The content for this article was taken from Q&A #20 of my article Q&A on Faith. While this article is mainly about the metonymical use of the word faith, I provide a more detailed explanation on the nature of faith in several of my articles, such as those available here, here, here, here, and here. In addition, I recommend the explanations on the meaning of the word belief in the Routledge Encyclopedia of Philosophy and Stanford Encyclopedia of Philosophy available here and here, respectively. What follows is an excerpt from the Q&A mentioned above.
Q20: Are you sure that faith always refers to the persuasion of the mind that some proposition is true?
A20: The word faith means belief. And belief is just the faculty of the mind that assents to propositions. This is all faith is, strictly speaking, and this is by far its most common usage in every realm of life. However, the word faith is also used in figures of speech, especially what is called metonymy, where one noun is used in place of another. Since faith represents its propositional object, it is only natural that we should find many examples of its use in metonymy. For example, Johannes Wollebius (1589-1629) in his Compendium Theologiae Christianae, a work that was regarded as an accurate reflection of Reformed theology in its day, said this about justification: "The expression 'We are justified by faith' is a metonymy, and has the same meaning as 'We are justified by the merit of Christ, which is apprehended by faith.'" (see p. 258 here). E.W. Bullinger in his Figures of Speech Used in the Bible gives several examples of faith being used metonymically, such as in Acts 17:31 and Galatians 3:23, as well as many others (see p. 599 here). As some more examples where faith is used as a metonymy for Christ, please consider: (1) "We live by faith:" Gal. 2:20; "by Christ," John 6:57. (2) "We have remission of sins by faith," Acts 13:38, 39; "by Christ," Eph. 1:7, Col. 1:14. (3) "We are justified by faith," Rom. 3:28, Gal. 3:24; "by Christ," Isa. 53:11, Rom. 5:9. (4) "We have peace with God by faith," Rom. 5:1,2; "by Christ," Eph. 2:3 and 3:12. (5) "We are sanctified by faith," Acts 15:9; "by Christ," Heb. 10:14, 1 Cor. 1:30. (6) "We overcome the world by faith," 1 John 5:4,5; "by Christ," John 16:33. (7) "We are the sons of God by faith," Gal. 3:26; "by Christ," Eph. 1:5. (8) We "have a heavenly inheritance by faith," Acts 26:18; "by Christ," Gal. 4:7. And (9) "We have eternal life by faith," John 3:16 and 5:24 and 6:47; "by Christ," 1 John 5:11,12. (Credit goes to the person with the screen name "eternomade" on the CARM Forum for compiling this list. See here). As explained in Hebrews 11:1, faith is the substance or assurance of things hoped for (or expected) and the conviction of things not seen. Christ accomplished justification and He reveals it to His elect by faith, by which they apprehend what Christ did and have fellowship with the substance of it. Therefore, when one of the elect says that he is justified by faith, what he really means is that He is justified by what Christ accomplished, as was pointed out by Wollebius above. This is why faith can be no more than a persuasion of the mind that its propositional object is true. Denying this inevitably adds an ethical element to justification that requires Christ's work in sinners to supplement Christ's work for sinners. In other words, it is a denial of the sufficiency of Christ's work on the cross. Theologians reveal that they don't understand the nature of faith or the Gospel when they say that faith is an appropriating instrument that accepts, receives, or procures what they suppose to be Christ's merits (which from their point of view does not include justification, for they don't believe Christ justified anyone on the cross) or that faith is not only mental assent to propositions but also includes a mystical and elusive "heart work" they call fiducia (see my article here). In contrast, a correct understanding of faith necessitates that Christ's merits belong to the elect even before they believe, and it is by believing that they come to know it.