of the Father (Mt. 10,20: "It is the Spirit of the Father that speaketh in you"; cf. John 15,26 : I Cor. 2, II et seq.), but also the Spirit of the Son (Gal. 4, 6: "God sent the Spirit of His Son into your hearts"), the Spirit of Jesus (Apostles 16, 7: "And the spirit of Jesus suffered them not"), the Spirit of Christ (Romans 8, 9: "Now if any man have not the Spirit of Christ he is none of His"). If the designation "spirit of the Father" expresses an original reference to the Father (spiramen Patris or spiratus a Patre), as the Greeks admit, then the expression "Spirit of the Son" must analogously express an original connection with the Son ( spiramen Filii or spiratus a Filio).
b) The Holy Spirit is sent not only from the Father (John 14, 16, 26), but also from the Son, John 15, 26: "The Paraclete Whom I will send you from the Father" ; cf. John 16, 7; Luke 24, 49; John 20, 22. This external mission (ad extra) is to a certain extent the continuation of the Eternal Procession in time. From the mission one can therefore infer the Eternal Procession. The external production corresponds to the mission, and the eternal being produced corresponds to the being sent. As, according to the testimony of Holy Scripture, the Holy Spirit is sent from the Father and from the Son, it must be inferred that He is produced by the Father and by the Son.
c) The Holy Spirit receives His knowledge from the Son. John 16, 12 et seq. : "What things soever He shall hear He shall speak. He shall glorify me ; because He shall receive of mine and shall show it to you." The hearing and receiving of knowledge can be understood of a Divine Person only in the sense that He receives the Divine Knowledge and, with it, the identical Divine Essence from all eternity from another Divine Person through communication of Essence. As the Holy Spirit receives His knowledge from the Son he must proceed from the Son as the Son, who receives His knowledge from the Father (John 8, 26 et seq.), proceeds from the Father. St. Augustine comments on this passage: "from each He will hear it, from whom He proceeds. Hearing is for Him knowing, but knowing is Being." (In Ioan. tr. 99, 4.)
That the Holy Spirit proceeds from the Father and from the Son as from One Single Principle and through One Single Spiration, is clear from John 16, 15: "All that the Father has, is mine." If the Son, by virtue of His eternal generation from the Father, possesses 'everything that the Father possesses except the Fatherhood and the ungeneratedness which are not communicable, then He msut also possess the power of spiration (vis spirativa) and with it the being a Principle in relation to the Holy Spirit.
Proof from Tradition:
The Latin Fathers preferred the co-ordination formula: ex Patre et Filio (filioque), the Greek the subordinating formula: ex Patre per Filium. Tertullian employs both forms, but explains the co-ordinating formula in the sense of the subordinating one. Adv. Prax. 4 : "I do not derive the Spirit other wise than from the Father through the Son (a Patre per Filium). Op. cit. 8 : "the Third is the Spirit proceeding from God (the Father) and from the Son (a Deo et filii), as the third from the root through the bud is the fruit." St. Hilary, under Greek influence, uses the subordinating formula : "From thee (the Father) through him (the Son) is thy Holy Spirit" (De Trin. XII 56). St. Ambrose teaches that "the Holy Spirit, since he proceeds from the Father and the Son, cannot be separated from the Father nor from the Son" (De Spiritu Sancto I 120). St. Augustine establishes the procession of the Holy Spirit from the Father and the son (de utroque) by a detailed scriptural proof (In Ioan. tr. 99, 6 ; De Trin. XV 27, 48).
Origen uses the subordination phrase : "the Holy Spirit is the first of everything by the Father through the Son" ; "The Son gives to His hypostatis not only that he is, but also that he is wise, understanding and just" (Comm. in Ioan. II 10 (6), 75-76). St. Athanasius declares: "The same peculiar relationship in which we know the Son to be with the Father, governs, as wel shall find, also that which is between the Spirit and the Son. And so the Son speaks : 'All things whatsoever the Father hath are mine (John 16, 15), so we shall find, that all this is also through the Son in the Spirit" (Ep. ad Serap. 3, I). St. Basil teaches that "the goodness and the sanctity and the kingly dignity characteristic of God the Father is transmitted from the Father through the Only-begotten to the Spirit" (De Spiritu Sancto 18, 47). The three Cappadocians (Basil, Gregory Naxianzus, Gregory of Nyssa) compare the relationship of the three Persons to each other with the links of a chain. The example is based on the subordinating formula "from the Father through the Son."
St. Didymus of Alexandria, St. Ephiphanius of Salamis and St. Cyril of Alexandria employ, even if not exclusively the co-ordinating formula (filioque). Cf. St. Epiphanius, Ancoratus 7 : "the Holy Spirit is from the same Essence of the Father and of the Son.: 16.8 : "From the Father and the Son, the third according to his name." Cf. Didymus, De Spiritu Sancto 34; Cyril of Al ; Thes. de sancta et consbust. Trin. 34.
St. John of Damascus rejects the notion that the Holy Spirit is from the Son, nevertheless he teaches that He is the spirit of the Son and that He proceeds through the Son from the Father (De fide orth. 18, 12). In saying this he does not deny that the Son is a Principle of the Holy Spirit, but only that unlike the Father He is not the Primitive Principle.
The co-ordinating formula (filioque) and the subordinating formula (per filium) concur essentially, in so far as they both attest that both the Father and the Son are the Principle of the Holy Spirit and they also complement each other. While in the former the unicity and the indivisibility of the Principle ar above all expressed, the latter effectively stresses that the Father is the Primitive Principle (cf. St. Augustine, De Trin. XV 17, 29: de quo procedit principaliter), and that the Son as "God from God" is the Derived Principle, in so far as He, with His Essence, receives the power of spiration from the Father. Cf. D 691.
This is all pre-Vatican II and is the teaching of the Roman Catholic Church. Personally, I see nothing wrong with it. I accept it as the broadest understand of the Faith from the Fathers. It excludes nothing that the Fathers have taught and offers the teaching humbly and with care. I think the Orthodox, in the modern day, have blinded themselves by Polemics against the Roman Church and have, in this case, drawn too narrowly the doctrine.
I cannot deny the 'orthodoxy' of the doctrine but I can recognize criticism in changing the Creed without the acceptance of the whole Church. I don't think the Western Church doesn't share in this belief and I could see a day when it is removed so that we might share 'one' Creed.